×

zbMATH — the first resource for mathematics

Wave propagation algorithms for multidimensional hyperbolic systems. (English) Zbl 0872.76075
A class of high resolution multidimensional wave-propagation algorithms is described for general time-dependent hyperbolic systems. The methods are based on solving Riemann problems and applying limiter functions to the resulting waves, which are then propagated in a multidimensional manner. For nonlinear systems of conservation laws the methods are conservative and yield good shock resolution. Several examples are included describing gas dynamics, acoustics in a heterogeneous medium, and advection in a stratified flow on curvilinear grids. The software package CLAWPACK implements these algorithms in FORTRAN.

MSC:
76M25 Other numerical methods (fluid mechanics) (MSC2010)
76L05 Shock waves and blast waves in fluid mechanics
35L67 Shocks and singularities for hyperbolic equations
Software:
CLAWPACK; AMRCLAW
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] L. Adams, 1995, A multigrid algorithm for immersed interface problems, Seventh Copper Mountain Conference on Multigrid Methods, 1, National Aeronautic and Space Administrator, Washington, DC
[2] L. M. Adams, M. J. Berger, R. J. LeVeque, Z. Li, K. M. Shyue
[3] Bell, J.B; Dawson, C.N; Shubin, G.R, An unsplit, higher order Godunov method for scalar conservation laws in multiple dimensions, J. comput. phys., 74, 1, (1988) · Zbl 0684.65088
[4] M. Berger, 1982, Computer Science Department, Stanford Univeristy
[5] Berger, M, Adaptive finite difference methods in fluid dynamics, Von karman institute for fluid dynamics lecture series, (1987)
[6] M. Berger, R. J. LeVeque, 1990, Cartesian meshes and adaptive mesh refinement for hyperbolic partial differential equations, Proc. Third Int’l Conf. Hyperbolic Problems, Uppsala · Zbl 0825.76502
[7] Berger, M; LeVeque, R.J, Stable boundary conditions for Cartesian grid calculations, Comput. systems eng., 1, 305, (1990)
[8] Berger, M; Oliger, J, Adaptive mesh refinement for hyperbolic partial differential equations, J. comput. phys., 53, 484, (1984) · Zbl 0536.65071
[9] Berger, M.J, On conservation at grid interfaces, SIAM J. num. anal., 24, 967, (1987) · Zbl 0633.65086
[10] Berger, M.J; Colella, P, Local adaptive mesh refinement for shock hydrodynamics, J. comput. phys., 82, 64, (1989) · Zbl 0665.76070
[11] M. J. Berger, R. J. LeVeque, Adaptive mesh refinement for two-dimensional hyperbolic systems and the AMRCLAW software
[12] M. J. Berger, R. J. LeVeque, AMRCLAW software, http://www.amath.washington.edu/ rjl/amrclaw/
[13] Colella, P, Multidimensional upwind methods for hyperbolic conservation laws, J. comput. phys., 87, 171, (1990) · Zbl 0694.65041
[14] Courant, R; Friedrichs, K.O, Supersonic flow and shock waves, (1948), Springer-Verlag New York/Berlin · Zbl 0041.11302
[15] Deconinck, H; Hirsch, C; Peuteman, J, Characteristic decomposition methods for the multidimensional Euler equations, 10th int. conf. on num. meth. in fluid dyn., (1986), Springer-Verlag New York/Berlin, p. 216- · Zbl 0624.76088
[16] Deconinck, H; Struijs, R; Roe, P.L, Fluctuation splitting for multidimensional convection problems: an alternative to finite volume and finite element methods, VKI lecture series 1990-3, (1990), Von Karman Institute Brussels
[17] M. Fey, 1995, The Method of Transport for Solving the Euler Equations, ETH-Zürich
[18] Fey, M; Jeltsch, R; Morel, A.-T, Multidimensional schemes for nonlinear systems of hyperbolic conservation laws, () · Zbl 0843.65064
[19] M. Fey, A.-T. Morel, 1995, Multidimensional Method of Transport for the Shallow Water Equations, ETH-Zürich
[20] Goodman, J.B; LeVeque, R.J, A geometric approach to high resolution TVD schemes, SIAM J. num. anal., 25, 268, (1988) · Zbl 0645.65051
[21] Harten, A; Hyman, J.M, Self-adjusting grid methods for one-dimensional hyperbolic conservation laws, J. comput. phys., 50, 235, (1983) · Zbl 0565.65049
[22] C. Hirsch, C. Lacor, H. Deconinck, 1987, AIAA Paper 87-1163
[23] J. O. Langseth, R. J. LeVeque, 1995, Three-dimensional Euler computations using CLAWPACK, Conf. on Numer. Meth. for Euler and Navier-Stokes Eq. P. Arminjon, Montreal
[24] J. O. Langseth, R. J. LeVeque
[25] Lax, P.D, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, SIAM regional conference series in applied mathematics, 11, (1972), SIAM Philadelphia
[26] P. D. Lax, X. D. Liu, Solution of two dimensional Riemann problem of gas dynamics by positive schemes, SIAM J. Sci. Comput. · Zbl 0952.76060
[27] B. P. Leonard, M. K. MacVean, A. P. Lock, 1993, NASA Technical Memorandum 106055, ICOMP-93-05
[28] R. J. LeVeque, CLAWPACK software, http://www.amath.washington.edu/ rjl/clawpack.html
[29] R. J. LeVeque, CLAWPACK User Notes, http://www.amath.washington.edu/ rjl/clawpack.html
[30] LeVeque, R.J, High resolution finite volume methods on arbitrary grids via wave propagation, J. comput. phys., 78, 36, (1988) · Zbl 0649.65050
[31] R. J. LeVeque, 1990, Hyperbolic Conservation Laws and Numerical Methods · Zbl 0723.65067
[32] LeVeque, R.J, Numerical methods for conservation laws, (1990), Birkhäuser Basel · Zbl 0682.76053
[33] LeVeque, R.J, Simplified multi-dimensional flux limiter methods, (), 175 · Zbl 0801.76069
[34] R. J. LeVeque, 1994, {\scCLAWPACK}, Proceedings of the Fifth International Conference on Hyperbolic Problems: Theory, Numerics, Applications, J. Glimmet al. 188, World Scientific, Singapore
[35] LeVeque, R.J, High-resolution conservative algorithms for advection in incompressible flow, SIAM J. numer. anal., 33, 627, (1996) · Zbl 0852.76057
[36] LeVeque, R.J; Li, Z, Immersed interface methods for Stokes flow with elastic boundaries or surface tension, SIAM J. sci. comput., (1997) · Zbl 0879.76061
[37] LeVeque, R.J; Li, Z, The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. numer. anal., 31, 1019, (1994) · Zbl 0811.65083
[38] LeVeque, R.J; Zhang, C, Immersed interface methods for wave equations with discontinuous coefficients, Wave motion, (1997)
[39] Z. Li, 1994, The Immersed Interface Method—A Numerical Approach for Partial Differential Equations with Interfaces, University of Washington
[40] Liu, X.D; Lax, P.D, Positive schemes for solving multi-dimensional hyperbolic systems of conservation laws, Comput. fluid dynamics J., 5, 133, (July 1996)
[41] Y. B. Radvogin, 1991, Soviet Academy of Sciences, preprint
[42] Roe, P.L, Approximate Riemann solvers, parameter vectors, and difference schemes, J. comput. phys., 43, 357, (1981) · Zbl 0474.65066
[43] Roe, P.L, Fluctuations and signals—A framework for numerical evolution problems, (), 219 · Zbl 0569.76072
[44] Roe, P.L, Upwind scemes using various formulations of the Euler equations, (), 14
[45] Roe, P.L, Sonic flux formulae, SIAM J. sci. stat. comput., 13, 611, (1992) · Zbl 0747.65073
[46] J. Saltzman, 1987, Los Alamos Report LA-UR-87-2479
[47] Saltzmann, J, An unsplit 3-D upwind method for hyperbolic conservation laws, J. comput. phys., 115, 153, (1994)
[48] Schultz-Rinne, C.W, Classification of the Riemann problem for two-dimensional gas dynamics, SIAM J. sci. comput., 14, 1394, (1993) · Zbl 0785.76050
[49] Schultz-Rinne, C.W; Collins, J.P; Glaz, H.M, Numerical solution of the Riemann problem for two-dimensional gas dynamics, SIAM J. sci. comput., 14, 1394, (1993) · Zbl 0785.76050
[50] Sidilkover, D, A genuinely multidimensional upwind scheme for the compressible Euler equations, (), 447 · Zbl 0948.76581
[51] D. Sidilkover, 1994, ICASE Report No. 94-84, NASA Langley Research Center
[52] D. Sidilkover, June 19-22 1995, Multidimensional upwinding and multigrid, AIAA 95-175912th AIAA CFD meeting, 12th AIAA CFD meeting, San Diego
[53] Smolarkiewicz, P.K; Clark, T.L, The multidimensional positive definite advection transport algorithm: further development and applications, J. comput. phys., 67, 396, (1986) · Zbl 0601.76098
[54] Smolarkiewicz, P.K; Margolin, L.G, On forward-in-time differencing for fluids: extension to a curvilinear framework, Mon. weather rev., 121, 1847, (1993)
[55] Strang, G, On the construction and comparison of difference schemes, SIAM J. num. anal., 5, 506, (1968) · Zbl 0184.38503
[56] Strikwerda, J.C, Finite difference schemes and partial differential equations, (1989), Wadsworth & Brooks/Cole · Zbl 0681.65064
[57] R. Struijs, H. Deconinck, P. de, Palma, P. L. Roe, K. G. Powell, 1991, Progress on multidimensional upwind Euler solvers for unstructured grids, AIAA Conference on Computational Fluid Dynamics
[58] Sweby, P.K, High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. num. anal., 21, 995, (1984) · Zbl 0565.65048
[59] J. A. Trangenstein, 1993
[60] Leer, B.van, Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov’s method, J. comput. phys., 32, 101, (1979) · Zbl 1364.65223
[61] Zalesak, S.T, Fully multidimensional flux corrected transport algorithms for fluids, J. comput. phys., 31, 335, (1979) · Zbl 0416.76002
[62] Zalesak, S.T, A preliminary comparison of modern shock-capturing schemes: linear advection, (), 15
[63] C. Zhang, 1996, Immersed Interface Methods for Wave Equations, University of Washington
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.