Original article
Numerical solution of the ‘classical’ Boussinesq system

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Abstract

We consider the ‘classical’ Boussinesq system of water wave theory, which belongs to the class of Boussinesq systems modelling two-way propagation of long waves of small amplitude on the surface of water in a horizontal channel. (We also consider its completely symmetric analog.) We discretize the initial-boundary-value problem for these systems, corresponding to homogeneous Dirichlet boundary conditions on the velocity variable at the endpoints of a finite interval, using fully discrete Galerkin-finite element methods of high accuracy. We use the numerical schemes as exploratory tools to study the propagation and interactions of solitary-wave solutions of these systems, as well as other properties of their solutions.

Introduction

We consider the so-called ‘classical’ Boussinesq system (CB)ηt+ux+(ηu)x=0,ut+ηx+uux13uxxt=0,for x, t > 0, supplemented by the initial conditionsη(x,0)=η0(x),u(x,0)=u0(x),where η0, u0 are given real functions on . The system (1.1) is a Boussinesq-type approximation of the two-dimensional Euler equations that models two-way propagation of long waves of small amplitude on the surface of an incompressible, inviscid fluid in a uniform horizontal channel of finite depth. The variables in (1.1), (1.2) are nondimensional and unscaled: x and t are proportional to position along the channel and time, respectively, and η(x, t) and u(x, t) are proportional to the elevation of the free surface above the level of rest y = 0, and to the horizontal velocity of the fluid at a height y=1+1+η(x,t)/3, respectively. (In terms of these variables the bottom of the channel is at y =  1.) The CB system is a member of a general family of Boussinesq systems derived in [11], that are approximations to the Euler equations of the same order and written in nondimensional, unscaled variables asηt+ux+(ηu)x+auxxxbηxxt=0,ut+ηx+uux+cηxxxduxxt=0,where a = (θ2  1/3)ν/2, b = (θ2  1/3)(1  ν)/2, c = (1  θ2)μ/2, d = (1  θ2)(1  μ)/2, where ν and μ are modelling parameters and 0  θ  1. In the case of the CB we have μ = 0 and θ2 = 1/3.

The CB system was formally derived from the Euler equations, in the appropriate parameter regime, in [27], [29], [33]. Specifically, if ɛ = A/h0 and σ = h0/λ, where A is a typical wave amplitude, h0 is the depth of the channel and λ a typical wavelength, and it is assumed that ɛ  1, σ  1, with ɛ = σ2 for definiteness (so that the Stoke's number S = ɛ/σ2 = 1), appropriate nondimensionalization and scaling of the Euler equations followed by expansions in powers of ɛ yield formally thatηt+ux+ɛ(ηu)x=O(ɛ2),ut+ηx+ɛuux13ɛuxxt=O(ɛ2),in nondimensional, scaled variables, denoted again by η, u, x, t. Replacing the right-hand side of the above equations by zero we obtain the scaled CB systemηt+ux+ɛ(ηu)x=0,ut+ηx+ɛuux13ɛuxxt=0,for x, t > 0. As pointed out in [33], [24], in his work in the 1870s Boussinesq derived a system very close to (1.4) but with the term −ɛuxxt/3 replaced (in our notation) by ɛηxtt/3. In view of the lower-order relation ηt =  ux + O(ɛ) implied by (1.4), one may then derive the system in its present form, which is mathematically more tractable.

Existence and uniqueness of solution of the initial-value problem (ivp) (1.1), (1.2) was studied by Schonbek [30] and Amick [3]. In these papers global existence and uniqueness was established for infinitely differentiable initial data of compact support. In [12] the theory of [30], [3] was used to establish that given initial data (η0, u0)  Hs × Hs+1, where s  1, and such that infxη0(x)>1, there is a unique solution (η, u), which, for any T > 0, lies in C(0, T ; Hs) × C(0, T ; Hs+1). (Here Hs=Hs() is the usual, L2-based Sobolev space of functions on and C(0, T ; X) denotes the space of functions ϕ = ϕ(t) that for each t  [0, T] have values in the Banach space X and are such that the map [0, T]   ϕ  X is continuous.) As a consequence of this global existence-uniqueness result and of the theory in [13], [2], one may rigorously formulate a precise sense in which the scaled system (1.4) is an O(ɛ2) approximation to the Euler equations. Specifically, one may prove that solutions of (1.4) with suitable initial data approximate, in appropriate norms, analogous solutions of the Euler equations, in the same scaling, with an error of O(ɛ2t) uniformly for t  [0, Tɛ], where Tɛ is of O(1/ɛ).

It is well known that the CB system possesses classical solitary-wave solutions. These are travelling wave solutions of (1.1) of the form η(x, t) = ηs(x + x0  cst), u(x, t) = us(x + x0  cst), where x0 and cs is constant. The functions ηs = ηs(ξ), us = us(ξ), ξ, will be assumed to be smooth, positive, even, and decaying monotonically to zero, along with their derivatives, as ξ ± ∞. Substituting into (1.1), integrating once and setting the integration constants equal to zero, one obtains the ode'scsη+u+ηu=0,csu+η+12u2cs3u=0,where we have denoted ηs, us simply by η, u and ′ = d/. (We may assume that cs > 0 since if (η, u) is a solution of this system of ode's for some cs > 0, then (η, − u) is also a solution propagating with speed −cs.) As opposed to the case of a more general a b c d system (1.3), wherein the resulting system of second-order nonlinear ode's is coupled, the above ode system decouples and may be written in the form η=ucsu,csu+ucsu+12u2cs3u=0. The second equation may be studied by a straightforward phase plane analysis, cf. [24], [18], and yields existence and uniqueness of solitary waves for any value of cs > 1.

Solitary waves play an important role in the evolution and long-time asymptotic behavior of solutions of the ivp (1.1), (1.2) that emanate from initial data that decays sufficiently fast at infinity. Such solutions are resolved as t increases into series of solitary waves followed by oscillatory dispersive tails of small amplitude. This resolution property has been rigorously proved by inverse scattering theory for integrable, one-way models such as the KdV equation, and has been observed numerically in many other examples of nonlinear dispersive wave propagation, and in particular in the case of Boussinesq systems, cf. e.g. [10], [4], [5], [26], [20], [8]. It may be viewed as a manifestation of the stability of solitary waves. It is well known, cf. e.g. [24], [20], that the classical variational theory for studying orbital stability of solitary waves does not work in the case of the Boussinesq systems, and that there does not exist yet a rigorous proof of their asymptotic stability. However, linearized ‘convective’ asymptotic stability of the solitary waves of the CB was established in [24]. A detailed numerical study of various stability properties of the solitary waves of the Bona-Smith family of Boussinesq systems has been carried out in [20]. (Similar numerical experiments have been performed by the authors of the present paper in the case of the solitary waves of the CB system, which appear again to be asymptotically stable under a variety of perturbations.)

In this paper we shall solve numerically the following initial-boundary-value problem (ibvp) for CB: For some L > 0 we seek η = η(x, t), u = u(x, t) defined for 0  x  L, t  0, that satisfy (1.1) for 0  x  L, t  0, the initial conditions (1.2) for 0  x  L, and the boundary conditionsu(0,t)=u(L,t)=0,t0.This ibvp has been recently analyzed by Adamy [1], who showed that it has weak (distributional) solutions (η, u) in L+,L1(0,L)×H01(0,L) provided that η0  L1(0, L), u0H01(0,L) with inf x∈[0,L]η0(x) >  1. (Here H01(0,L) denotes the subspace of H1(0, L) consisting of those elements of H1(0, L) that vanish at x = 0 and x = L.) The proof follows the parabolic regularization technique of Schonbek [30]. It should be noted that (1.6) is one kind of boundary conditions that lead to well posed ibvp's in the case of the linearized CB system [22]. The CB system needs only two boundary conditions (b.c.'s) for well-posedness as opposed to the four b.c.'s (for example, Dirichlet b.c.'s on η and u at each end,) required in the case of other Boussinesq systems such as the BBM-BBM [10], and the Bona-Smith systems [9]. Considering the homogeneous Dirichlet boundary conditions (1.6) may be viewed as a first step towards studying their nonhomogeneous analog, where u(0, t) and u(L, t) are given functions of t  0, and which would correspond to known measurement of the velocity variable at the two ends of a channel of finite length. This data may be used then as boundary conditions for a numerical scheme, whose results in [0, L] may be compared with experimental values of η and u in order to assess the accuracy of the CB model. This cannot be done if the numerical solution is computed with periodic b.c.'s. (The latter are adequate for the numerical simulation of interactions of solitary waves in the interior of the domain but may not be used e.g. to study the interactions of solitary waves with the boundary.)

In [13], Bona et al. introduced another type of Boussinesq systems that they called completely symmetric, obtained from the usual systems of the type (1.3) by a nonlinear change of variables, and having the mathematical advantage that their Cauchy problem is always locally nonlinearly well posed (provided the linearized system is well posed). In addition, solutions of their scaled analogs are O(ɛ2t) approximations to appropriate solutions of the Euler equations for t  [0, Tɛ], with Tɛ = O(1/ɛ). The completely symmetric analog of CB is the systemηt+ux+12(ηu)x=0,ut+ηx+32uux+12ηηx13uxxt=0,which we will call SB in the sequel. Its scaled analog isηt+ux+12ɛ(ηu)x=0,ut+ηx+32ɛuux+12ɛηηx13ɛuxxt=0.If we consider (1.7) on the finite spatial interval [0, L] with the homogeneous boundary conditions (1.6), it turns out that the solution of the resulting ibvp satisfies the conservation propertyη(t)2+u(t)2+13ux(t)2=η02+u02+13u02,for t  0, which simplifies the study of its well-posedness and the estimation of the errors of its numerical approximations, cf. [7]. The SB system has solitary waves, that are solutions of the ode's η=2u2csu,csu+η+34u2+14η2cs3u=0.

In the paper at hand we shall study, by numerical means, various properties of the solutions of the CB and the SB systems, paying particular attention to simulations of the generation, propagation, and interactions of their solitary waves. Numerical simulations of solitary wave interactions of the CB system were also performed with spectral methods in [25], [26], and with finite element methods in [4], [5]. (For numerical work on other Boussinesq systems of the type (1.3) we refer the reader to e.g. [25], [10], [6], [4], [26], [23], [15], [16], [20].) It should be noted that the CB system has been extensively used in the engineering fluid mechanics literature for dispersive, nonlinear wave modelling and computations, starting with the pioneering papers of Peregrine [27], [28].

In Section 2 of the paper at hand we describe the numerical scheme that will be used in our simulations. We consider the ibvp's for CB and SB with the homogeneous boundary conditions (1.6) and we discretize them in space using the standard Galerkin-finite element method with continuous, piecewise linear functions and cubic splines. (In the computations of Section 3 we use cubic splines.) The resulting semidiscrete systems of ode's are discretized in the temporal variable using the classical, explicit, four-stage Runge–Kutta scheme of order four. It turns out that these ode systems are mildly stiff, so that a condition of the form k = O(h) is needed for stability, where k is the time step and h the spatial meshlength. This stability condition is not severe and it allows using an explicit time-stepping procedure, thus avoiding the more costly implicit methods that require solving nonlinear systems of equations at each time step.

The error analysis of these numerical schemes is of considerable interest due to the loss of optimal order of accuracy in the discretization of the ‘hyperbolic’ first p.d.e. of the systems (1.1) or (1.7) on a finite interval in the presence of non-periodic boundary conditions. (The loss of optimal order of accuracy in standard Galerkin approximations of first-order hyperbolic problems is well known, cf. e.g. [21].) In our case, the facts that the other pde of both systems has the dispersive term −uxxt, and that u vanishes at the endpoints of [0, L] help us to prove, in the case of uniform spatial meshlength h, improved error estimates e.g. of the form max0tTη(t)ηh(t)Ch3.5ln1/h, max0tTu(t)uh(t)Ch4ln1/h, where ηh, uh are the standard Galerkin cubic spline semidiscrete approximations to η, u, respectively, ∥· ∥ denotes the L2 norm on [0, L], and C is a constant independent of h. In Section 2 we state this and other error estimates for our numerical schemes; their proofs will appear elsewhere, but the interested reader may consult [7].

In Section 3 we present the results of numerical experiments that we performed using the cubic spline – Runge–Kutta fully discrete scheme as an exploratory tool for investigating various properties of solutions of CB and SB. In several numerical experiments involving solitary waves and their interactions, we compute on fairly large spatial intervals [0, L] so that there are no unwanted interactions with the boundaries. In Section 3.1 we study the resolution into solitary waves propagating in both directions emanating from an initial Gaussian ‘heap’ of water, and make some remarks on the number of solitary waves that are produced. We recall the process of iterative ‘cleaning’ used to generate solitary waves numerically [10], [20], and find analytic relations between the speed cs and the amplitude As of solitary waves for CB and SB.

In Section 3.2 we study numerically head-on collisions between solitary waves of equal and unequal height for both systems. The results are qualitatively the same as those observed in similar numerical experiments in the case of other Boussinesq systems and the Euler equations [10], [8], [19]. For both CB and SB there seems to occur a slight, permanent loss of height of the solitary wave of larger amplitude after the interaction. In Section 3.3 we turn to overtaking collisions of solitary waves travelling in the same direction. Such collisions have been studied in detail for one-way models numerically, and by the inverse scattering transform in the integrable case. The results in the case of CB and SB qualitatively agree with those of similar interactions of solitary waves of other Boussinesq systems, including the appearance of a large-wavelength, small-amplitude, dispersive wavelet that travels in a direction opposite to that of the motion of the solitary waves.

In Section 3.4 we study numerically what happens when a solitary wave of the CB or the SB system collides with the boundary point, say at x = L, where u = 0 holds. Although the condition ηx = 0 is not imposed as a boundary condition on the pde system or the numerical scheme, it appears that the numerical approximation of ηx(L, t) tends to zero as h  0. Hence, in the case of this type of Boussinesq systems as well, it appears that reflection of a solitary wave from a vertical wall (where only u = 0 is imposed now) is equivalent to (one half) of the head-on collision of the solitary wave with its symmetric image of opposite velocity, at the center of which ηx = 0 and u = 0 hold due to symmetry.

In Section 3.5 we present the results of numerical experiments indicating that solutions of the ibvp's for the CB and SB systems with the boundary conditions (1.6) may blow up in finite time provided the initial conditions are suitable large, but still satisfy 1 + η0 > 0 on [0, L]. Thus, it appears that although the Cauchy problem for these systems is well posed in classes of functions that decay fast enough at infinity, solutions of the ibvp on a finite interval may develop singularities in finite time, something which is not precluded by the existing well-posedness theory [1]. Finally, in Section 3.6 we compare solutions of the ibvp's for the scaled CB and SB systems for appropriate initial data and small values of the parameter ɛ, as t grows, in order to test the consistency of the theory of [13], [2] (which is properly valid for the Cauchy problem) in the case of a finite interval under the b.c.'s (1.6).

Preliminary results of some numerical experiments included in this paper originally appeared in [4], [5].

Section snippets

Numerical schemes

For the convenience of the reader we rewrite here the ibvp's that will be solved numerically. In the case of the CB system, we seek η = η(x, t), u = u(x, t) for 0  x  L, t  0, such thatηt+ux+(ηu)x=0,ut+ηx+uux13uxxt=0,0xL,t0,η(x,0)=η0(x),u(x,0)=u0(x),0xL,u(0,t)=u(L,t)=0,t0,where η0, u0 are given real functions on [0, L] with u0(0) = u0(L) = 0. The analogous ibvp for the SB system isηt+ux+12(ηu)x=0,ut+ηx+32uux+12ηηx13uxxt=0,0xL,t0,η(x,0)=η0(x),u(x,0)=u0(x),0xL,u(0,t)=u(L,t)=0,t0.We shall

Numerical experiments

In this section we present a series of numerical experiments illustrating the behavior of solutions of the ‘classical’ Boussinesq system CB and its symmetric counterpart SB in some examples of interest. Unless otherwise stated, we solve throughout the ibvp's (2.1), (2.2) numerically, using the standard Galerkin method with cubic splines on a uniform mesh for the spatial discretization and the classical, four-stage, fourth-order RK scheme for the time-stepping. As was mentioned in Section 1,

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