A spectral collocation method to solve Helmholtz problems with boundary conditions involving mixed tangential and normal derivatives

doi:10.1016/j.jcp.2004.03.004

Journal of Computational Physics

Volume 200, Issue 1, 10 October 2004, Pages 34-49

https://doi.org/10.1016/j.jcp.2004.03.004 Get rights and content

Abstract

The performing spectral method, developed by Haldenwang et al. [J. Comput. Phys. 55 (1984) 115], to solve multi-dimensional Helmholtz equations, associated to mixed boundary conditions with constant coefficients, is extended to boundary conditions mixing a first order normal derivative with a second order tangential derivative. The accuracy of the proposed algorithm is evaluated on two test cases for which analytical solutions exist: an academic problem and a physical configuration including an interface with shear viscosity. The procedure is also applied to the research of the Rayleigh–Bénard instability thresholds in closed cavities with thin diffusive walls.

Introduction

The Navier–Stokes and energy equations associated with boundary conditions mixing a normal derivative to a second order tangential derivative are encountered in many physical systems such as fluids confined within walls of arbitrary thermal conductivity [5] or in flows involving an interface with shear viscosity [10]. This type of condition arises whenever a boundary is subjected to a coupling between flux and intrinsic interfacial dissipation of some physical quantity.

Conservation equations are often transformed into Helmholtz problems which can be solved using a direct Chebyshev collocation method as developed by Dang-Vu and Delcarte [8]. This method leads to the resolution of quasi-tridiagonal systems. Though avoiding any iterative algorithm, and offering a very accurate solution, this procedure is slower than the one proposed by Haidvogel and Zang [4] in the case of time dependent problems for which a Helmholtz equation has to be solved at each time step. Haldenwang et al. [6] have proposed a performing 3D spectral solver for the Helmholtz equation in case of Dirichlet, Neumann or Robin (mixed) boundary conditions with constant coefficients. This method, based on successive diagonalisations of the second order derivatives of the Helmholtz operator, allows to reduce efficiently the computing time and is relevant to time dependent problems. In this paper, we propose an extension of this method to boundary conditions mixing a normal derivative to a second order tangential derivative.

The general formulation of the method is presented in Section 2, the corresponding algorithm is detailed in Section 3, and validated in Section 4 through comparisons with analytical solutions of mathematical and physical configurations. Results of the literature on the thermal stability of a fluid contained in a rectangular enclosure heated from below are reproduced and extended. The CPU cost of the proposed algorithm is estimated on this time dependent problem. Section 5 presents the conclusions.

Section snippets

Description of the problem and method of solution

We consider the 2D Helmholtz problem in Cartesian coordinates $∂^{2} u ∂ x^{2} + ∂^{2} u ∂ z^{2} −au=f, x,z,∈[−1,1], a>0,$ where u and f are, respectively, the solution and a source term, x and z being the horizontal and vertical directions. The following boundary conditions are imposed: $u∈ ∂ Ω α_{1±} ∂^{2} u ∂ z^{2} +β_{1±} ∂ u ∂ x =γ_{1±} (z) at x=±1 (a) α_{2±} u=γ_{2±} (x) at z=±1 (b).$ α_1±, α_2± and β_1± are constant (Fig. 1).

For clarity's sake, the method is developed in case of Dirichlet conditions along one set of parallel boundaries with γ_2± depending on the

The discrete problem

We use a Chebyshev collocation spectral method on a Gauss–Lobatto grid [2], [3]. The collocation points along $e →_{x}$ and $e →_{z}$ are given by $x_{i} =− cos i π N_{x}, i∈{0,…,N_{x}},$ $z_{j} =− cos j π N_{z}, j∈{0,…,N_{z}}.$ The discrete Helmholtz problem reads $∑^{N_{x}}_{k=0} (D_{x}^{2})_{ik} u_{kj} +∑^{N_{z}}_{l=0} (D_{z}^{2})_{jl} u_{il} −au_{ij} =f_{ij}, i∈{0,…,N_{x}} and j∈{0,…,N_{z}},$ associated with the set of boundary conditions: $u∈ ∂ Ω α_{1±} ∑^{N_{z}}_{l=0} (D_{z}^{2})_{jl} u_{|_{0}^{N_{x}}l} +β_{1±} ∑^{N_{x}}_{k=0} (D_{x})_{|_{0}^{N_{x}}k} u_{kj} =γ_{1±} (j) for j∈{0,…,N_{z}} (a) α_{2±} u_{i|_{0}^{N_{z}}} =γ_{2±} (i) for i∈{0,…,N_{x}} (b)$ with γ_1±(j)=γ_1±(z_j) and γ_2±(i)=γ_2±(x_i). $D_{_{z}^{x}}$ denotes the first

An analytical test case

The algorithm is now applied to the Helmholtz problem , where the source, f, and the coefficients, α_1±, β_1±, γ_1±(z), α_2±, β_2±, γ_2±(x), are given by $f=−(2 π^{2} +a)(sin (π x) sin (π z)+ cos (π x) cos (π z))−a C,$ $α_{1±} =− 12,$ $β_{1±} =±1,$ $γ_{1±} (z)=− π^{2} 2 cos (π z)∓ π sin (π z),$ $α_{2±} =1,$ $γ_{2±} (x)=− cos (π x)+C.$ Of course, the chosen coefficients insure the compatibility of the boundary conditions in the corners of the domain. The analytical solution is the trigonometric function of x and z: $u= sin (π x) sin (π z)+ cos (π x) cos (π z)+C.$ This function does not

Conclusion

We have extended the successive diagonalisation technique solving the Helmholtz problem with homogeneous Robin boundary conditions to the case of boundary conditions mixing a second order tangential derivative with a first order normal derivative. The algorithm does not increase the computation time but requires more memory storage. The spectral accuracy is preserved. The application of the procedure to the determination of the transition thresholds between the conductive and convective regimes

Acknowledgements

The authors thank the CRI (Centre de Ressources Informatiques de l'Université d'Orsay) for the use of its computing facilities and the support they received.

References (13)

D.B. Haidvogel et al.
The accurate solution of Poisson's equation by expansion in Chebyshev polynomials
J. Comput. Phys
(1979)
P. Haldenwang et al.
Chebyshev 3-D spectral and 2-D pseudospectral solvers for the Helmholtz equation
J. Comput. Phys
(1984)
H. Dang-Vu et al.
An accurate solution of the Poisson equation by the Chebyshev collocation method
J. Comput. Phys
(1993)
V.C. Regnier et al.
Numerical simulations of interface viscosity effects on thermoconvective motion in two-dimensional rectangular boxes
Int. J. Heat Mass Transfer
(1995)
S.H. Davis
Convection in a box
J. Fluid Mech
(1967)
C. Canuto et al.
Spectral Methods in Fluids Dynamics
(1977)

There are more references available in the full text version of this article.

Cited by (16)

Jacobi-Gauss-Lobatto collocation method for solving nonlinear reaction-diffusion equations subject to Dirichlet boundary conditions
2016, Applied Mathematical Modelling
Citation Excerpt :
The processes of diffusion and reaction play an essential role in the dynamics of many systems, e.g., in plasma [2], or semiconductor physics [3]. Spectral methods (see, e.g., [4,5] and the references therein) are techniques used in applied mathematics and scientific computing to numerically solve both linear and nonlinear differential equations. The Galerkin, tau and collocation methods are three well-known types of spectral methods.
This paper extends the application of the spectral Jacobi–Gauss–Lobatto collocation (J-GL-C) method based on Gauss–Lobatto nodes to obtain semi-analytical solutions of nonlinear time-dependent reaction–diffusion equations (RDEs) subject to Dirichlet boundary conditions. This approach has the advantage of allowing us to obtain the solution in terms of the Jacobi parameters α and β, which therefore means that the method holds a number of collocation methods as a special case. In addition, the problem is reduced to the solution of system of ordinary differential equations (SODEs) in the time variable, which may then be solved by any standard numerical technique. We consider five applications of the general method to concrete examples. In each of the examples considered, the numerical results show that the proposed method is of high accuracy and is efficient for solving nonlinear time-dependent RDEs.
On the regularity of some thermocapillary convection models
2008, European Journal of Mechanics, B/Fluids
The classical model of confined thermocapillary convection is analyzed. Its vorticity singularity, independent of the contact angle, leads to infinite pressure values at contact lines, forbidding any numerical use of the Laplace equation to calculate free surface shapes. Four models are explored to overcome this difficulty: an explicit polynomial filtering, a Navier slip at the solid boundaries, an interface viscosity model and the combination of slip and interface viscosity. Regular solutions are obtained with the first and last approaches. Only the last one is based on physical considerations and, by the introduction of physical length scales, avoids infinite pressure values at the contact line.
The Trefftz method for the Helmholtz equation with degeneracy
2008, Applied Numerical Mathematics
The Trefftz method (TM) [E. Trefftz, Ein Gegenstuck zum Ritz'schen Verfahren, in: Proc. 2nd Ind. Congr. Appl. Mech., Zurich, 1926, pp. 131–137] (i.e., Boundary approximation method) is developed to solve the Helmholtz equation, $Δ u + k^{2} u = 0$ , where $k^{2}$ is not exactly equal (but may be very close) to an eigenvalue of the operator −Δ. Piecewise particular solutions are chosen and then matched together in order to satisfy the exterior and interior boundary conditions. Error analysis is presented to estimate error bounds for the Helmholtz solutions in the entire solution domain. Let δ be the smallest relative distance between $k^{2}$ and the eigenvalues of −Δ. We prove that the error asymptote of the solutions by the TM is $O (\frac{1}{δ})$ as $δ \to 0$ , which is called degeneracy in this paper. Such an asymptote $O (\frac{1}{δ})$ has been verified by the numerical computations. We also explain why the exponential convergence rates of solutions can be obtained easily by splitting the solution domain into smaller subdomains. Numerical experiments of both smooth and singular solutions are provided, to support the TM algorithms and the error analysis made.
About the ellipticity of the Chebyshev-Gauss-Radau discrete Laplacian with Neumann condition
2010, Journal of Computational Physics
The Chebyshev–Gauss–Radau discrete version of the polar-diffusion operator, $(\frac{1}{r} \frac{\partial}{\partial r} (r \frac{\partial}{\partial r}) - \frac{k^{2}}{r^{2}}), k$ being the azimuthal wave number, presents complex conjugate eigenvalues, with negative real parts, when it is associated with a Neumann boundary condition imposed at r = 1. It is shown that this ellipticity marginal violation of the original continuous problem is genuine and not due to some round-off error amplification. This situation, which does not lead per se to any particular computational difficulty, is taken here as an opportunity to numerically check the sensitivity of the quoted ellipticity to slight changes in the mesh. A particular mapping is chosen for that purpose. The impact of this option on the ellipticity and on the numerical accuracy of a computed flow is evaluated.
Numerical Solution of Helmholtz Equation using the Galerkin and Collocation Methods
2024, AIP Conference Proceedings
Numerical simulation of fractional-order two-dimensional Helmholtz equations
2023, AIMS Mathematics

View all citing articles on Scopus

View full text

A spectral collocation method to solve Helmholtz problems with boundary conditions involving mixed tangential and normal derivatives

Abstract

Introduction

Section snippets

Description of the problem and method of solution

The discrete problem

An analytical test case

Conclusion

Acknowledgements

J. Comput. Phys

J. Comput. Phys

J. Comput. Phys

Int. J. Heat Mass Transfer

Convection in a box

J. Fluid Mech

Spectral Methods in Fluids Dynamics