Comparison of solvers for the generalized Riemann problem for hyperbolic systems with source terms

Journal of Computational Physics

Volumn 231, Issue 19, 2012, Pages 6472-6494

  Montecinos, G ^a   Castro, C E ^b   Dumbser, M ^a   Toro, E F ^a  

^a UNIVERSITY OF TRENTO   (Italy)

^b UNIVERSITY OF HAMBURG   (Germany)

Author keywords

ADER method; Cauchy Kowalewski procedure; Cauchy Kowalewski theorem; Discontinuous Galerkin methods; Euler equations; Hyperbolic equations; Riemann problem

Indexed keywords

COMPUTATION THEORY; COMPUTATIONAL FLUID DYNAMICS; EULER EQUATIONS; FINITE ELEMENT METHOD; GALERKIN METHODS; INITIAL VALUE PROBLEMS; LINEAR SYSTEMS; NUMERICAL METHODS; PARTIAL DIFFERENTIAL EQUATIONS;

ADER METHOD; CAUCHY-KOWALEWSKI PROCEDURE; CAUCHY-KOWALEWSKI THEOREM; DISCONTINOUS GALERKIN METHODS; EULER'S EQUATION; HYPERBOLIC EQUATIONS; HYPERBOLIC SYSTEMS WITH SOURCES; NON LINEAR SYSTEM; RIEMANN PROBLEM; SOURCE TERMS;

COMPUTATIONAL EFFICIENCY;

EID: 84863993566     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2012.06.011     Document Type: Article

References (41)

1. Harten A., Engquist B., Osher S., Chakravarthy S.R. Uniformly high order accuracy essentially non-oscillatory schemes III. Journal of Computational Physics 1987, 71:231-303.
2. Siviglia A., Toro E.F. The WAF method and splitting procedure for simulating hydro and thermal peaking waves in open channels flows. Journal of fluids engineering 2009, 135:651-662.
3. van Leer B. Towards the the ultimate conservation difference scheme V: a second order sequel to Godunov method. Journal of Computational Physics 1979, 32:101-136.
4. Castro C.E., Toro E.F. Solvers for the high-order Riemann problem for hyperbolic balance laws. Journal of Computational Physics 2008, 227:2481-2513.
5. Colella P. Multidimensional upwind methods for hyperbolic conservation laws. Journal of Computational Physics 1990, 87:171-200.
6. Dragos L. Mathematical Methods in Aerodynamics 2004, Springer, New York.
7. E.F. Toro, A Weighted Average Flux Method for Hyperbolic Conservation Laws. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 423(1865, 1989) 401-418.
8. Toro E.F. Riemann problems and the WAF method for solving the two dimensional shallow water equations. Philosophical transcription of the Royal Society of London Series A: Physical sciences and engineering 1992, 338:43-68.
9. E.F. Toro, Multidimensional WAF-type schemes for model conservation laws, Cranfield College of Aeronautics report, College of Aeronautics, Cranfield Institute of Technology, 1993.
10. Toro E.F. Direct Riemann solvers for the time dependent Euler equations Shock Waves 1995, 5:75-80.
11. E.F. Toro, Exact and approximate Riemann solvers for the artificial compressibility equations, Technical report MMU-970 Department of Mathematics and Physics, Manchester Metropolitan University, 1997.
12. Toro E.F. Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction 2009, Springer, Verlag. third ed.
13. Toro E.F., Titarev V.A. Solution of the generalised Riemann problem for advection-reaction equations. Proceeding of the Royal Society of London Series A Mathematical and Physical Sciences 2002, 458:271-281.
14. Toro E.F., Titarev V.A. Derivative Riemann solvers for systems of conservation laws and ADER methods. Journal of Computational Physics 2006, 212:150-165.
15. Toro E.F., Millington R.C., Nejad L.A.M. Towards very high-order Godunov schemes. Godunov Methods: Theory and Applications Edited Review 2001, 905-938. Kluwer Academic Plenum Publishers. E.F. Toro (Ed.).
16. Folland G. Introduction to Partial Differential Equations 1995, Princeton University Press, USA.
17. Jian G.S., Shu C.W. Efficient implementation of weighted ENO scheme. Journal of Computational Physics 1996, 126:202-212.
18. P. Garabedian, Partial Differential Equations, second ed., Chelsea, New York, 1986.
19. Qiu J., Dumbser M., Shu C.W. The Discontinuous Galerkin method with Lax-Wendroff type time discretizations. Computer Methods in Applied Mechanics and Engineering 2005, 194:4528-4543.
20. Ben-Artzi M., Falcovitz J. A second-order Godunov-type scheme for compressible fluid dynamics. Journal of Computational Physics 1984, 55:1-32.
21. Dumbser M., Munz C.D. Building blocks for arbitrary high order discontinuous Galerkin schemes. Journal of Scientific Computing 2006, 27:215-230.
22. Dumbser M., Kaser M. Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems. Journal of Computational Physics 2007, 221:693-723.
23. Dumbser M., Zanotti O. Very high orderPNPM schemes on unstructured meshes for the resistive relativistic MHD equations. Journal of Computational Physics 2009, 228:6991-7006.
24. Dumbser M., Enaux C., Toro E.F. Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. Journal of Computational Physics 2008, 227:3971-4001.
25. Dumbser M., Balsara D., Toro E.F., Munz C.D. A unified framework for the construction of one-step finite-volume and discontinuous Galerkin schemes. Journal of Computational Physics 2008, 227:8209-8253.
26. Dumbser M., Kaser M., Titarev V.A., Toro E.F. Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems. Journal of Computational Physics 2007, 226:204-243.
27. Men'Shov I.S. Increasing the order of approximation of Godunov's schemes using solution of the generalised Riemann problem. USSR J. Comput. Math. and Math. Phys. 1990, 30(5):54-65.
28. Courant R., Friedrichs K., Lewy H. On the partial difference equations of mathematical physics. IBM Journal 1970, 215-234.
29. Krantz S.G., Parks H.R. A Primer of Real Analytic Functions 2002, Birkhuser, Boston. second ed.
30. Godunov S.K. A finite difference method for the computation of discontinuous solutions of the equations of fluids dynamics. Mathematics Sbornik 1959, 47:357-393.
31. Godunov S.K. A difference scheme for numerical solution of discontinuous solution of hydrodynamic equations. Mathematics Sbornik 1959, 47:271-306. translated US Joint Publ. Res. Service, JPRS 7226, 1969.
32. S.K. Godunov, A.V. Zabrodin, M.Y. Ivanov, A.N. Kraiko, G.P. Prokopov, Numerical Solution of Multidimensional Problems in Gas Dynamics, Moscow, 1976.
33. S.K. Godunov, A.V. Zabrodin, M.Y. Ivanov, A.N. Kraiko, G.P. Prokopov, Resolution numerique des problemes multidimensionnels de la dynamique des gaz, Moscow, 1979.
34. Schwartzkopff T., Munz C.D., Toro E.F. ADER: High-order approach for linear hyperbolic systems in 2D. Journal of Scientific Computing 2002, 17(1-4):231-240.
35. E.F. Toro, TVD regions for the Weighted Average Flux (WAF) method as applied to a model hyperbolic conservation law, Cranfield College of Aeronautics report College of Aeronautics Cranfield Institute of Technology, 1989.
36. E.F. Toro, Approximate Riemann solvers for one dimensional magneto gasdynamics, Department of Aerospace Science, technical report College of Aeronautics, Cranfield Institute of Technology, 1992.
37. Treves F. An abstract non linear Cachy-Kowalewski theorem. American Mathematical Society 1970, 150:77-92.
38. Titarev V.A., Toro E.F. ADER: arbitrary high order Godunov approach. Journal of Scientific Computing 2002, 17:609-618.
39. Titarev V.A., Toro E.F. ADER schemes for three-dimensional non-linear hyperbolic systems. Journal of Computational Physics 2005, 204:715-736.
40. Titarev V.A., Toro E.F. Analysis of ADER and ADER-WAF schemes. IMA Journal of Numerical Analysis 2007, 27:616-630.
41. Titarev V.A., Romenski E., Toro E.F. MUSTA-type upwind fluxes for non-linear elasticity. International Journal of Numerical Methods in Engineering 2008, 73:897-926.