×

Stable and accurate hybrid finite volume methods based on pure convexity arguments for hyperbolic systems of conservation law. (English) Zbl 1039.35010

The author presents an approach for the numerical solution of hyperbolic systems of conservation laws, which avoids upwind processes and computation of derivatives of mean Jacobian matrices. The stability in this method is obtained using convexity considerations. In order to obtain accuracy, the author proposes a combination of the second-order Lax-Wendroff scheme and the first-order modified Lax-Friedrichs scheme. With a certain definition of “local dissipation by convexity” it is possible to overcome the difficulty of nonexistence of classical entropy-flux pairs for certain systems. The applicability of the method is demonstrated with a lot of demanding numerical simulations.

MSC:

35A35 Theoretical approximation in context of PDEs
35L60 First-order nonlinear hyperbolic equations
35L65 Hyperbolic conservation laws
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
76M12 Finite volume methods applied to problems in fluid mechanics

Software:

SHASTA; HLLE; CATHARE
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] F. Benkhaldoun, Analysis and validation of a new finite volume scheme for nonhomogeneous systems, in: R. Herbin, D. Kröner (Eds.), Proceedings of the Third International Symposium on Finite Volumes for Complex Applications FVCA 3, HPS, 2002, pp. 269-276; F. Benkhaldoun, Analysis and validation of a new finite volume scheme for nonhomogeneous systems, in: R. Herbin, D. Kröner (Eds.), Proceedings of the Third International Symposium on Finite Volumes for Complex Applications FVCA 3, HPS, 2002, pp. 269-276 · Zbl 1177.76208
[2] N. Andrianov, R. Saurel, G. Warnecke, A simple method for compressible multiphase mixtures and interfaces, INRIA Technical Report RR-4247, 2001. Available from <ftp://ftp.inria.fr/INRIA/publication/publi-pdf/RR/RR-4247.pdf; N. Andrianov, R. Saurel, G. Warnecke, A simple method for compressible multiphase mixtures and interfaces, INRIA Technical Report RR-4247, 2001. Available from <ftp://ftp.inria.fr/INRIA/publication/publi-pdf/RR/RR-4247.pdf
[3] M. Baudin, C. Berthon, F. Coquel, Ph. Hoche, R. Masson, Q.H. Tran, A relaxation method for two-phase flow models with hydrodynamic closure law (to appear); M. Baudin, C. Berthon, F. Coquel, Ph. Hoche, R. Masson, Q.H. Tran, A relaxation method for two-phase flow models with hydrodynamic closure law (to appear) · Zbl 1204.76025
[4] Boris, J. P.; Book, D. L., Flux corrected transport I, SHASTA, a fluid transport algorithm that works, J. Comp. Phys., 11, 38-89 (1973) · Zbl 0251.76004
[5] Bestion, D., The physical closure laws in the CATHARE code, Nucl. Eng. Des., 124, 229-245 (1990)
[6] F. Bouchut, An antidiffusive entropy scheme for monotone scalar conservation laws (2002). Available from <http://www.math.ntnu.no/conservation/2002/051.html; F. Bouchut, An antidiffusive entropy scheme for monotone scalar conservation laws (2002). Available from <http://www.math.ntnu.no/conservation/2002/051.html
[7] Cooke, C. H.; Chen, T. J., On shock capturing for pure water with general equation of state, Commun. Appl. Numer. Methods, 8, 219-233 (1992) · Zbl 0756.76049
[8] Collela, P., A direct Eulerian MUSCL scheme for gas dynamics, SIAM J. Sci. Stat. Comput., 6, 104-117 (1985)
[9] F. Coquel, M.S. Liou, Hybrid upwind splitting (HUS) by a field-by-field decomposition, NASA TM-106843, Icomp-95-2, 1995; F. Coquel, M.S. Liou, Hybrid upwind splitting (HUS) by a field-by-field decomposition, NASA TM-106843, Icomp-95-2, 1995
[10] Cortes, J., On the construction of upwind schemes for nonequilibrium transient two-phase flows, Comput. Fluids, 159-182 (2002) · Zbl 1007.76046
[11] P.H. Cournede, B. Desjardins, A. Llor, A TVD Lagrange plus remap scheme for the simulation of two-fluid flows, in: R. Herbin, D. Kröner (Eds.), Proceedings of the Third International Symposium on Finite Volumes for Complex Applications FVCA 3, HPS, 2002, pp. 495-502; P.H. Cournede, B. Desjardins, A. Llor, A TVD Lagrange plus remap scheme for the simulation of two-fluid flows, in: R. Herbin, D. Kröner (Eds.), Proceedings of the Third International Symposium on Finite Volumes for Complex Applications FVCA 3, HPS, 2002, pp. 495-502 · Zbl 1177.76217
[12] Coquel, F.; Perthame, B., Relaxation of energy and approximate Riemann solvers for general pressure laws in fluid dynamics, SIAM J. Numer. Anal., 35, 6, 2223-2249 (1998) · Zbl 0960.76051
[13] F. De Vuyst, J.M. Ghidaglia, G. Le Coq, On the numerical simulation of multiphase water flows with changes of phase and strong gradients using the homogeneous equilibrium model, Int. J. Mult. Flows (submitted). Available from <http://www.math.ntnu.no/conservation/2002/057.html; F. De Vuyst, J.M. Ghidaglia, G. Le Coq, On the numerical simulation of multiphase water flows with changes of phase and strong gradients using the homogeneous equilibrium model, Int. J. Mult. Flows (submitted). Available from <http://www.math.ntnu.no/conservation/2002/057.html
[14] Godunov, S. K., A difference scheme for numerical computation of discontinuous solutions of equation of fluid dynamics, Mat. Sbornik, 47, 89, 271-306 (1959) · Zbl 0171.46204
[15] K. Halaoua, Quelques solveurs pour les opérateurs de convection et leurs applications à la mécanique des fludies diphasiques, Ph.D. Thesis and publication du CMLA, ENS de Cachan, France, 1998; K. Halaoua, Quelques solveurs pour les opérateurs de convection et leurs applications à la mécanique des fludies diphasiques, Ph.D. Thesis and publication du CMLA, ENS de Cachan, France, 1998
[16] Harten, A.; Lax, P. D., A random choice finite difference scheme for hyperbolic conservation laws, SIAM J. Numer. Anal., 18, 289-315 (1981) · Zbl 0467.65038
[17] Harten, A.; Lax, P. D.; van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25, 35-61 (1983) · Zbl 0565.65051
[18] Hou, T.; Le Floch, Ph., Why nonconservative scheme do not converge to weak solutions, Math. Comp., 62, 206, 497-530 (1994) · Zbl 0809.65102
[19] Harten, A.; Zwas, G., Seft-adjusting hybrid schemes for shock computation, J. Comp. Phys., 9, 568 (1972) · Zbl 0244.76033
[20] Gallouët, T.; Hérard, J. M.; Seguin, N., Some recent finite volume schemes to compute Euler equations using real gas EOS, Int. J. Numer. Meth. Fluids, 39-12, 1073-1138 (2002) · Zbl 1053.76044
[21] Ghidaglia, J. M.; Kumbaro, A.; Le Coq, G., On the numerical solution of two fluid models via a cell centered finite volume method, Eur. J. Mech. B, 20, 6, 841-867 (2001) · Zbl 1059.76041
[22] In, A., Numerical evaluation of an energy relaxation method for inviscid real fluids, SIAM J. Sci. Comput., 21, 1, 340-365 (1999) · Zbl 0953.76066
[23] Ivings, M. J.; Causon, D. M.; Toro, E. F., Riemann solvers for compressible water, Proceedings of the ECCOMAS (1996), Wiley: Wiley New York · Zbl 0918.76047
[24] Jin, S.; Xin, Z., The relaxation scheme for systems of conservation laws in arbitrary space dimension, Commun. Pure Appl. Math., 45, 235-276 (1995) · Zbl 0826.65078
[25] Lax, P. D., Shock waves and entropy, (Zarantonello, E. A., Contributions to Nonlinear Functional Analysis (1971), Academic Press: Academic Press New York), 603-634
[26] Lax, P. D.; Wendroff, B., Systems of conservation laws, Commun. Pure Appl. Math., 13, 217-237 (1960) · Zbl 0152.44802
[27] R. Liska, B. Wendroff, Comparison of several difference schemes on 1D and 2D test problems for the Euler equations, SISC SIAM J. Sci. Comput. (submitted); R. Liska, B. Wendroff, Comparison of several difference schemes on 1D and 2D test problems for the Euler equations, SISC SIAM J. Sci. Comput. (submitted) · Zbl 1096.65089
[28] Liska, R.; Wendroff, B., Composite schemes for conservation laws, SIAM J. Numer. Anal., 35, 6, 2250-2271 (1998) · Zbl 0920.65054
[29] R.W. MacCormack, The effect of viscosity in hypervelocity impact cratering, AIAA Paper 69-354, 1969; R.W. MacCormack, The effect of viscosity in hypervelocity impact cratering, AIAA Paper 69-354, 1969
[30] Majda, A.; Osher, S., A systematic approach for correcting nonlinear instabilities, Numer. Math., 30, 429-452 (1978) · Zbl 0368.65048
[31] Ransom, V. H., Numerical tenchmark tests, (Hewitt, G. F.; Delhaye, J. M.; Zuber, N., Multiphase Science and Technology, 3 (1987), Hemisphere Publishing Corporation: Hemisphere Publishing Corporation Washington, DC)
[32] Richtmyer, R. D.; Morton, K. W., Difference Methods for Initial-value Problems (1967), Wiley-Interscience: Wiley-Interscience New York · Zbl 0155.47502
[33] Roe, P. L., Approximate Riemann solvers, parameter vectors and difference schemes, J. Comp. Phys., 43, 357-372 (1981) · Zbl 0474.65066
[34] Rusanov, V. V., Calculation of interaction of non-steady shock waves with obstacles, J. Comp. Math. Phys. USSR, 1, 267-279 (1961)
[35] Sod, G., A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys., 27, 1-31 (1978) · Zbl 0387.76063
[36] Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comp. Phys., 54, 115-173 (1984) · Zbl 0573.76057
[37] Zalesak, S. T., Fully multidimensional flux corrected transport algorithms for fluids, J. Comp. Phys., 31, 335-362 (1979) · Zbl 0416.76002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.