High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems

By Manuel Castro, José M. Gallardo, Carlos Parés

Abstract

This paper is concerned with the development of high order methods for the numerical approximation of one-dimensional nonconservative hyperbolic systems. In particular, we are interested in high order extensions of the generalized Roe methods introduced by I. Toumi in 1992, based on WENO reconstruction of states. We also investigate the well-balanced properties of the resulting schemes. Finally, we will focus on applications to shallow-water systems.

1. Introduction

The motivating question of this paper was the design of well-balanced high order numerical schemes for PDE systems that can be written under the form

StartLayout 1st Row with Label left-parenthesis 1.1 right-parenthesis EndLabel StartFraction partial-differential w Over partial-differential t EndFraction plus StartFraction partial-differential upper F Over partial-differential x EndFraction left-parenthesis w right-parenthesis equals script upper B left-parenthesis w right-parenthesis StartFraction partial-differential w Over partial-differential x EndFraction plus upper S left-parenthesis w right-parenthesis StartFraction d sigma Over d x EndFraction comma EndLayout

where the unknown w left-parenthesis x comma t right-parenthesis takes values on an open convex subset upper D of double-struck upper R Superscript upper N , upper F is a regular function from upper D to double-struck upper R Superscript upper N , script upper B is a regular matrix-valued function from upper D to script upper M Subscript upper N times upper N Baseline left-parenthesis double-struck upper R right-parenthesis , upper S is a function from upper D to double-struck upper R Superscript upper N , and sigma left-parenthesis x right-parenthesis is a known function from double-struck upper R to double-struck upper R .

System Equation1.1 includes as particular cases: systems of conservation laws ( script upper B equals 0 , upper S equals 0 ), systems of conservation laws with source term or balance laws ( script upper B equals 0 ), and coupled systems of conservation laws.

More precisely, the discretization of the shallow-water systems that govern the flow of one layer or two superposed layers of immiscible homogeneous fluids was focused. The corresponding systems can be written respectively as a balance law or a coupled system of two conservation laws. Systems with similar characteristics also appear in other flow models, such as boiling flows and two-phase flows (see Reference13).

It is well known that standard methods that correctly solve systems of conservation laws can fail in solving Equation1.1, especially when approaching equilibria or near to equilibria solutions. In the context of shallow-water equations, Bermúdez and Vázquez-Cendón introduced in Reference2 the condition called conservation property or script upper C -property: a scheme is said to satisfy this condition if it correctly solves the steady state solutions corresponding to water at rest. This idea of constructing numerical schemes that preserve some equilibria, which are called in general well-balanced schemes, has been extended in different ways; see, e.g., Reference3, Reference6, Reference7, Reference8, Reference12, Reference14, Reference17, Reference18, Reference21, Reference22, Reference26, Reference28, Reference29, Reference30, Reference35.

Among the main techniques used in the derivation of well-balanced numerical schemes, one of them consists in first choosing a standard conservative scheme for the discretization of the flux terms and then discretizing the source and the coupling terms in order to obtain a consistent scheme which correctly solves a predetermined family of equilibria. This was the approach in Reference2 where the authors proved, in the context of shallow-water equations, that numerical schemes based on Roe methods for the discretization of the flux terms and upwinding the source term exacly solve equilibria corresponding to water at rest. In Reference12 it was shown that the technique of modified equations can be helpful in the deduction of well-balanced numerical schemes.

This procedure has been succesfully applied to obtain high order numerical schemes for some particular cases of Equation1.1 (see, for instance, Reference4, Reference38 and Reference39). The main disadvantage of this first technique is its lack of generality: the calculation of the correct discretization of the source and the coupling terms depends on both the specific problem and the conservative numerical scheme chosen.

Another technique to obtain well-balanced first order schemes for solving Equation1.1 consists in considering piecewise constant approximations of the solutions that are updated by means of Approximate Riemann Solvers at the intercells. In particular, Godunov’s methods, i.e., methods based on Exact Riemann Solvers, have been used in the context of shallow-water systems in Reference1, Reference9, Reference10, Reference15, Reference21, Reference22. This approach was also used in Reference5, where the flux and the coupling terms of a bilayer shallow-water system were treated together by using a generalized Roe linearization.

If this second procedure is followed, the main difficulty both from the mathematical and the numerical points of view comes from the presence of nonconservative products, which makes difficult even the definition of weak solutions: in general, the product script upper B left-parenthesis w right-parenthesis w Subscript x does not make sense as a distribution for discontinuous solutions. This is also the case for the product upper S left-parenthesis w right-parenthesis sigma Subscript x when piecewise constant approximations of sigma are considered.

A helpful strategy in solving these difficulties consists in considering system Equation1.1 as a particular case of a one-dimensional quasilinear hyperbolic system:

StartLayout 1st Row with Label left-parenthesis 1.2 right-parenthesis EndLabel StartFraction partial-differential upper W Over partial-differential t EndFraction plus script upper A left-parenthesis upper W right-parenthesis StartFraction partial-differential upper W Over partial-differential x EndFraction equals 0 comma x element-of double-struck upper R comma t greater-than 0 period EndLayout

In effect, adding to Equation1.1 the trivial equation

StartFraction partial-differential sigma Over partial-differential t EndFraction equals 0 comma

system Equation1.1 can be easily rewritten under this form (see Reference17, Reference18, Reference21, Reference22).

In Reference11, Dal Maso, LeFloch, and Murat proposed an interpretation of non-conservative products as Borel measures, based on the choice of a family of paths in the phases space. After this theory it is possible to give a rigorous definition of weak solutions of Equation1.2. Together with the definition of weak solutions, a notion of entropy has to be chosen as the usual Lax’s concept or one related to an entropy pair. Once this choice has been done, the classical theory of simple waves of hyperbolic systems of conservation laws and the results concerning the solutions of Riemann problems can be extended to systems of the form Equation1.2.

The introduction of a family of paths does not only give a way to properly define the concept of weak solution for nonconservative systems, it also allows us to extend to this framework some basic concepts related to the numerical approximation of weak solutions of conservation laws. For instance, in Reference36 a general definition of Roe linearizations was introduced, also based on the use of a family of paths. In Reference28 a general definition of well-balanced schemes for solving Equation1.2 was introduced. It was shown that the well-balanced properties of these generalized Roe methods depend on the choice of the family of paths. Moreover, this general methodology was applied to some systems of the form Equation1.1 related to shallow-water flows, recovering some known well-balanced schemes, or resulting in new schemes.

The goal of this paper is to obtain the general expression of a well-balanced high order method for Equation1.2 based on the use of a first order Roe scheme and reconstruction of states. The interest of such a general expression is that, once obtained, particular schemes can be deduced for any system of the form Equation1.1, where the numerical treatment of source and coupling terms is automatically derived. To our knowledge, the present work is the first attempt to obtain well-balanced high order numerical schemes following this procedure.

The paper is organized as follows. In Section 2 we give some basic definitions and results about nonconservative systems, Roe linearizations and generalized Roe schemes, for which we will follow Reference28 closely. High order versions of the Roe schemes, based on reconstruction operators, are introduced in Section 3. Next, Section 4 is devoted to the analysis of the well-balanced properties of the high order schemes previously constructed. In Section 6, the WENO method is applied to build the reconstruction operators. Applications to a family of systems that generalize Equation1.1 are presented in Section 5, with particular interest in some shallow-water systems with one and two layers of fluid. Finally, Section 7 contains numerical results to test the performances of our high-order schemes. In particular, the high order well-balanced property is numerically verified.

2. Roe methods for nonconservative hyperbolic systems

Consider the system in nonconservative form

StartLayout 1st Row with Label left-parenthesis 2.1 right-parenthesis EndLabel upper W Subscript t Baseline plus script upper A left-parenthesis upper W right-parenthesis upper W Subscript x Baseline equals 0 comma x element-of double-struck upper R comma t greater-than 0 comma EndLayout

where we suppose that the range of upper W left-parenthesis x comma t right-parenthesis is contained inside an open convex subset normal upper Omega of double-struck upper R Superscript upper N , and upper W element-of normal upper Omega right-arrow from bar script upper A left-parenthesis upper W right-parenthesis element-of script upper M Subscript upper N Baseline left-parenthesis double-struck upper R right-parenthesis is a smooth locally bounded map. The system Equation2.1 is assumed to be strictly hyperbolic: for each upper W element-of normal upper Omega the matrix script upper A left-parenthesis upper W right-parenthesis has upper N real distinct eigenvalues lamda 1 left-parenthesis upper W right-parenthesis less-than midline-horizontal-ellipsis less-than lamda Subscript upper N Baseline left-parenthesis upper W right-parenthesis . We also suppose that the j th characteristic field upper R Subscript j is either genuinely nonlinear:

upper R Subscript j Baseline left-parenthesis upper W right-parenthesis dot nabla lamda Subscript j Baseline left-parenthesis upper W right-parenthesis not-equals 0 comma for-all upper W element-of normal upper Omega comma

or linearly degenerate:

upper R Subscript j Baseline left-parenthesis upper W right-parenthesis dot nabla lamda Subscript j Baseline left-parenthesis upper W right-parenthesis equals 0 comma for-all upper W element-of normal upper Omega period

For discontinuous solutions upper W , the nonconservative product script upper A left-parenthesis upper W right-parenthesis upper W Subscript x does not make sense as a distribution. However, the theory developed by Dal Maso, LeFloch and Murat (Reference11) allows us to give a rigorous definition of nonconservative products, associated to the choice of a family of paths in normal upper Omega .

Definition 2.1

A family of paths in normal upper Omega subset-of double-struck upper R Superscript upper N is a locally Lipschitz map

normal upper Phi colon left-bracket 0 comma 1 right-bracket times normal upper Omega times normal upper Omega right-arrow normal upper Omega

that satisfies the following properties:

(1)

normal upper Phi left-parenthesis 0 semicolon upper W Subscript upper L Baseline comma upper W Subscript upper R Baseline right-parenthesis equals upper W Subscript upper L and normal upper Phi left-parenthesis 1 semicolon upper W Subscript upper L Baseline comma upper W Subscript upper R Baseline right-parenthesis equals upper W Subscript upper R , for any upper W Subscript upper L Baseline comma upper W Subscript upper R Baseline element-of normal upper Omega .

(2)

Given an arbitrary bounded set script upper B subset-of normal upper Omega , there exists a constant k such that StartAbsoluteValue StartFraction partial-differential normal upper Phi Over partial-differential s EndFraction left-parenthesis s semicolon upper W Subscript upper L Baseline comma upper W Subscript upper R Baseline right-parenthesis EndAbsoluteValue less-than-or-equal-to k StartAbsoluteValue upper W Subscript upper L Baseline minus upper W Subscript upper R Baseline EndAbsoluteValue comma

for any upper W Subscript upper L Baseline comma upper W Subscript upper R Baseline element-of script upper B and for almost every s element-of left-bracket 0 comma 1 right-bracket .

(3)

For every bounded set script upper B subset-of normal upper Omega , there exists a constant upper K such that StartAbsoluteValue StartFraction partial-differential normal upper Phi Over partial-differential s EndFraction left-parenthesis s semicolon upper W Subscript upper L Superscript 1 Baseline comma upper W Subscript upper R Superscript 1 Baseline right-parenthesis minus StartFraction partial-differential normal upper Phi Over partial-differential s EndFraction left-parenthesis s semicolon upper W Subscript upper L Superscript 2 Baseline comma upper W Subscript upper R Superscript 2 Baseline right-parenthesis EndAbsoluteValue less-than-or-equal-to upper K left-parenthesis StartAbsoluteValue upper W Subscript upper L Superscript 1 Baseline minus upper W Subscript upper L Superscript 2 Baseline EndAbsoluteValue plus StartAbsoluteValue upper W Subscript upper R Superscript 1 Baseline minus upper W Subscript upper R Superscript 2 Baseline EndAbsoluteValue right-parenthesis comma

for each upper W Subscript upper L Superscript 1 Baseline comma upper W Subscript upper R Superscript 1 Baseline comma upper W Subscript upper L Superscript 2 Baseline comma upper W Subscript upper R Superscript 2 Baseline element-of script upper B and for almost every s element-of left-bracket 0 comma 1 right-bracket .

Suppose that a family of paths normal upper Phi in normal upper Omega has been chosen. Then, for upper W element-of left-parenthesis upper L Superscript normal infinity Baseline left-parenthesis double-struck upper R times double-struck upper R Superscript plus Baseline right-parenthesis intersection upper B upper V left-parenthesis double-struck upper R times double-struck upper R Superscript plus Baseline right-parenthesis right-parenthesis Superscript upper N , the nonconservative product can be interpreted as a Borel measure denoted by left-bracket script upper A left-parenthesis upper W right-parenthesis upper W Subscript x Baseline right-bracket Subscript normal upper Phi . When no confusion arises, we will drop the dependency on normal upper Phi .

A weak solution of system Equation2.1 is defined as a function upper W element-of left-parenthesis upper L Superscript normal infinity Baseline left-parenthesis double-struck upper R times double-struck upper R Superscript plus Baseline right-parenthesis intersection upper B upper V left-parenthesis double-struck upper R times double-struck upper R Superscript plus Baseline right-parenthesis right-parenthesis Superscript upper N that satisfies the equality

upper W Subscript t Baseline plus left-bracket script upper A left-parenthesis upper W right-parenthesis upper W Subscript x Baseline right-bracket Subscript normal upper Phi Baseline equals 0 period

In particular, a piecewise script upper C Superscript 1 function upper W is a weak solution of Equation2.1 if and only if the two following conditions are satisfied:

(i)

upper W is a classical solution in the domains where it is script upper C Superscript 1 .

(ii)

Along a discontinuity upper W satisfies the jump condition StartLayout 1st Row with Label left-parenthesis 2.2 right-parenthesis EndLabel integral Subscript 0 Superscript 1 Baseline left-parenthesis xi script upper I minus script upper A left-parenthesis normal upper Phi left-parenthesis s semicolon upper W Superscript minus Baseline comma upper W Superscript plus Baseline right-parenthesis right-parenthesis right-parenthesis StartFraction partial-differential normal upper Phi Over partial-differential s EndFraction left-parenthesis s semicolon upper W Superscript minus Baseline comma upper W Superscript plus Baseline right-parenthesis d s equals 0 comma EndLayout

where script upper I is the identity matrix, xi is the speed of propagation of the discontinuity, and upper W Superscript minus , upper W Superscript plus are the left and right limits of the solution at the discontinuity.

Note that in the particular case of a system of conservation laws (that is, script upper A left-parenthesis upper W right-parenthesis is the Jacobian matrix of some flux function upper F left-parenthesis upper W right-parenthesis ) the jump condition Equation2.2 is independent of the family of paths, and it reduces to the usual Rankine-Hugoniot condition:

StartLayout 1st Row with Label left-parenthesis 2.3 right-parenthesis EndLabel upper F left-parenthesis upper W Superscript plus Baseline right-parenthesis minus upper F left-parenthesis upper W Superscript minus Baseline right-parenthesis equals xi left-parenthesis upper W Superscript plus Baseline minus upper W Superscript minus Baseline right-parenthesis period EndLayout

In the general case, the selection of the family of paths has to be based on the physical background of the problem under consideration. Nevertheless, it is natural from the mathematical point of view to require this family to satisfy some hypotheses concerning the relation of the paths with the integral curves of the characteristic fields. For instance, if upper W Subscript upper L and upper W Subscript upper R are linked by an integral curve of a linearly degenerate field, the natural choice of the path is a parametrization of that curve, as this choice assures that the contact discontinuity

StartLayout 1st Row with Label left-parenthesis 2.4 right-parenthesis EndLabel upper W left-parenthesis x comma t right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column upper W Subscript upper L Baseline 2nd Column if x less-than xi t comma 2nd Row 1st Column upper W Subscript upper R Baseline 2nd Column if x greater-than xi t comma EndLayout EndLayout

where xi is the (constant) value of the corresponding eigenvalue through the integral curve, is a weak solution of the problem, as would be the case for a system of conservation laws.

Due to these requirements, the explicit calculation of the path linking two given states upper W Subscript upper L and upper W Subscript upper R can be difficult: in most cases, the explicit expression of the solution of the Riemann problem related to the states is needed (see Reference28).

Together with this definition of weak solutions, a notion of entropy has to be chosen, either as the usual Lax’s concept or one related to an entropy pair (see Reference16 for details). Once this choice has been done, the theory of simple waves of hyperbolic systems of conservation laws and the results concerning the solutions of Riemann problems can be naturally extended to systems of the form Equation2.1 (see Reference11).

Some of the usual numerical schemes designed for conservation laws can be adapted to the discretization of the more general system Equation2.1. This is the case of Roe schemes (see Reference31): in Reference36 a general definition of Roe linearization was introduced, based again on the use of a family of paths.

Definition 2.2

Given a family of paths normal upper Psi , a function script upper A Subscript normal upper Psi Baseline colon normal upper Omega times normal upper Omega right-arrow script upper M Subscript upper N Baseline left-parenthesis double-struck upper R right-parenthesis is called a Roe linearization of system Equation2.1 if it verifies the following properties:

(1)

For each upper W Subscript upper L Baseline comma upper W Subscript upper R Baseline element-of normal upper Omega , script upper A Subscript normal upper Psi Baseline left-parenthesis upper W Subscript upper L Baseline comma upper W Subscript upper R Baseline right-parenthesis has upper N distinct real eigenvalues.

(2)

script upper A Subscript normal upper Psi Baseline left-parenthesis upper W comma upper W right-parenthesis equals script upper A left-parenthesis upper W right-parenthesis , for every upper W element-of normal upper Omega .

(3)

For any upper W Subscript upper L Baseline comma upper W Subscript upper R Baseline element-of normal upper Omega , StartLayout 1st Row with Label left-parenthesis 2.5 right-parenthesis EndLabel script upper A Subscript normal upper Psi Baseline left-parenthesis upper W Subscript upper L Baseline comma upper W Subscript upper R Baseline right-parenthesis left-parenthesis upper W Subscript upper R Baseline minus upper W Subscript upper L Baseline right-parenthesis equals integral Subscript 0 Superscript 1 Baseline script upper A left-parenthesis normal upper Psi left-parenthesis s semicolon upper W Subscript upper L Baseline comma upper W Subscript upper R Baseline right-parenthesis right-parenthesis StartFraction partial-differential normal upper Psi Over partial-differential s EndFraction left-parenthesis s semicolon upper W Subscript upper L Baseline comma upper W Subscript upper R Baseline right-parenthesis d s period EndLayout

Note again that if script upper A left-parenthesis upper W right-parenthesis is the Jacobian matrix of a smooth flux function upper F left-parenthesis upper W right-parenthesis , Equation2.5 is independent of the family of paths and reduces to the usual Roe property:

StartLayout 1st Row with Label left-parenthesis 2.6 right-parenthesis EndLabel script upper A Subscript normal upper Psi Baseline left-parenthesis upper W Subscript upper R Baseline minus upper W Subscript upper L Baseline right-parenthesis equals upper F left-parenthesis upper W Subscript upper R Baseline right-parenthesis minus upper F left-parenthesis upper W Subscript upper L Baseline right-parenthesis period EndLayout

Once a Roe linearization script upper A Subscript normal upper Psi has been chosen, in order to construct numerical schemes for solving Equation2.1, computing cells upper I Subscript i Baseline equals left-bracket x Subscript i minus 1 slash 2 Baseline comma x Subscript i plus 1 slash 2 Baseline right-bracket are considered; let us suppose for simplicity that the cells have constant size normal upper Delta x and that x Subscript i plus one-half Baseline equals i normal upper Delta x . Define x Subscript i Baseline equals left-parenthesis i minus 1 slash 2 right-parenthesis normal upper Delta x , the center of the cell upper I Subscript i . Let normal upper Delta t be the constant time step and define t Superscript n Baseline equals n normal upper Delta t . Denote by upper W Subscript i Superscript n the approximation of the cell averages of the exact solution provided by the numerical scheme, that is,

upper W Subscript i Superscript n Baseline approximately-equals StartFraction 1 Over normal upper Delta x EndFraction integral Subscript x Subscript i minus 1 slash 2 Baseline Superscript x Subscript i plus 1 slash 2 Baseline Baseline upper W left-parenthesis x comma t Superscript n Baseline right-parenthesis d x period

Then, the numerical scheme advances in time by solving linear Riemann problems at the intercells at time t Superscript n and taking the averages of their solutions on the cells at time t Superscript n plus 1 . Under usual CFL conditions, it can be written as follows (see Reference28):

StartLayout 1st Row with Label left-parenthesis 2.7 right-parenthesis EndLabel upper W Subscript i Superscript n plus 1 Baseline equals upper W Subscript i Superscript n Baseline minus StartFraction normal upper Delta t Over normal upper Delta x EndFraction left-parenthesis script upper A Subscript i minus 1 slash 2 Superscript plus Baseline left-parenthesis upper W Subscript i Superscript n Baseline minus upper W Subscript i minus 1 Superscript n Baseline right-parenthesis plus script upper A Subscript i plus 1 slash 2 Superscript minus Baseline left-parenthesis upper W Subscript i plus 1 Superscript n Baseline minus upper W Subscript i Superscript n Baseline right-parenthesis right-parenthesis period EndLayout Here, the intermediate matrices are defined by

script upper A Subscript i plus 1 slash 2 Baseline equals script upper A Subscript normal upper Psi Baseline left-parenthesis upper W Subscript i Superscript n Baseline comma upper W Subscript i plus 1 Superscript n Baseline right-parenthesis comma

and, as usual,

script upper L Subscript i plus 1 slash 2 Superscript plus-or-minus Baseline equals Start 3 By 3 Matrix 1st Row 1st Column left-parenthesis lamda 1 Superscript i plus 1 slash 2 Baseline right-parenthesis Superscript plus-or-minus Baseline 2nd Column midline-horizontal-ellipsis 3rd Column 0 2nd Row 1st Column vertical-ellipsis 2nd Column down-right-diagonal-ellipsis 3rd Column vertical-ellipsis 3rd Row 1st Column 0 2nd Column midline-horizontal-ellipsis 3rd Column left-parenthesis lamda Subscript upper N Superscript i plus 1 slash 2 Baseline right-parenthesis Superscript plus-or-minus Baseline EndMatrix comma script upper A Subscript i plus 1 slash 2 Superscript plus-or-minus Baseline equals script upper K Subscript i plus 1 slash 2 Baseline script upper L Subscript i plus 1 slash 2 Superscript plus-or-minus Baseline script upper K Subscript i plus 1 slash 2 Superscript negative 1 Baseline comma

where script upper L Subscript i plus 1 slash 2 is the diagonal matrix whose coefficients are the eigenvalues of script upper A Subscript i plus 1 slash 2 :

lamda 1 Superscript i plus 1 slash 2 Baseline less-than lamda 2 Superscript i plus 1 slash 2 Baseline less-than midline-horizontal-ellipsis less-than lamda Subscript upper N Superscript i plus 1 slash 2 Baseline comma

and script upper K Subscript i plus 1 slash 2 is an upper N times upper N matrix whose columns are associated eigenvectors.

In the particular case of a system of conservation laws, Equation2.7 can be written under the usual form of a conservative numerical scheme. First, the numerical flux is defined by

upper G Subscript i plus 1 slash 2 Baseline equals upper G left-parenthesis upper W Subscript i Superscript n Baseline comma upper W Subscript i plus 1 Superscript n Baseline right-parenthesis equals one-half left-parenthesis upper F left-parenthesis upper W Subscript i Superscript n Baseline right-parenthesis plus upper F left-parenthesis upper W Subscript i plus 1 Superscript n Baseline right-parenthesis right-parenthesis minus one-half StartAbsoluteValue script upper A Subscript i plus 1 slash 2 Baseline EndAbsoluteValue left-parenthesis upper W Subscript i plus 1 Superscript n Baseline minus upper W Subscript i Superscript n Baseline right-parenthesis comma

where

StartAbsoluteValue script upper A Subscript i plus 1 slash 2 Baseline EndAbsoluteValue equals script upper A Subscript i plus 1 slash 2 Superscript plus Baseline minus script upper A Subscript i plus 1 slash 2 Superscript minus Baseline period

Then, the following equalities can be easily verified:

StartLayout 1st Row with Label left-parenthesis 2.8 right-parenthesis EndLabel 1st Column upper F left-parenthesis upper W Subscript i plus 1 Superscript n Baseline right-parenthesis minus upper G Subscript i plus 1 slash 2 2nd Column equals 3rd Column script upper A Subscript i plus 1 slash 2 Superscript plus Baseline left-parenthesis upper W Subscript i plus 1 Superscript n Baseline minus upper W Subscript i Superscript n Baseline right-parenthesis comma 2nd Row with Label left-parenthesis 2.9 right-parenthesis EndLabel 1st Column upper G Subscript i plus 1 slash 2 Baseline minus upper F left-parenthesis upper W Subscript i Superscript n Baseline right-parenthesis 2nd Column equals 3rd Column script upper A Subscript i plus 1 slash 2 Superscript minus Baseline left-parenthesis upper W Subscript i plus 1 Superscript n Baseline minus upper W Subscript i Superscript n Baseline right-parenthesis period EndLayout

Finally using these equalities in Equation2.7 we obtain:

upper W Subscript i plus 1 Superscript n Baseline equals upper W Subscript i Superscript n Baseline plus StartFraction normal upper Delta t Over normal upper Delta x EndFraction left-parenthesis upper G Subscript i minus 1 slash 2 Baseline minus upper G Subscript i plus 1 slash 2 Baseline right-parenthesis period

The best choice of the family of paths normal upper Psi appearing in the definition of Roe linearization is the family normal upper Phi selected for the definition of weak solutions. In effect, Roe methods based on the family of paths normal upper Phi can correctly solve discontinuities in the following sense: let us suppose that upper W Subscript i Superscript n and upper W Subscript i plus 1 Superscript n can be linked by an entropic discontinuity propagating at speed xi ; then, from Equation2.5 and Equation2.2 we deduce that

script upper A Subscript i plus 1 slash 2 Baseline left-parenthesis upper W Subscript i plus 1 Superscript n Baseline minus upper W Subscript i Superscript n Baseline right-parenthesis equals xi left-parenthesis upper W Subscript i plus 1 Superscript n Baseline minus upper W Subscript i Superscript n Baseline right-parenthesis comma

i.e., xi is an eigenvalue of the intermediate matrix and upper W Subscript i plus 1 Superscript n Baseline minus upper W Subscript i Superscript n is an associated eigenvector. As a consequence, the solution of the linear Riemann problem corresponding to the intercell x Subscript i plus 1 slash 2 ,

StartLayout Enlarged left-brace 1st Row upper U Subscript t Baseline plus script upper A Subscript i plus 1 slash 2 Baseline upper U Subscript x Baseline equals 0 comma 2nd Row upper U left-parenthesis x comma t Superscript n Baseline right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column upper W Subscript i Superscript n Baseline 2nd Column if x less-than x Subscript i plus 1 slash 2 Baseline comma 2nd Row 1st Column upper W Subscript i plus 1 Superscript n Baseline 2nd Column if x greater-than x Subscript i plus 1 slash 2 Baseline comma EndLayout EndLayout

coincides with the solution of the Riemann problem

StartLayout Enlarged left-brace 1st Row upper U Subscript t Baseline plus script upper A left-parenthesis upper U right-parenthesis upper U Subscript x Baseline equals 0 comma 2nd Row upper U left-parenthesis x comma t Superscript n Baseline right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column upper W Subscript i Superscript n Baseline 2nd Column if x less-than x Subscript i plus 1 slash 2 Baseline comma 2nd Row 1st Column upper W Subscript i plus 1 Superscript n Baseline 2nd Column if x greater-than x Subscript i plus 1 slash 2 Baseline period EndLayout EndLayout

Both solutions consist of a discontinuity linking the states and propagating at velocity xi .

Nevertheless, the construction of these schemes with normal upper Psi equals normal upper Phi can be difficult or very costly in practice. In this case, a simpler family of paths normal upper Psi has to be chosen as the family of segments

StartLayout 1st Row with Label left-parenthesis 2.10 right-parenthesis EndLabel normal upper Psi left-parenthesis s semicolon upper W Subscript upper L Baseline comma upper W Subscript upper R Baseline right-parenthesis equals upper W Subscript upper L Baseline plus s left-parenthesis upper W Subscript upper R Baseline minus upper W Subscript upper L Baseline right-parenthesis comma s element-of left-bracket 0 comma 1 right-bracket period EndLayout

In Reference28 it was remarked that, in this case, the convergence of the numerical scheme can fail when the weak solution to approach involves discontinuities connecting states upper W Superscript minus and upper W Superscript plus such that the paths of the families normal upper Phi and normal upper Psi linking them are different.

As in the case of a system of conservation laws, the scheme Equation2.7 has to be used with a CFL condition:

max left-brace StartAbsoluteValue lamda Subscript l Superscript i plus 1 slash 2 Baseline EndAbsoluteValue comma 1 less-than-or-equal-to l less-than-or-equal-to upper N comma i element-of double-struck upper Z right-brace StartFraction normal upper Delta t Over normal upper Delta x EndFraction less-than-or-equal-to gamma comma

with 0 less-than gamma less-than-or-equal-to 1 . An entropy fix technique, as the Harten-Hyman one (Reference24, Reference25), also has to be included.

3. High order schemes based on reconstruction of states

In the case of systems of conservation laws

StartLayout 1st Row with Label left-parenthesis 3.1 right-parenthesis EndLabel upper W Subscript t Baseline plus upper F left-parenthesis upper W right-parenthesis Subscript x Baseline equals 0 comma EndLayout

there are several methods to obtain higher order schemes based on the use of a reconstruction operator. In particular, methods based on the reconstruction of states are built using the following procedure: given a first order conservative scheme with numerical flux function upper G left-parenthesis upper U comma upper V right-parenthesis , a reconstruction operator of order p is considered, that is, an operator that associates to a given sequence StartSet upper W Subscript i Baseline EndSet two new sequences, StartSet upper W Subscript i plus 1 slash 2 Superscript minus Baseline EndSet and StartSet upper W Subscript i plus 1 slash 2 Superscript plus Baseline EndSet , in such a way that, whenever

upper W Subscript i Baseline equals StartFraction 1 Over normal upper Delta x EndFraction integral Underscript upper I Subscript i Baseline Endscripts upper W left-parenthesis x right-parenthesis d x

for some smooth function upper W , we have that

upper W Subscript i plus 1 slash 2 Superscript plus-or-minus Baseline equals upper W left-parenthesis x Subscript i plus 1 slash 2 Baseline right-parenthesis plus upper O left-parenthesis normal upper Delta x Superscript p Baseline right-parenthesis period

Once the first order method and the reconstruction operator have been chosen, the method of lines can be used to develop high order methods for Equation3.1: the idea is to discretize only in space, leaving the problem continuous in time. This procedure leads to a system of ordinary differential equations which is solved using a standard numerical method. In particular, we assume here that the first order scheme is a Roe method.

Let ModifyingAbove upper W With bar Subscript j Baseline left-parenthesis t right-parenthesis denote the cell average of a regular solution upper W of Equation3.1 over the cell upper I Subscript i at time t :

ModifyingAbove upper W With bar Subscript i Baseline left-parenthesis t right-parenthesis equals StartFraction 1 Over normal upper Delta x EndFraction integral Subscript x Subscript i minus 1 slash 2 Baseline Superscript x Subscript i plus 1 slash 2 Baseline Baseline upper W left-parenthesis x comma t right-parenthesis d x period

The following equation can be easily obtained for the cell averages:

ModifyingAbove upper W With bar prime Subscript i Baseline left-parenthesis t right-parenthesis equals StartFraction 1 Over normal upper Delta x EndFraction left-parenthesis upper F left-parenthesis upper W left-parenthesis x Subscript i minus 1 slash 2 Baseline comma t right-parenthesis right-parenthesis minus upper F left-parenthesis upper W left-parenthesis x Subscript i plus 1 slash 2 Baseline comma t right-parenthesis right-parenthesis right-parenthesis period

This system is now approached by

StartLayout 1st Row with Label left-parenthesis 3.2 right-parenthesis EndLabel upper W prime Subscript i Baseline left-parenthesis t right-parenthesis equals StartFraction 1 Over normal upper Delta x EndFraction left-parenthesis upper G overTilde Subscript i minus 1 slash 2 Baseline minus upper G overTilde Subscript i plus 1 slash 2 Baseline right-parenthesis comma EndLayout

with

upper G overTilde Subscript i plus 1 slash 2 Baseline equals upper G left-parenthesis upper W Subscript i plus 1 slash 2 Superscript minus Baseline left-parenthesis t right-parenthesis comma upper W Subscript i plus 1 slash 2 Superscript plus Baseline left-parenthesis t right-parenthesis right-parenthesis comma

where upper W Subscript i Baseline left-parenthesis t right-parenthesis is the approximation to ModifyingAbove upper W With bar Subscript i Baseline left-parenthesis t right-parenthesis provided by the scheme, and upper W Subscript i plus 1 slash 2 Superscript plus-or-minus Baseline left-parenthesis t right-parenthesis is the reconstruction associated to the sequence StartSet upper W Subscript j Baseline left-parenthesis t right-parenthesis EndSet .

Let us now generalize this semi-discrete method for a nonconservative system Equation2.1. Observe that, in Section 2, the key point to generalize both the Rankine-Hugoniot condition Equation2.3 and the Roe property Equation2.6 to system Equation2.1 was to replace a difference of fluxes by an integral along a path. Let us apply the same technique here. First of all, as the first order scheme is a Roe method, using the equalities Equation2.8 and Equation2.9 (replacing upper W Subscript i Superscript n and upper W Subscript i plus 1 Superscript n by upper W Subscript i plus 1 slash 2 Superscript minus Baseline left-parenthesis t right-parenthesis and upper W Subscript i plus 1 slash 2 Superscript plus Baseline left-parenthesis t right-parenthesis , respectively) we can rewrite Equation3.2 as follows:

StartLayout 1st Row with Label left-parenthesis 3.3 right-parenthesis EndLabel StartLayout 1st Row 1st Column upper W prime Subscript i Baseline left-parenthesis t right-parenthesis equals minus StartFraction 1 Over normal upper Delta x EndFraction left-parenthesis 2nd Column script upper A Subscript i minus 1 slash 2 Superscript plus Baseline left-parenthesis upper W Subscript i minus 1 slash 2 Superscript plus Baseline left-parenthesis t right-parenthesis minus upper W Subscript i minus 1 slash 2 Superscript minus Baseline left-parenthesis t right-parenthesis right-parenthesis 2nd Row 1st Column Blank 2nd Column plus script upper A Subscript i plus 1 slash 2 Superscript minus Baseline left-parenthesis upper W Subscript i plus 1 slash 2 Superscript plus Baseline left-parenthesis t right-parenthesis minus upper W Subscript i plus 1 slash 2 Superscript minus Baseline left-parenthesis t right-parenthesis right-parenthesis minus upper F left-parenthesis upper W Subscript i minus 1 slash 2 Superscript plus Baseline left-parenthesis t right-parenthesis right-parenthesis plus upper F left-parenthesis upper W Subscript i plus 1 slash 2 Superscript minus Baseline left-parenthesis t right-parenthesis right-parenthesis right-parenthesis comma EndLayout EndLayout

where script upper A Subscript i plus 1 slash 2 is the intermediate matrix corresponding to the states upper W Subscript i plus 1 slash 2 Superscript minus Baseline left-parenthesis t right-parenthesis and upper W Subscript i plus 1 slash 2 Superscript plus Baseline left-parenthesis t right-parenthesis .

Let us now introduce, at every cell upper I Subscript i , any regular function upper P Subscript i Superscript t such that

StartLayout 1st Row with Label left-parenthesis 3.4 right-parenthesis EndLabel limit Underscript x right-arrow x Subscript i minus 1 slash 2 Superscript plus Baseline Endscripts upper P Subscript i Superscript t Baseline left-parenthesis x right-parenthesis equals upper W Subscript i minus 1 slash 2 Superscript plus Baseline left-parenthesis t right-parenthesis comma limit Underscript x right-arrow x Subscript i plus 1 slash 2 Superscript minus Baseline Endscripts upper P Subscript i Superscript t Baseline left-parenthesis x right-parenthesis equals upper W Subscript i plus 1 slash 2 Superscript minus Baseline left-parenthesis t right-parenthesis period EndLayout

Then, Equation3.3 can now be written under the form

StartLayout 1st Row with Label left-parenthesis 3.5 right-parenthesis EndLabel StartLayout 1st Row 1st Column upper W prime Subscript i Baseline left-parenthesis t right-parenthesis equals minus StartFraction 1 Over normal upper Delta x EndFraction left-parenthesis 2nd Column script upper A Subscript i minus 1 slash 2 Superscript plus Baseline left-parenthesis upper W Subscript i minus 1 slash 2 Superscript plus Baseline left-parenthesis t right-parenthesis minus upper W Subscript i minus 1 slash 2 Superscript minus Baseline left-parenthesis t right-parenthesis right-parenthesis 2nd Row 1st Column Blank 2nd Column plus script upper A Subscript i plus 1 slash 2 Superscript minus Baseline left-parenthesis upper W Subscript i plus 1 slash 2 Superscript plus Baseline left-parenthesis t right-parenthesis minus upper W Subscript i plus 1 slash 2 Superscript minus Baseline left-parenthesis t right-parenthesis right-parenthesis plus integral Subscript x Subscript i minus 1 slash 2 Baseline Superscript x Subscript i plus 1 slash 2 Baseline Baseline StartFraction d Over d x EndFraction upper F left-parenthesis upper P Subscript i Superscript t Baseline left-parenthesis x right-parenthesis right-parenthesis d x right-parenthesis period EndLayout EndLayout

Note now that Equation3.5 can be easily generalized to obtain a numerical scheme for solving Equation2.1:

StartLayout 1st Row with Label left-parenthesis 3.6 right-parenthesis EndLabel StartLayout 1st Row 1st Column upper W prime Subscript i Baseline left-parenthesis t right-parenthesis equals minus StartFraction 1 Over normal upper Delta x EndFraction left-parenthesis 2nd Column script upper A Subscript i minus 1 slash 2 Superscript plus Baseline left-parenthesis upper W Subscript i minus 1 slash 2 Superscript plus Baseline left-parenthesis t right-parenthesis minus upper W Subscript i minus 1 slash 2 Superscript minus Baseline left-parenthesis t right-parenthesis right-parenthesis 2nd Row 1st Column Blank 2nd Column plus script upper A Subscript i plus 1 slash 2 Superscript minus Baseline left-parenthesis upper W Subscript i plus 1 slash 2 Superscript plus Baseline left-parenthesis t right-parenthesis minus upper W Subscript i plus 1 slash 2 Superscript minus Baseline left-parenthesis t right-parenthesis right-parenthesis plus integral Subscript x Subscript i minus 1 slash 2 Baseline Superscript x Subscript i plus 1 slash 2 Baseline Baseline script upper A left-parenthesis upper P Subscript i Superscript t Baseline left-parenthesis x right-parenthesis right-parenthesis StartFraction d Over d x EndFraction upper P Subscript i Superscript t Baseline left-parenthesis x right-parenthesis d x right-parenthesis comma EndLayout EndLayout

where the intermediate matrices are defined by means of a Roe linearization based on a family of paths normal upper Psi and upper P Subscript i Superscript t is again a regular function satisfying Equation3.4.

Remark 3.1

It is important to note that for conservative problems, the numerical scheme Equation3.6 is equivalent to the conservative numerical scheme Equation3.3 if, and only if, the integral term is computed exactly. However, the formulation Equation3.6 is useless when working with conservative problems, as we would get involved with a more complex expression of the numerical scheme. The numerical scheme Equation3.6 is useful only for problems with nonconservative products, as it allows us to deduce numerical schemes for particular problems, using numerical quadratures if necessary.

There is an important difference between the conservative and the nonconservative case: in the conservative case the numerical scheme is independent of the functions upper P Subscript i Superscript t chosen at the cells, but this is not the case for nonconservative problems. As a consequence, while the numerical scheme Equation3.2 has the same order of the reconstruction operator, in the case of the scheme Equation3.6 it seems clear that, in order to have a high order scheme, together with a high order reconstruction operator, the functions upper P Subscript i Superscript t and their derivatives have to be high order approximations of upper W left-parenthesis dot comma t right-parenthesis and its partial derivative upper W left-parenthesis dot comma t right-parenthesis Subscript x .

In practice, the definition of the reconstruction operator gives the natural choice of the function upper P Subscript i Superscript t , as the usual procedure to define a reconstruction operator is the following: given a sequence StartSet upper W Subscript i Baseline EndSet of values at the cells, first an approximation function is constructed at every cell upper I Subscript i , based on the values of upper W Subscript i at some of the neighbor cells (the stencil):

upper P Subscript i Baseline left-parenthesis x semicolon upper W Subscript i minus l Baseline comma ellipsis comma upper W Subscript i plus r Baseline right-parenthesis comma

for some natural numbers l comma r . These approximations functions are calculated by means of an interpolation or approximation procedure. Once these functions have been constructed, upper W Subscript i plus 1 slash 2 Superscript minus (resp. upper W Subscript i plus 1 slash 2 Superscript plus ) is calculated by taking the limit of upper P Subscript i (resp. upper P Subscript i plus 1 ) to the left (resp. to the right) of x Subscript i plus 1 slash 2 . If the reconstruction operator is built following this procedure (as we will assume in the sequel), the natural choice of upper P Subscript i Superscript t is

upper P Subscript i Superscript t Baseline left-parenthesis x right-parenthesis equals upper P Subscript i Baseline left-parenthesis x semicolon upper W Subscript i minus l Baseline left-parenthesis t right-parenthesis comma ellipsis comma upper W Subscript i plus r Baseline left-parenthesis t right-parenthesis right-parenthesis period

Let us now investigate the order of the numerical scheme Equation3.6. Note first that, for regular solutions upper W of Equation2.1, the cell averages at the cells StartSet ModifyingAbove upper W With bar Subscript j Baseline left-parenthesis t right-parenthesis EndSet satisfy

StartLayout 1st Row with Label left-parenthesis 3.7 right-parenthesis EndLabel ModifyingAbove upper W With bar prime Subscript i Baseline left-parenthesis t right-parenthesis equals minus StartFraction 1 Over normal upper Delta x EndFraction integral Subscript x Subscript i minus 1 slash 2 Baseline Superscript x Subscript i plus 1 slash 2 Baseline Baseline script upper A left-parenthesis upper W left-parenthesis x comma t right-parenthesis right-parenthesis upper W Subscript x Baseline left-parenthesis x comma t right-parenthesis d x period EndLayout

Thus, Equation3.6 is expected to be a good approximation of Equation3.7. This fact is stated in the following result:

Theorem 3.2

Let us assume that script upper A is of class script upper C squared with bounded derivatives and script upper A Subscript normal upper Psi is bounded. Let us also suppose that the p -order reconstruction operator is such that, given a sequence defined by

upper W Subscript i Baseline equals StartFraction 1 Over normal upper Delta x EndFraction integral Underscript upper I Subscript i Baseline Endscripts upper W left-parenthesis x right-parenthesis d x

for some smooth function upper W , we have that

StartLayout 1st Row 1st Column Blank 2nd Column upper P Subscript i Baseline left-parenthesis x semicolon upper W Subscript i minus l Baseline comma ellipsis comma upper W Subscript i plus r Baseline right-parenthesis equals upper W left-parenthesis x right-parenthesis plus upper O left-parenthesis normal upper Delta x Superscript q Baseline right-parenthesis comma for-all x element-of upper I Subscript i Baseline comma 2nd Row 1st Column Blank 2nd Column StartFraction d Over d x EndFraction upper P Subscript i Baseline left-parenthesis x right-parenthesis equals upper W prime left-parenthesis x right-parenthesis plus upper O left-parenthesis normal upper Delta x Superscript r Baseline right-parenthesis comma for-all x element-of upper I Subscript i Baseline period EndLayout

Then Equation3.6 is an approximation of order at least gamma equals min left-parenthesis p comma q plus 1 comma r plus 1 right-parenthesis to the system Equation3.7 in the following sense:

StartLayout 1st Row with Label left-parenthesis 3.8 right-parenthesis EndLabel StartLayout 1st Row 1st Column Blank 2nd Column script upper A Subscript i minus 1 slash 2 Superscript plus Baseline left-parenthesis upper W Subscript i minus 1 slash 2 Superscript plus Baseline left-parenthesis t right-parenthesis minus upper W Subscript i minus 1 slash 2 Superscript minus Baseline left-parenthesis t right-parenthesis right-parenthesis 2nd Row 1st Column Blank 2nd Column plus script upper A Subscript i plus 1 slash 2 Superscript minus Baseline left-parenthesis upper W Subscript i plus 1 slash 2 Superscript plus Baseline left-parenthesis t right-parenthesis minus upper W Subscript i plus 1 slash 2 Superscript minus Baseline left-parenthesis t right-parenthesis right-parenthesis 3rd Row 1st Column Blank 2nd Column plus integral Subscript x Subscript i minus 1 slash 2 Baseline Superscript x Subscript i plus 1 slash 2 Baseline script upper A left-parenthesis upper P Subscript i Superscript t Baseline left-parenthesis x right-parenthesis right-parenthesis StartFraction d Over d x EndFraction upper P Subscript i Superscript t Baseline left-parenthesis x right-parenthesis d x 4th Row 1st Column Blank 2nd Column equals integral Subscript x Subscript i minus 1 slash 2 Baseline Superscript x Subscript i plus 1 slash 2 Baseline Baseline script upper A left-parenthesis upper W left-parenthesis x comma t right-parenthesis right-parenthesis upper W Subscript x Baseline left-parenthesis x comma t right-parenthesis d x plus upper O left-parenthesis normal upper Delta x Superscript gamma Baseline right-parenthesis comma EndLayout EndLayout

for every smooth enough solution upper W , upper W Subscript i plus 1 slash 2 Superscript plus-or-minus Baseline left-parenthesis t right-parenthesis being the associated reconstructions and upper P Subscript i Superscript t the approximation functions corresponding to the sequence

ModifyingAbove upper W With bar Subscript i Baseline left-parenthesis t right-parenthesis equals StartFraction 1 Over normal upper Delta x EndFraction integral Subscript x Subscript i minus 1 slash 2 Baseline Superscript x Subscript i plus 1 slash 2 Baseline Baseline upper W left-parenthesis x comma t right-parenthesis d x period

Proof.

On the one hand, as the reconstruction operator is of order p , we have

script upper A Subscript i minus 1 slash 2 Superscript plus Baseline left-parenthesis upper W Subscript i minus 1 slash 2 Superscript plus Baseline left-parenthesis t right-parenthesis minus upper W Subscript i minus 1 slash 2 Superscript minus Baseline left-parenthesis t right-parenthesis right-parenthesis plus script upper A Subscript i plus 1 slash 2 Superscript minus Baseline left-parenthesis upper W Subscript i plus 1 slash 2 Superscript plus Baseline left-parenthesis t right-parenthesis minus upper W Subscript i plus 1 slash 2 Superscript minus Baseline left-parenthesis t right-parenthesis right-parenthesis equals upper O left-parenthesis normal upper Delta x Superscript p Baseline right-parenthesis period

On the other hand,

StartLayout 1st Row 1st Column Blank 2nd Column integral Subscript x Subscript i minus 1 slash 2 Baseline Superscript x Subscript i plus 1 slash 2 Baseline script upper A left-parenthesis upper P Subscript i Superscript t Baseline left-parenthesis x right-parenthesis right-parenthesis StartFraction d Over d x EndFraction upper P Subscript i Superscript t Baseline left-parenthesis x right-parenthesis d x minus integral Subscript x Subscript i minus 1 slash 2 Baseline Superscript x Subscript i plus 1 slash 2 Baseline script upper A left-parenthesis upper W left-parenthesis x comma t right-parenthesis right-parenthesis upper W Subscript x Baseline left-parenthesis x comma t right-parenthesis d x 2nd Row 1st Column Blank 2nd Column equals integral Subscript x Subscript i minus 1 slash 2 Baseline Superscript x Subscript i plus 1 slash 2 Baseline Baseline left-parenthesis script upper A left-parenthesis upper P Subscript i Superscript t Baseline left-parenthesis x right-parenthesis right-parenthesis minus script upper A left-parenthesis upper W left-parenthesis x comma t right-parenthesis right-parenthesis right-parenthesis StartFraction d Over d x EndFraction upper P Subscript i Superscript t Baseline left-parenthesis x right-parenthesis d x 3rd Row 1st Column Blank 2nd Column plus integral Subscript x Subscript i minus 1 slash 2 Baseline Superscript x Subscript i plus 1 slash 2 Baseline Baseline script upper A left-parenthesis upper W left-parenthesis x comma t right-parenthesis right-parenthesis left-parenthesis upper W Subscript x Baseline left-parenthesis x comma t right-parenthesis minus StartFraction d Over d x EndFraction upper P Subscript i Superscript t Baseline left-parenthesis x right-parenthesis right-parenthesis d x 4th Row 1st Column Blank 2nd Column equals upper O left-parenthesis normal upper Delta x Superscript r plus 1 Baseline right-parenthesis plus upper O left-parenthesis normal upper Delta x Superscript q plus 1 Baseline right-parenthesis period EndLayout

The equality Equation3.8 is easily deduced from the above equalities.

Remark 3.3

For the usual reconstruction operators one has r less-than-or-equal-to q less-than-or-equal-to p , and thus the order of Equation3.6 is r plus 1 for nonconservative systems and p for conservation laws. Therefore a loss of accuracy can be observed when a technique of reconstruction giving order p for systems of conservation laws is applied to a nonconservative problem.

4. Well-balanced property

In this paragraph we investigate the well-balanced properties of schemes of the form Equation3.6. Well-balancing is related with the numerical approximation of equilibria, i.e., steady state solutions. System Equation2.1 can only have nontrivial steady state solutions if it has linearly degenerate fields: if upper W left-parenthesis x right-parenthesis is a regular steady state solution it satisfies

script upper A left-parenthesis upper W left-parenthesis x right-parenthesis right-parenthesis dot upper W prime left-parenthesis x right-parenthesis equals 0 comma for-all x element-of double-struck upper R comma

and then 0 is an eigenvalue of script upper A left-parenthesis upper W left-parenthesis x right-parenthesis right-parenthesis for all x and upper W prime left-parenthesis x right-parenthesis is an eigenvector. Therefore, the solution can be interpreted as a parametrization of an integral curve of a linearly degenerate characteristic field whose corresponding eigenvalue takes the value 0 through the curve. In order to define the concept of well-balancing, let us introduce the set normal upper Gamma of all the integral curves gamma of a linearly degenerate field of script upper A left-parenthesis upper W right-parenthesis such that the corresponding eigenvalue vanishes on normal upper Gamma . According to Reference28, given a curve gamma element-of normal upper Gamma , a numerical scheme is said to be exactly well-balanced (respectively well-balanced with order k ) for gamma if it solves exactly (respectively up to order upper O left-parenthesis normal upper Delta x Superscript k Baseline right-parenthesis ) regular stationary solutions upper W satisfying upper W left-parenthesis x right-parenthesis element-of gamma for every x . The numerical scheme is said to be exactly well-balanced or well-balanced with order k if these properties are satisfied for any curve gamma of normal upper Gamma (see Reference28 for details).

In the cited article, it has been shown that a Roe scheme Equation2.7 based on a family of paths normal upper Psi is exactly well balanced for a curve gamma element-of normal upper Gamma if, given two states upper W Subscript upper L and upper W Subscript upper R in gamma , the path normal upper Psi left-parenthesis s semicolon upper W Subscript upper L Baseline comma upper W Subscript upper R Baseline right-parenthesis is a parametrization of the arc of gamma linking the states. In particular, if the family of paths normal upper Psi coincides with the one used in the definition of weak solutions normal upper Phi , the numerical scheme is exactly well balanced. On the other hand, the numerical scheme is well balanced with order p if normal upper Psi left-parenthesis s semicolon upper W Subscript upper L Baseline comma upper W Subscript upper R Baseline right-parenthesis approximates with order p a regular parametrization of the arc of gamma linking the states. In particular, a Roe scheme based on the family of segments Equation2.10 is well balanced with order 2. Moreover, it is exactly well balanced for curves of normal upper Gamma that are straight lines.

The definition of a well-balanced scheme introduced in Reference28 can be easily extended for semi-discrete methods.

Definition 4.1

Let us consider a semi-discrete method for solving Equation2.1:

StartLayout 1st Row with Label left-parenthesis 4.1 right-parenthesis EndLabel StartLayout Enlarged left-brace 1st Row upper W prime Subscript i Baseline left-parenthesis t right-parenthesis equals StartFraction 1 Over normal upper Delta x EndFraction script upper H left-parenthesis bold upper W left-parenthesis t right-parenthesis semicolon i right-parenthesis comma 2nd Row bold upper W left-parenthesis 0 right-parenthesis equals bold upper W 0 comma EndLayout EndLayout

where bold upper W left-parenthesis t right-parenthesis equals StartSet upper W Subscript i Baseline left-parenthesis t right-parenthesis EndSet represent the vector of approximations to the cell averages of the exact solution, and bold upper W 0 equals StartSet upper W Subscript i Superscript 0 Baseline EndSet the vector of initial data. Let gamma be a curve of normal upper Gamma . The numerical method Equation4.1 is said to be exactly well balanced for gamma if, given a regular stationary solution upper W such that

upper W left-parenthesis x right-parenthesis element-of gamma comma for-all x element-of double-struck upper R comma

the vector bold upper W equals StartSet upper W left-parenthesis x Subscript i Baseline right-parenthesis EndSet is a critical point for the system of differential equations in Equation4.1, i.e.,

script upper H left-parenthesis bold upper W semicolon i right-parenthesis equals 0 comma for-all i period

Also, it is said to be well balanced with order p if

script upper H left-parenthesis bold upper W semicolon i right-parenthesis equals upper O left-parenthesis normal upper Delta x Superscript p Baseline right-parenthesis comma for-all i period

Finally, the semi-discrete method Equation4.1 is said to be exactly well balanced or well balanced with order p if these properties are satisfied for every curve gamma of the set normal upper Gamma .

For the particular case of the numerical schemes based on reconstruction of states Equation3.6 we have

StartLayout 1st Row script upper H left-parenthesis bold upper W semicolon i right-parenthesis equals script upper A Subscript i minus 1 slash 2 Superscript plus Baseline left-parenthesis upper W Subscript i minus 1 slash 2 Superscript plus Baseline minus upper W Subscript i minus 1 slash 2 Superscript minus Baseline right-parenthesis plus script upper A Subscript i plus 1 slash 2 Superscript minus Baseline left-parenthesis upper W Subscript i plus 1 slash 2 Superscript plus Baseline minus upper W Subscript i plus 1 slash 2 Superscript minus Baseline right-parenthesis 2nd Row plus integral Subscript x Subscript i minus 1 slash 2 Baseline Superscript x Subscript i plus 1 slash 2 Baseline Baseline script upper A left-parenthesis upper P Subscript i Baseline left-parenthesis x right-parenthesis right-parenthesis StartFraction d Over d x EndFraction upper P Subscript i Baseline left-parenthesis x right-parenthesis d x comma EndLayout

where upper W Subscript i plus 1 slash 2 Superscript plus-or-minus are the reconstructions associated to the sequence bold upper W and upper P Subscript i the corresponding approximation functions. Hereafter, we give two results concerning the well-balanced property of this scheme, but first we introduce a new definition.

Definition 4.2

A reconstruction operator based on smooth approximation functions is said to be exactly well balanced for a curve gamma element-of normal upper Gamma if, given a sequence StartSet upper W Subscript i Baseline EndSet in gamma , the approximation functions satisfy

StartLayout 1st Row with Label left-parenthesis 4.2 right-parenthesis EndLabel upper P Subscript i Baseline left-parenthesis x right-parenthesis element-of gamma comma for-all x element-of left-bracket x Subscript i minus 1 slash 2 Baseline comma x Subscript i plus 1 slash 2 Baseline right-bracket comma EndLayout

for every i .

Theorem 4.3

Let gamma belong to normal upper Gamma . Let us suppose that both the generalized Roe method and the reconstruction operator chosen are exactly well balanced for gamma . Then the numerical scheme Equation3.6 is also exactly well balanced for gamma .

Proof.

Let upper W be a regular stationary solution satisfying

upper W left-parenthesis x right-parenthesis element-of gamma comma for-all x comma

and bold upper W equals StartSet upper W left-parenthesis x Subscript i Baseline right-parenthesis EndSet . From Equation4.2 and the exactly well-balanced character of the generalized Roe method, we obtain

script upper A Subscript i plus 1 slash 2 Superscript plus-or-minus Baseline left-parenthesis upper W Subscript i plus 1 slash 2 Superscript plus Baseline minus upper W Subscript i plus 1 slash 2 Superscript minus Baseline right-parenthesis equals 0 period

On the other hand, using Equation4.2, upper P Subscript i can be understood as a parametrization of an arc of gamma , which is an integral curve of a linearly degenerate field whose corresponding eigenvalue is zero. Therefore,

integral Subscript x Subscript i minus 1 slash 2 Baseline Superscript x Subscript i plus 1 slash 2 Baseline Baseline script upper A left-parenthesis upper P Subscript i Baseline left-parenthesis x right-parenthesis right-parenthesis StartFraction d Over d x EndFraction upper P Subscript i Baseline left-parenthesis x right-parenthesis d x equals 0 period

The proof is easily deduced from the two equalities above.

Theorem 4.4

Under the hypotheses of Theorem 3.2, the scheme Equation3.6 is well balanced with order at least gamma equals min left-parenthesis p comma q plus 1 comma r plus 1 right-parenthesis .

Proof.

The proof is similar to that of Theorem 3.2

Remark 4.5

Note that well-balanced properties for the Roe scheme or the reconstruction operator are not required in this latter result.

5. Applications

We consider in this section systems of the form

StartLayout 1st Row with Label left-parenthesis 5.1 right-parenthesis EndLabel StartFraction partial-differential upper W Over partial-differential t EndFraction plus StartFraction partial-differential upper F Over partial-differential x EndFraction left-parenthesis upper W comma sigma right-parenthesis equals script upper B left-parenthesis upper W comma sigma right-parenthesis dot StartFraction partial-differential upper W Over partial-differential x EndFraction plus ModifyingAbove upper S With tilde left-parenthesis upper W comma sigma right-parenthesis StartFraction d sigma Over d x EndFraction comma EndLayout

where

upper W left-parenthesis x comma t right-parenthesis equals Start 4 By 1 Matrix 1st Row w 1 left-parenthesis x comma t right-parenthesis 2nd Row w 2 left-parenthesis x comma t right-parenthesis 3rd Row vertical-ellipsis 4th Row w Subscript upper N Baseline left-parenthesis x comma t right-parenthesis EndMatrix element-of double-struck upper R Superscript upper N Baseline period

Here, sigma left-parenthesis x right-parenthesis is a known function from double-struck upper R to double-struck upper R , upper F is a regular function from normal upper Omega times double-struck upper R to double-struck upper R Superscript upper N , normal upper Omega is an open convex subset of double-struck upper R Superscript upper N , script upper B is a regular matrix-valued function from normal upper Omega times double-struck upper R to script upper M Subscript upper N Baseline left-parenthesis double-struck upper R right-parenthesis , and upper S overTilde is a function from normal upper Omega to double-struck upper R Superscript upper N . We can assume without loss of generality that upper S overTilde has the form

ModifyingAbove upper S With tilde left-parenthesis upper W comma sigma right-parenthesis equals upper S left-parenthesis upper W comma sigma right-parenthesis plus StartFraction partial-differential upper F Over partial-differential sigma EndFraction left-parenthesis upper W comma sigma right-parenthesis comma

for some regular function upper S .

We denote by script upper J left-parenthesis upper W comma sigma right-parenthesis the Jacobian matrix of upper F left-parenthesis dot comma sigma right-parenthesis :

script upper J left-parenthesis upper W comma sigma right-parenthesis equals StartFraction partial-differential upper F Over partial-differential upper W EndFraction left-parenthesis upper W comma sigma right-parenthesis period

System Equation5.1 includes as particular cases systems of conservation laws ( script upper B equals 0 , upper S equals 0 ) whose flux function may depend on x via the function sigma , systems of conservation laws with source term or balance laws ( script upper B equals 0 ), or coupled systems of conservation laws as defined in Reference5. In this latter case, script upper J is block-diagonal and the blocks of script upper B corresponding to the nonzero diagonal blocks of script upper J are zero.

Following the idea developed in Reference17, Reference18 for conservation laws with source terms, if we add to Equation5.1 the trivial equation

StartFraction partial-differential sigma Over partial-differential t EndFraction equals 0 comma

the problem can be written under the form Equation2.1:

StartLayout 1st Row with Label left-parenthesis 5.2 right-parenthesis EndLabel upper W overTilde Subscript t Baseline plus ModifyingAbove script upper A With script tilde left-parenthesis upper W overTilde right-parenthesis dot upper W overTilde Subscript x Baseline equals 0 comma EndLayout

where upper W overTilde is the augmented vector

upper W overTilde equals StartBinomialOrMatrix upper W Choose sigma EndBinomialOrMatrix comma

and the block structure of the left-parenthesis upper N plus 1 right-parenthesis times left-parenthesis upper N plus 1 right-parenthesis matrix ModifyingAbove script upper A With script tilde left-parenthesis upper W overTilde right-parenthesis is given by

ModifyingAbove script upper A With script tilde left-parenthesis upper W overTilde right-parenthesis equals Start 2 By 2 Matrix 1st Row 1st Column script upper A left-parenthesis upper W overTilde right-parenthesis 2nd Column minus ModifyingAbove upper S With tilde left-parenthesis upper W overTilde comma sigma right-parenthesis 2nd Row 1st Column 0 2nd Column 0 EndMatrix period

Here script upper A left-parenthesis upper W overTilde right-parenthesis represents the upper N times upper N matrix

script upper A left-parenthesis upper W overTilde right-parenthesis equals script upper J left-parenthesis upper W comma sigma right-parenthesis minus script upper B left-parenthesis upper W comma sigma right-parenthesis period

We assume that the matrix script upper A left-parenthesis upper W overTilde right-parenthesis has upper N real distinct eigenvalues

lamda 1 left-parenthesis upper W overTilde right-parenthesis less-than midline-horizontal-ellipsis less-than lamda Subscript upper N Baseline left-parenthesis upper W overTilde right-parenthesis

and associated eigenvectors upper R Subscript j Baseline left-parenthesis upper W overTilde right-parenthesis , j equals 1 comma ellipsis comma upper N . If these eigenvalues do not vanish, Equation5.2 is a strictly hyperbolic system: ModifyingAbove script upper A With script tilde left-parenthesis upper W overTilde right-parenthesis has upper N plus 1 distinct real eigenvalues

lamda 1 left-parenthesis upper W overTilde right-parenthesis comma ellipsis comma lamda Subscript upper N Baseline left-parenthesis upper W overTilde right-parenthesis comma 0 comma

with associated eigenvectors

ModifyingAbove upper R With tilde Subscript 1 Baseline left-parenthesis upper W overTilde right-parenthesis comma ellipsis comma ModifyingAbove upper R With tilde Subscript upper N plus 1 Baseline left-parenthesis upper W overTilde right-parenthesis comma

given by

ModifyingAbove upper R With tilde Subscript i Baseline left-parenthesis upper W overTilde right-parenthesis equals StartBinomialOrMatrix upper R Subscript i Baseline left-parenthesis upper W overTilde right-parenthesis Choose 0 EndBinomialOrMatrix comma i equals 1 comma ellipsis comma upper N comma ModifyingAbove upper R With tilde Subscript upper N plus 1 Baseline left-parenthesis upper W overTilde right-parenthesis equals StartBinomialOrMatrix script upper A left-parenthesis upper W overTilde right-parenthesis Superscript negative 1 Baseline dot upper S left-parenthesis upper W overTilde right-parenthesis Choose 1 EndBinomialOrMatrix period

Clearly, the left-parenthesis upper N plus 1 right-parenthesis -th field is linearly degenerate and, for the sake of simplicity, we will suppose that it is the only one. The integral curves of the linearly degenerate field are given by those of the ODE system

StartFraction d upper W overTilde Over d s EndFraction equals ModifyingAbove upper R With tilde Subscript upper N plus 1 Baseline left-parenthesis upper W overTilde right-parenthesis period

Remark 5.1

Note that, in this case, the set normal upper Gamma defined in the previous section is simply the set of all the integral curves of the linearly degenerate field, as the corresponding eigenvalues always take the value 0. Let us illustrate in this case the relation between these integral curves and the stationary solutions. Let gamma be an integral curve of the linearly degenerate field and let us suppose that it can be described implicitly by a system of upper N equations:

StartLayout 1st Row with Label left-parenthesis 5.3 right-parenthesis EndLabel g Subscript j Baseline left-parenthesis w 1 comma ellipsis comma w Subscript upper N Baseline comma sigma right-parenthesis equals 0 comma 1 less-than-or-equal-to j less-than-or-equal-to upper N period EndLayout

As sigma is a known function, for every x , Equation5.3 is a system of upper N equations with upper N unknowns w 1 comma ellipsis comma w Subscript upper N Baseline . The stationary solutions associated to the curve gamma are obtained by searching solutions StartSet w 1 left-parenthesis x right-parenthesis comma ellipsis comma w Subscript upper N Baseline left-parenthesis x right-parenthesis EndSet of system Equation5.3 which depend smoothly on x .

For the definition of weak solutions of system Equation5.2 and the choice of the family of paths, we refer the interested reader to Reference28 and the references therein. Let us only mention that the complete definition of the path linking two states is not easy, as it requires the explicit knowledge of the solution of the corresponding Riemann problem. Therefore, the construction of Roe schemes based on the family of paths used in the definition of weak solutions is, in general, a difficult task.

Thus we consider the general case in which the family of paths normal upper Psi overTilde used for the construction of Roe matrices is different to that used in the definition of weak solutions. In particular, in the applications the family of segments Equation2.10 has been considered.

The following notation will be used:

ModifyingAbove normal upper Psi With tilde left-parenthesis s semicolon upper W overTilde Subscript i Superscript n Baseline comma upper W overTilde Subscript i plus 1 Superscript n Baseline right-parenthesis equals StartBinomialOrMatrix normal upper Psi left-parenthesis s semicolon upper W overTilde Subscript i Superscript n Baseline comma upper W overTilde Subscript i plus 1 Superscript n Baseline right-parenthesis Choose normal upper Psi Subscript upper N plus 1 Baseline left-parenthesis s semicolon upper W overTilde Subscript i Superscript n Baseline comma upper W overTilde Subscript i plus 1 Superscript n Baseline right-parenthesis EndBinomialOrMatrix equals Start 4 By 1 Matrix 1st Row normal upper Psi 1 left-parenthesis s semicolon upper W overTilde Subscript i Superscript n Baseline comma upper W overTilde Subscript i plus 1 Superscript n Baseline right-parenthesis 2nd Row vertical-ellipsis 3rd Row normal upper Psi Subscript upper N Baseline left-parenthesis s semicolon upper W overTilde Subscript i Superscript n Baseline comma upper W overTilde Subscript i plus 1 Superscript n Baseline right-parenthesis 4th Row normal upper Psi Subscript upper N plus 1 Baseline left-parenthesis s semicolon upper W overTilde Subscript i Superscript n Baseline comma upper W overTilde Subscript i plus 1 Superscript n Baseline right-parenthesis EndMatrix period

Let us suppose that, for any fixed value of sigma , Roe matrices can be calculated for the system of conservation laws corresponding to script upper B equals 0 and upper S equals 0 , i.e., we assume that, given upper W Subscript i Superscript n , upper W Subscript i plus 1 Superscript n and sigma , we can calculate a matrix script upper J Subscript i plus 1 slash 2 Superscript sigma such that

script upper J Subscript i plus 1 slash 2 Superscript sigma Baseline dot left-parenthesis upper W Subscript i plus 1 Superscript n Baseline minus upper W Subscript i Superscript n Baseline right-parenthesis equals upper F left-parenthesis upper W Subscript i plus 1 Superscript n Baseline comma sigma right-parenthesis minus upper F left-parenthesis upper W Subscript i Superscript n Baseline comma sigma right-parenthesis period

Let us also suppose that it is possible to calculate a value sigma Subscript i plus 1 slash 2 of sigma , a upper N times upper N matrix script upper B Subscript i plus 1 slash 2 , and a vector upper S Subscript i plus 1 slash 2 , such that the following identities hold:

StartLayout 1st Row with Label left-parenthesis 5.4 right-parenthesis EndLabel upper F left-parenthesis upper W Subscript i plus 1 Superscript n Baseline comma sigma Subscript i plus 1 Baseline right-parenthesis minus upper F left-parenthesis upper W Subscript i plus 1 Superscript n Baseline comma sigma Subscript i plus 1 slash 2 Baseline right-parenthesis plus upper F left-parenthesis upper W Subscript i Superscript n Baseline comma sigma Subscript i plus 1 slash 2 Baseline right-parenthesis minus upper F left-parenthesis upper W Subscript i Superscript n Baseline comma sigma Subscript i Baseline right-parenthesis 2nd Row equals integral Subscript 0 Superscript 1 Baseline StartFraction partial-differential upper F Over partial-differential sigma EndFraction left-parenthesis ModifyingAbove normal upper Psi With tilde left-parenthesis s semicolon upper W overTilde Subscript i Superscript n Baseline comma upper W overTilde Subscript i plus 1 Superscript n Baseline right-parenthesis right-parenthesis dot StartFraction partial-differential normal upper Psi Subscript upper N plus 1 Baseline Over partial-differential s EndFraction left-parenthesis s semicolon upper W overTilde Subscript i Superscript n Baseline comma upper W overTilde Subscript i plus 1 Superscript n Baseline right-parenthesis d s comma EndLayout

script upper B Subscript i plus 1 slash 2 Baseline dot left-parenthesis upper W Subscript i plus 1 Superscript n Baseline minus upper W Subscript i Superscript n Baseline right-parenthesis equals integral Subscript 0 Superscript 1 Baseline script upper B left-parenthesis ModifyingAbove normal upper Psi With tilde left-parenthesis s semicolon upper W overTilde Subscript i Superscript n Baseline comma upper W overTilde Subscript i plus 1 Superscript n Baseline right-parenthesis right-parenthesis dot StartFraction partial-differential normal upper Psi Over partial-differential s EndFraction left-parenthesis s semicolon upper W overTilde Subscript i Superscript n Baseline comma upper W overTilde Subscript i plus 1 Superscript n Baseline right-parenthesis d s comma

upper S Subscript i plus 1 slash 2 Baseline left-parenthesis sigma Subscript i plus 1 Baseline minus sigma Subscript i Baseline right-parenthesis equals integral Subscript 0 Superscript 1 Baseline upper S left-parenthesis ModifyingAbove normal upper Psi With tilde left-parenthesis s semicolon upper W overTilde Subscript i Superscript n Baseline comma upper W overTilde Subscript i plus 1 Superscript n Baseline right-parenthesis right-parenthesis dot StartFraction partial-differential normal upper Psi Subscript upper N plus 1 Baseline Over partial-differential s EndFraction left-parenthesis s semicolon upper W overTilde Subscript i Superscript n Baseline comma upper W overTilde Subscript i plus 1 Superscript n Baseline right-parenthesis d s period

Then, it can be easily shown (see Reference28) that the matrix

script upper A overTilde Subscript i plus 1 slash 2 Baseline equals Start 3 By 2 Matrix 1st Row 1st Column script upper A Subscript i plus 1 slash 2 Baseline 2nd Column minus upper S Subscript i plus 1 slash 2 Baseline 2nd Row 1st Column Blank 3rd Row 1st Column 0 2nd Column 0 EndMatrix comma

where

script upper A Subscript i plus 1 slash 2 Baseline equals script upper J Subscript i plus 1 slash 2 Superscript sigma Super Subscript i plus 1 slash 2 Superscript Baseline minus script upper B Subscript i plus 1 slash 2 Baseline comma

is a Roe matrix provided that it has upper N plus 1 distinct real eigenvalues.

Once the Roe matrices have been calculated, the reconstructions are added to go to higher order. We will use the following notation:

upper P overTilde Subscript i Superscript t Baseline equals StartBinomialOrMatrix upper P Subscript i Superscript t Baseline Choose p Subscript i comma upper N plus 1 Superscript t Baseline EndBinomialOrMatrix equals Start 4 By 1 Matrix 1st Row p Subscript i comma 1 Superscript t Baseline 2nd Row vertical-ellipsis 3rd Row p Subscript i comma upper N Superscript t Baseline 4th Row p Subscript i comma upper N plus 1 Superscript t Baseline EndMatrix period

Some straightforward calculations allow us to rewrite the scheme Equation3.6 under a form closer to that of WENO-Roe methods for conservation laws:

StartLayout 1st Row with Label left-parenthesis 5.5 right-parenthesis EndLabel StartLayout 1st Row 1st Column upper W prime Subscript i Baseline equals 2nd Column StartFraction normal upper Delta t Over normal upper Delta x EndFraction left-parenthesis upper G overTilde Subscript i minus 1 slash 2 Baseline minus upper G overTilde Subscript i plus 1 slash 2 Baseline right-parenthesis 2nd Row 1st Column Blank 2nd Column plus StartFraction normal upper Delta t Over 2 normal upper Delta x EndFraction left-parenthesis script upper B Subscript i minus 1 slash 2 Baseline dot left-parenthesis upper W Subscript i minus 1 slash 2 Superscript plus Baseline minus upper W Subscript i minus 1 slash 2 Superscript minus Baseline right-parenthesis plus script upper B Subscript i plus 1 slash 2 Baseline dot left-parenthesis upper W Subscript i plus 1 slash 2 Superscript plus Baseline minus upper W Subscript i plus 1 slash 2 Superscript minus Baseline right-parenthesis right-parenthesis 3rd Row 1st Column Blank 2nd Column plus StartFraction normal upper Delta t Over normal upper Delta x EndFraction left-parenthesis script upper P Subscript i minus 1 slash 2 Superscript plus Baseline upper S Subscript i minus 1 slash 2 Baseline left-parenthesis sigma Subscript i minus 1 slash 2 Superscript plus Baseline minus sigma Subscript i minus 1 slash 2 Superscript minus Baseline right-parenthesis plus script upper P Subscript i plus 1 slash 2 Superscript minus Baseline upper S Subscript i plus 1 slash 2 Baseline left-parenthesis sigma Subscript i plus 1 slash 2 Superscript plus Baseline minus sigma Subscript i plus 1 slash 2 Superscript minus Baseline right-parenthesis right-parenthesis 4th Row 1st Column Blank 2nd Column plus StartFraction normal upper Delta t Over 2 normal upper Delta x EndFraction left-parenthesis upper V Subscript i minus 1 slash 2 Baseline plus upper V Subscript i plus 1 slash 2 Baseline right-parenthesis 5th Row 1st Column Blank 2nd Column plus StartFraction normal upper Delta t Over normal upper Delta x EndFraction left-parenthesis script upper I Subscript upper B comma i Baseline plus script upper I Subscript upper S comma i Baseline right-parenthesis comma EndLayout EndLayout

where

StartLayout 1st Row 1st Column upper G overTilde Subscript i plus 1 slash 2 Baseline equals 2nd Column one-half left-parenthesis upper F left-parenthesis upper W Subscript i plus 1 slash 2 Superscript minus Baseline comma sigma Subscript i plus 1 slash 2 Superscript minus Baseline right-parenthesis plus upper F left-parenthesis upper W Subscript i plus 1 slash 2 Superscript plus Baseline comma sigma Subscript i plus 1 slash 2 Superscript plus Baseline right-parenthesis right-parenthesis 2nd Row 1st Column Blank 2nd Column minus one-half StartAbsoluteValue script upper A Subscript i plus 1 slash 2 Baseline EndAbsoluteValue dot left-parenthesis upper W Subscript i plus 1 slash 2 Superscript plus Baseline minus upper W Subscript i plus 1 slash 2 Superscript minus Baseline right-parenthesis EndLayout

and

script upper P Subscript i plus 1 slash 2 Superscript plus-or-minus Baseline equals one-half left-parenthesis script upper I plus-or-minus StartAbsoluteValue script upper A Subscript i plus 1 slash 2 Baseline EndAbsoluteValue script upper A Subscript i plus 1 slash 2 Superscript negative 1 Baseline right-parenthesis period

These latter matrices can be also be written under the form

script upper P Subscript i plus 1 slash 2 Superscript plus-or-minus Baseline equals one-half script upper K Subscript i plus 1 slash 2 Baseline left-parenthesis script upper I plus-or-minus s g n left-parenthesis script upper L right-parenthesis Subscript i plus 1 slash 2 Baseline right-parenthesis script upper K Subscript i plus 1 slash 2 Superscript negative 1 Baseline comma

where script upper K Subscript i plus 1 slash 2 is the upper N times upper N matrix whose columns are the eigenvectors upper R Subscript i plus 1 slash 2 comma 1 , …, upper R Subscript i plus 1 slash 2 comma upper N and s g n left-parenthesis script upper L right-parenthesis Subscript i plus 1 slash 2 is the diagonal matrix whose coefficients are the signs of the eigenvalues lamda Subscript i plus 1 slash 2 comma 1 ,…, lamda Subscript i plus 1 slash 2 comma upper N . Besides,

StartLayout 1st Row 1st Column upper V Subscript i plus 1 slash 2 Baseline equals 2nd Column upper F left-parenthesis upper W Subscript i plus 1 slash 2 Superscript plus Baseline comma sigma Subscript i plus 1 slash 2 Superscript plus Baseline right-parenthesis minus upper F left-parenthesis upper W Subscript i plus 1 slash 2 Superscript plus Baseline comma sigma Subscript i plus 1 slash 2 Baseline right-parenthesis 2nd Row 1st Column Blank 2nd Column plus upper F left-parenthesis upper W Subscript i plus 1 slash 2 Superscript minus Baseline comma sigma Subscript i plus 1 slash 2 Baseline right-parenthesis minus upper F left-parenthesis upper W Subscript i plus 1 slash 2 Superscript minus Baseline comma sigma Subscript i plus 1 slash 2 Superscript minus Baseline right-parenthesis comma EndLayout

or, equivalently (see Equation5.4),

upper V Subscript i plus 1 slash 2 Baseline equals integral Subscript 0 Superscript 1 Baseline StartFraction partial-differential upper F Over partial-differential sigma EndFraction left-parenthesis ModifyingAbove normal upper Psi With tilde left-parenthesis s semicolon upper W overTilde Subscript i plus 1 slash 2 Superscript minus Baseline comma upper W overTilde Subscript i plus 1 slash 2 Superscript plus Baseline right-parenthesis right-parenthesis dot StartFraction partial-differential normal upper Psi Subscript upper N plus 1 Baseline Over partial-differential s EndFraction left-parenthesis s semicolon upper W overTilde Subscript i plus 1 slash 2 Superscript minus Baseline comma upper W overTilde Subscript i plus 1 slash 2 Superscript plus Baseline right-parenthesis d s period

Finally,

StartLayout 1st Row 1st Column Blank 2nd Column script upper I Subscript upper B comma i Baseline equals integral Subscript x Subscript i minus 1 slash 2 Baseline Superscript x Subscript i plus 1 slash 2 Baseline Baseline script upper B left-parenthesis ModifyingAbove upper P With tilde Subscript i Superscript t Baseline left-parenthesis x right-parenthesis right-parenthesis StartFraction d Over d x EndFraction upper P Subscript i Superscript t Baseline left-parenthesis x right-parenthesis d x comma 2nd Row 1st Column Blank 2nd Column script upper I Subscript upper S comma i Baseline equals integral Subscript x Subscript i minus 1 slash 2 Baseline Superscript x Subscript i plus 1 slash 2 Baseline Baseline upper S left-parenthesis ModifyingAbove upper P With tilde Subscript i Superscript t Baseline left-parenthesis x right-parenthesis right-parenthesis StartFraction d Over d x EndFraction p Subscript i comma upper N plus 1 Superscript t Baseline left-parenthesis x right-parenthesis d x period EndLayout

Remark 5.2

In this context, the meaning of the well-balanced property of the reconstruction operator can be understood as follows: let us suppose, as in Remark 5.1, that an integral curve gamma of the linearly degenerate field can be described by a system of equations Equation5.3. Let us suppose that ModifyingAbove upper W With tilde left-parenthesis x right-parenthesis equals left-parenthesis w 1 left-parenthesis x right-parenthesis comma ellipsis comma w Subscript upper N Baseline left-parenthesis x right-parenthesis comma sigma left-parenthesis x right-parenthesis right-parenthesis is a stationary solution such that ModifyingAbove upper W With tilde left-parenthesis x right-parenthesis element-of gamma for all x , i.e.,

g Subscript j Baseline left-parenthesis w 1 left-parenthesis x right-parenthesis comma ellipsis comma w Subscript upper N Baseline left-parenthesis x right-parenthesis comma sigma left-parenthesis x right-parenthesis right-parenthesis equals 0 comma j equals 1 comma period period period upper N comma for-all x period

If we now apply a well-balanced reconstruction operator to the sequence StartSet ModifyingAbove upper W With tilde left-parenthesis x Subscript i Baseline right-parenthesis EndSet , then the approximation functions upper P overTilde Subscript i have to satisfy

g Subscript j Baseline left-parenthesis p Subscript i comma 1 Baseline left-parenthesis x right-parenthesis comma ellipsis comma p Subscript i comma upper N Baseline left-parenthesis x right-parenthesis comma p Subscript i comma upper N plus 1 Baseline left-parenthesis x right-parenthesis right-parenthesis equals 0 comma j equals 1 comma period period period upper N comma for-all x element-of upper I Subscript i Baseline period

6. WENO-Roe methods

In this section we consider numerical schemes of the form Equation3.6, in which the approximation functions used in the reconstruction operator are built by means of a WENO interpolation procedure using stencils with r points; we denote this method simply as r -WENO, and the resulting scheme as r -WENO-Roe. For the details about WENO interpolation, see Reference23, Reference27, Reference33, Reference34. The reconstructions proposed in the r -WENO method are as follows:

upper W Subscript i plus 1 slash 2 Superscript minus Baseline equals sigma-summation Underscript k equals 0 Overscript r minus 1 Endscripts omega Subscript k Superscript minus Baseline q Subscript k Baseline left-parenthesis x Subscript i plus 1 slash 2 Baseline right-parenthesis comma upper W Subscript i minus 1 slash 2 Superscript plus Baseline equals sigma-summation Underscript k equals 0 Overscript r minus 1 Endscripts omega Subscript k Superscript plus Baseline q Subscript k Baseline left-parenthesis x Subscript i minus 1 slash 2 Baseline right-parenthesis comma

where each q Subscript k is the derivative of an interpolation polynomial that uses the values of the sequence upper W Subscript i Superscript n at the stencil

upper S Subscript k Superscript r Baseline equals StartSet x Subscript i minus k Baseline comma ellipsis comma x Subscript i minus k plus r minus 1 Baseline EndSet period

The weights omega Subscript k Superscript plus-or-minus satisfy

w Subscript k Superscript plus-or-minus Baseline greater-than-or-equal-to 0 comma sigma-summation Underscript k equals 0 Overscript r minus 1 Endscripts w Subscript k Superscript plus-or-minus Baseline equals 1 period

These weights are calculated so that, on the one hand, the reconstruction operator is of order 2 r minus 1 and, on the other hand, the weight omega Subscript k is near to zero when the data on the stencil upper S Subscript k Superscript r indicate the presence of a discontinuity.

In order to construct the approximation function at the cells, let us first define

upper P Subscript i plus 1 slash 2 Superscript minus Baseline left-parenthesis x right-parenthesis equals sigma-summation Underscript k equals 0 Overscript r minus 1 Endscripts omega Subscript k Superscript minus Baseline q Subscript k Baseline left-parenthesis x right-parenthesis comma upper P Subscript i minus 1 slash 2 Superscript plus Baseline left-parenthesis x right-parenthesis equals sigma-summation Underscript k equals 0 Overscript r minus 1 Endscripts omega Subscript k Superscript plus Baseline q Subscript k Baseline left-parenthesis x right-parenthesis

(see Figure 1).

We have to define a function upper P Subscript i at the cell upper I Subscript i satisfying

limit Underscript x right-arrow x Subscript i minus 1 slash 2 Superscript plus Baseline Endscripts upper P Subscript i Baseline left-parenthesis x right-parenthesis equals upper W Subscript i minus 1 slash 2 Superscript plus Baseline comma limit Underscript x right-arrow x Subscript i plus 1 slash 2 Superscript minus Baseline Endscripts upper P Subscript i Baseline left-parenthesis x right-parenthesis equals upper W Subscript i plus 1 slash 2 Superscript minus Baseline period

A first possibility is given by

StartLayout 1st Row with Label left-parenthesis 6.1 right-parenthesis EndLabel upper P Subscript i Baseline left-parenthesis x right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column upper P Subscript i minus 1 slash 2 Superscript plus Baseline left-parenthesis x right-parenthesis 2nd Column if x element-of left-bracket x Subscript i minus 1 slash 2 Baseline comma x Subscript i Baseline right-parenthesis comma 2nd Row 1st Column upper P Subscript i plus 1 slash 2 Superscript minus Baseline left-parenthesis x right-parenthesis 2nd Column if x element-of left-parenthesis x Subscript i Baseline comma x Subscript i plus 1 slash 2 Baseline right-bracket period EndLayout EndLayout

This first definition does not fit into the framework defined in Section 3, as upper P Subscript i is, in general, discontinuous:

upper W Subscript i Superscript minus Baseline equals upper P Subscript i minus 1 slash 2 Superscript plus Baseline left-parenthesis x Subscript i Baseline right-parenthesis not-equals upper P Subscript i plus 1 slash 2 Superscript minus Baseline left-parenthesis x Subscript i Baseline right-parenthesis equals upper W Subscript i Superscript plus Baseline period

Due to this fact, if a WENO-Roe scheme Equation3.6 is used to design a high order numerical method for a problem of the form Equation5.1, when the numerical scheme is written under the form Equation5.5, an extra term has to be added at the right-hand side:

upper F left-parenthesis upper W Subscript i Superscript plus Baseline right-parenthesis minus upper F left-parenthesis upper W Subscript i Superscript minus Baseline right-parenthesis period

Nevertheless, this difference of fluxes is of order r , and it can be neglected.

A second definition avoiding this discontinuity is the following:

StartLayout 1st Row with Label left-parenthesis 6.2 right-parenthesis EndLabel upper P Subscript i Baseline left-parenthesis x right-parenthesis equals StartFraction 1 Over normal upper Delta x EndFraction left-parenthesis left-parenthesis x Subscript i plus 1 slash 2 Baseline minus x right-parenthesis upper P Subscript i minus 1 slash 2 Superscript plus Baseline left-parenthesis x right-parenthesis plus left-parenthesis x minus x Subscript i minus 1 slash 2 Baseline right-parenthesis upper P Subscript i plus 1 slash 2 Superscript minus Baseline left-parenthesis x right-parenthesis right-parenthesis period EndLayout

Due to the definition of the reconstruction operator, the functions upper P Subscript i given by Equation6.1 or Equation6.2 provide only approximations of order r at the interior points of the cells, while their derivatives give approximations of order r minus 1 . Therefore, applying Theorem 3.2, the method Equation3.6 has only order r , while it has order 2 r minus 1 when it is applied to systems of conservation laws.

Remark 6.1

If, instead of a WENO method the r -ENO reconstruction operator is chosen, the expected order of the numerical scheme is r , since in this case the approximation functions coincide with interpolation polynomials constructed on the basis of stencils with r points. Nevertheless, as commented in Reference34, the use of WENO approximations has several advantadges: it gives smoother operators, it is less sensible to round-off errors, and it avoids the use of conditionals in its practical implementation, being optimal for the vectorization of the algorithms.

It is however possible, performing some slight modifications on the WENO interpolation procedure, to obtain a method of order 2 r minus 1 . The idea is as follows: instead of choosing the usual WENO reconstructions we consider the functions

ModifyingAbove upper P With tilde Subscript i plus 1 slash 2 Superscript minus Baseline left-parenthesis x right-parenthesis equals sigma-summation Underscript k equals 0 Overscript r minus 1 Endscripts ModifyingAbove omega With tilde Subscript k Superscript minus Baseline left-parenthesis x right-parenthesis q Subscript k Baseline left-parenthesis x right-parenthesis comma ModifyingAbove upper P With tilde Subscript i minus 1 slash 2 Superscript plus Baseline left-parenthesis x right-parenthesis equals sigma-summation Underscript k equals 0 Overscript r minus 1 Endscripts ModifyingAbove omega With tilde Subscript k Superscript plus Baseline left-parenthesis x right-parenthesis q Subscript k Baseline left-parenthesis x right-parenthesis comma

where the weights now depend on x and are calculated following the usual procedure in WENO reconstruction, so that the order of approximation is 2 r minus 1 in the cell. Unfortunately, the derivatives of these approximation functions are not easy to obtain. Instead, we substitute these derivatives by new WENO approximation functions

ModifyingAbove upper Q With tilde Subscript i plus 1 slash 2 Superscript minus Baseline left-parenthesis x right-parenthesis equals sigma-summation Underscript k equals 0 Overscript r minus 1 Endscripts ModifyingAbove gamma With tilde Subscript k Superscript minus Baseline left-parenthesis x right-parenthesis q prime Subscript k Baseline left-parenthesis x right-parenthesis comma ModifyingAbove upper Q With tilde Subscript i minus 1 slash 2 Superscript plus Baseline left-parenthesis x right-parenthesis equals sigma-summation Underscript k equals 0 Overscript r minus 1 Endscripts ModifyingAbove gamma With tilde Subscript k Superscript plus Baseline left-parenthesis x right-parenthesis q prime Subscript k Baseline left-parenthesis x right-parenthesis comma

where, again, the weights are calculated, for every x , following the usual procedure in WENO reconstruction. Therefore, we again obtain order 2 r minus 2 in the cell.

Once these functions have been defined, we introduce the new approximation functions at the cells given either by

ModifyingAbove upper P With tilde Subscript i Baseline left-parenthesis x right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column ModifyingAbove upper P With tilde Subscript i minus 1 slash 2 Superscript plus Baseline left-parenthesis x right-parenthesis 2nd Column if x element-of left-bracket x Subscript i minus 1 slash 2 Baseline comma x Subscript i Baseline right-parenthesis comma 2nd Row 1st Column ModifyingAbove upper P With tilde Subscript i plus 1 slash 2 Superscript minus Baseline left-parenthesis x right-parenthesis 2nd Column if x element-of left-parenthesis x Subscript i Baseline comma x Subscript i plus 1 slash 2 Baseline right-bracket comma EndLayout

ModifyingAbove upper Q With tilde Subscript i Baseline left-parenthesis x right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column ModifyingAbove upper Q With tilde Subscript i minus 1 slash 2 Superscript plus Baseline left-parenthesis x right-parenthesis 2nd Column if x element-of left-bracket x Subscript i minus 1 slash 2 Baseline comma x Subscript i Baseline right-parenthesis comma 2nd Row 1st Column ModifyingAbove upper Q With tilde Subscript i plus 1 slash 2 Superscript minus Baseline left-parenthesis x right-parenthesis 2nd Column if x element-of left-parenthesis x Subscript i Baseline comma x Subscript i plus 1 slash 2 Baseline right-bracket comma EndLayoutor

ModifyingAbove upper P With tilde Subscript i Baseline left-parenthesis x right-parenthesis equals StartFraction 1 Over normal upper Delta x EndFraction left-parenthesis left-parenthesis x Subscript i plus 1 slash 2 Baseline minus x right-parenthesis ModifyingAbove upper P With tilde Subscript i minus 1 slash 2 Superscript plus Baseline left-parenthesis x right-parenthesis plus left-parenthesis x minus x Subscript i minus 1 slash 2 Baseline right-parenthesis ModifyingAbove upper P With tilde Subscript i plus 1 slash 2 Superscript minus Baseline left-parenthesis x right-parenthesis right-parenthesis comma

ModifyingAbove upper Q With tilde Subscript i Baseline left-parenthesis x right-parenthesis equals StartFraction 1 Over normal upper Delta x EndFraction left-parenthesis left-parenthesis x Subscript i plus 1 slash 2 Baseline minus x right-parenthesis ModifyingAbove upper Q With tilde Subscript i minus 1 slash 2 Superscript plus Baseline left-parenthesis x right-parenthesis plus left-parenthesis x minus x Subscript i minus 1 slash 2 Baseline right-parenthesis ModifyingAbove upper Q With tilde Subscript i plus 1 slash 2 Superscript minus Baseline left-parenthesis x right-parenthesis right-parenthesis commadepending on the chosen approach.

Once these functions have been defined, the integral appearing in Equation3.6 is replaced by

integral Subscript x Subscript i minus 1 slash 2 Baseline Superscript x Subscript i plus 1 slash 2 Baseline Baseline script upper A left-parenthesis ModifyingAbove upper P With tilde Subscript i Baseline left-parenthesis x right-parenthesis right-parenthesis ModifyingAbove upper Q With tilde Subscript i Baseline left-parenthesis x right-parenthesis d x period

In practice, this integral is approached by means of a Gaussian quadrature of order at least 2 r minus 1 . As a consequence, the weights omega Superscript plus-or-minus Baseline left-parenthesis x right-parenthesis and gamma Superscript plus-or-minus Baseline left-parenthesis x right-parenthesis have to be calculated only at the quadrature points.

Following the same steps as in the case of the r -WENO-Roe method, it can be easily shown that the resulting scheme (that will be denoted as modified r -WENO-Roe) is well balanced with order 2 r minus 1 .

The computational cost of this modified numerical scheme is higher than those corresponding to standard WENO reconstructions, as two set of weights have to be calculated at every quadrature point. Moreover, the positivity of the weights is only ensured at the intercells, due to the choice of stencils. Therefore, in some cases negative weights may appear at interior quadrature points giving rise to oscillations and instabilities. For handling these negative weights, if necessary, the splitting technique of Shi, Hu and Shu (Reference32) can be applied. However, in some cases (see, e.g., Section 7.7) this technique does not completely remove the oscillations, and the scheme eventually crashes. The causes of this problem are currently under investigation.

We finish this section with a remark about time-stepping. As is usual in WENO interpolation based schemes, in order to obtain a full high resolution scheme it is necessary to use a high order method to advance in time. In the schemes considered here we have taken optimal high order TVD Runge-Kutta schemes (Reference19, Reference33).

6.1. Shallow-water equations with depth variations

The equations governing the flow of a shallow-water layer of fluid through a straight channel with constant rectangular cross-section can be written as

StartLayout 1st Row with Label left-parenthesis 6.3 right-parenthesis EndLabel StartLayout Enlarged left-brace 1st Row StartFraction partial-differential h Over partial-differential t EndFraction plus StartFraction partial-differential q Over partial-differential x EndFraction equals 0 comma 2nd Row StartFraction partial-differential q Over partial-differential t EndFraction plus StartFraction partial-differential Over partial-differential x EndFraction left-parenthesis StartFraction q squared Over h EndFraction plus StartFraction g Over 2 EndFraction h squared right-parenthesis equals g h StartFraction d upper H Over d x EndFraction period EndLayout EndLayout

The variable x makes reference to the axis of the channel and t is time, q left-parenthesis x comma t right-parenthesis and h left-parenthesis x comma t right-parenthesis represent the mass-flow and the thickness, respectively, g is gravity, and upper H left-parenthesis x right-parenthesis is the depth function measured from a fixed level of reference. The fluid is supposed to be homogeneous and inviscid.

The system Equation6.3 can be rewritten under the form Equation5.1 with upper N equals 2 ,

upper W equals StartBinomialOrMatrix h Choose q EndBinomialOrMatrix comma upper F left-parenthesis upper W right-parenthesis equals StartBinomialOrMatrix q Choose StartFraction q squared Over h EndFraction plus StartFraction g Over 2 EndFraction h squared EndBinomialOrMatrix comma upper S left-parenthesis upper W right-parenthesis equals StartBinomialOrMatrix 0 Choose minus g h EndBinomialOrMatrix comma

script upper B equals 0 and sigma equals upper H . Observe that, in this case, the flux and the coefficients of the source term do not depend on sigma .

We can also write system Equation6.3 under the nonconservative form Equation5.2 with

upper W overTilde equals Start 3 By 1 Matrix 1st Row h 2nd Row q 3rd Row upper H EndMatrix comma ModifyingAbove script upper A With tilde left-parenthesis upper W overTilde right-parenthesis equals Start 3 By 3 Matrix 1st Row 1st Column 0 2nd Column 1 3rd Column 0 2nd Row 1st Column minus u squared plus c squared 2nd Column 2 u 3rd Column minus c squared 3rd Row 1st Column 0 2nd Column 0 3rd Column 0 EndMatrix comma

where u equals q slash h is the averaged velocity and c equals StartRoot g h EndRoot .

If the family of segments Equation2.10 is chosen as the family of paths, a family of Roe matrices for system Equation6.3 is given by (see Reference28)

ModifyingAbove script upper A With tilde left-parenthesis upper W overTilde Subscript 0 Baseline comma upper W overTilde Subscript 1 Baseline right-parenthesis equals Start 3 By 3 Matrix 1st Row 1st Column 1 2nd Column 0 3rd Column 0 2nd Row 1st Column minus u overTilde squared plus c overTilde squared 2nd Column 2 u overTilde 3rd Column minus c overTilde squared 3rd Row 1st Column 0 2nd Column 0 3rd Column 0 EndMatrix comma

where

u overTilde equals StartFraction StartRoot h 0 EndRoot u 0 plus StartRoot h 1 EndRoot u 1 Over StartRoot h 0 EndRoot plus StartRoot h 1 EndRoot EndFraction comma c overTilde equals StartRoot g StartFraction h 0 plus h 1 Over 2 EndFraction EndRoot period

For system Equation6.3, stationary solutions are given by

StartLayout 1st Row with Label left-parenthesis 6.4 right-parenthesis EndLabel q equals q 0 comma h plus StartFraction q 0 squared Over 2 g h squared EndFraction minus upper H equals upper C comma EndLayout

where q 0 and upper C are constants. In the particular case of water at rest, we have the solutions

StartLayout 1st Row with Label left-parenthesis 6.5 right-parenthesis EndLabel q equals 0 comma h minus upper H equals upper C period EndLayout

Therefore, solutions corresponding to still water define straight lines in the h - q - upper H space. As a consequence, Roe methods based on the family of segments are exactly well balanced for still-water solutions and well balanced with order 2 for general stationary solutions (see Reference2, Reference28).

The reconstruction operator proposed here to get higher order schemes is based on WENO reconstruction related to the variables q , upper H and eta equals h minus upper H (this variable represents the water surface elevation). That is, given a sequence left-parenthesis q Subscript i Baseline comma h Subscript i Baseline comma upper H Subscript i Baseline right-parenthesis we consider the new sequence left-parenthesis q Subscript i Baseline comma eta Subscript i Baseline comma upper H Subscript i Baseline right-parenthesis with eta Subscript i Baseline equals h Subscript i Baseline minus upper H Subscript i and apply the r -WENO reconstruction operator to obtain polynomials

p Subscript i plus 1 slash 2 comma q Superscript plus-or-minus Baseline comma p Subscript i plus 1 slash 2 comma eta Superscript plus-or-minus Baseline comma p Subscript i plus 1 slash 2 comma upper H Superscript plus-or-minus Baseline semicolonthen, we define

p Subscript i plus 1 slash 2 comma h Superscript plus-or-minus Baseline equals p Subscript i plus 1 slash 2 comma eta Superscript plus-or-minus Baseline plus p Subscript i plus 1 slash 2 comma upper H Superscript plus-or-minus Baseline period

This reconstruction is exactly well balanced for stationary solutions corresponding to water at rest. In effect, if the sequence left-parenthesis q Subscript i Baseline comma h Subscript i Baseline comma upper H Subscript i Baseline right-parenthesis lie on the curve defined by Equation6.5, then q Subscript i Baseline equals 0 and eta Subscript i Baseline equals upper C . As a consequence,

p Subscript i plus 1 slash 2 comma q Superscript plus-or-minus Baseline identical-to 0 comma p Subscript i plus 1 slash 2 comma eta Superscript plus-or-minus Baseline identical-to upper C comma

so we have

p Subscript i plus 1 slash 2 comma q Superscript plus-or-minus Baseline identical-to 0 comma p Subscript i plus 1 slash 2 comma h Superscript plus-or-minus Baseline minus p Subscript i plus 1 slash 2 comma upper H Superscript plus-or-minus Baseline equals upper C comma

and thus the reconstruction operator is well balanced (see Remark 5.2).

Applying Theorems 4.3 and 4.4, we deduce that the corresponding WENO-Roe schemes satisfy the script upper C -property, i.e., they are exactly well balanced for still-water solutions, and well balanced with order r for general stationary solutions. To obtain a well-balanced numerical scheme with order 2 r minus 1 , we have to add to the numerical scheme the modifications proposed in Section 6.

6.2. The two-layer shallow-water system

We now consider the equations of a one-dimensional flow of two superposed inmiscible layers of shallow-water fluids studied in Reference5:

StartLayout 1st Row with Label left-parenthesis 6.6 right-parenthesis EndLabel StartLayout Enlarged left-brace 1st Row StartFraction partial-differential h 1 Over partial-differential t EndFraction plus StartFraction partial-differential q 1 Over partial-differential x EndFraction equals 0 comma 2nd Row StartFraction partial-differential q 1 Over partial-differential t EndFraction plus StartFraction partial-differential Over partial-differential x EndFraction left-parenthesis StartFraction q 1 squared Over h 1 EndFraction plus StartFraction g Over 2 EndFraction h 1 squared right-parenthesis equals minus g h 1 StartFraction partial-differential h 2