A general approximate-state Riemann solver for hyperbolic systems of conservation laws with source terms

International Journal for Numerical Methods in Fluids

Volumn 53, Issue 9, 2007, Pages 1509-1540

  Lhomme, Julien ^a   Guinot, Vincent ^a  

^a UNIVERSITÉ DE MONTPELLIER   (France)

Author keywords

Godunov type schemes; Riemann solvers; Shallow water equations; Source terms; Water hammer equations

Indexed keywords

APPROXIMATION THEORY; COMPUTATIONAL EFFICIENCY; COMPUTATIONAL GEOMETRY; STEADY FLOW; VECTORS; WATER HAMMER;

CONSERVATION LAW; FLUX VECTOR; HYPERBOLIC SYSTEMS; RIEMANN SOLVER; SHALLOW WATER EQUATIONS; SOURCE TERM; WATER HAMMER EQUATIONS;

COMPUTATIONAL FLUID DYNAMICS;

APPROXIMATION THEORY; COMPUTATIONAL EFFICIENCY; COMPUTATIONAL FLUID DYNAMICS; COMPUTATIONAL GEOMETRY; STEADY FLOW; VECTORS; WATER HAMMER;

EID: 33947207493     PISSN: 02712091     EISSN: 10970363     Source Type: Journal    
DOI: 10.1002/fld.1374     Document Type: Article

References (29)

1. Godunov SK. A difference method for numerical calculation of discontinuous equations of hydrodynamic equations (in Russian). Matematicheskii Sbornik 1959; 47:271-300.
2. Toro EF. Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer: Berlin, 1997.
3. Guinot V. Godunov-type Schemes, An Introduction for Engineers. Elsevier: Amsterdam, 2003.
4. LeVeque RJ. Finite Volume Methods for Hyperbolic Problems. Cambridge University Press: Cambridge, MA, 2002.
5. Roe PL. Approximate Riemann solvers, parameter vectors and difference scheme. Journal of Computational Physics 1981; 43:357-372.
6. Toro EF. A linearised Riemann solver for the time-dependent Euler equations of gas dynamics. Proceedings of the Royal Society London, Series A 1991; 434:683-693.
7. Harten A, Lax PD, Van Leer B. On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. Journal of Computational Physics 1983; 50:235-269.
8. Davis SF. Simplified second-order Godunov-type methods. SIAM Journal on Scientific and Statistical Computing 1988; 9:445-473.
9. Einfeldt B. On Godunov-type methods for gas dynamics. SIAM Journal on Numerical Analysis 1988; 25(2):294-318.
10. Toro EF, Spruce M, Speares W. Restoration of the contact surface in the HLL-Riemann solver. Shock Waves 1994; 4:25-34.
11. Osher S, Solomon F. Upwind difference schemes for hyperbolic conservation laws. Mathematics of Computation 1982; 38:339-374.
12. Linde T. A practical, general-purpose, two-state HLL Riemann solver for hyperbolic conservation laws. International Journal for Numerical Methods in Fluids 2002; 40:391-402.
13. Dukowicz JK. A general, non-iterative Riemann solver for Godunov's method. Journal of Computational Physics 1985; 61:119-137.
14. Guinot V. Riemann solvers for water hammer simulations by Godunov method. International Journal for Numerical Methods in Engineering 2000; 49:851-870.
15. Lax PD. Hyperbolic systems of conservation laws. Communications on Pure and Applied Mathematics 1957; 10:537-566.
16. Erduran KS, Kutija V, Hewett JM. Performance of finite volume solutions to the shallow water equations with shock-capturing schemes. International Journal for Numerical Methods in Fluids 2002; 40:1237-1273.
17. LeVeque RJ. Balancing source terms and flux gradients in high-resolution Godunov methods. Journal of Computational Physics 1998; 146:346-365.
18. Bermudez A, Vazquez ME. Upwind methods for hyperbolic conservation laws with source terms. Computers and Fluids 1994; 8:1049-1071.
19. Greenberg JM, Le Roux AY. A well balanced scheme for the numerical processing of source terms in hyperbolic equation. SIAM Journal on Numerical Analysis 1996; 33(1):1-16.
20. Alcrudo F, Benkhaldoun F. Exact solutions to the Riemann problem of the shallow water equations with a bottom step. Computers and Fluids 2001; 30:643-671.
21. Gallouët T, Hérard JM, Seguin N. Some approximate Godunov schemes to compute shallow-water equations with topography. Computers and Fluids 2003; 32:479-513.
22. Garcia-Navarro P, Vazquez-Cendon ME. On numerical treatment of the source terms in the shallow water equations. Computers and Fluids 2000; 29:951-979.
23. Nujic M. Efficient implementation of non-oscillatory schemes for the computation of free-surface flows. Journal of Hydraulic Research 1995; 33:101-111.
24. Colella P. Glimm's method for gas dynamics. SIAM Journal on Scientific and Statistical Computing 1982; 3:76-110.
25. Toro EF. Direct Riemann solvers for the time-dependent Euler equations. Shock Waves 1995; 5:75-80.
26. Ivings MJ, Causon DM, Toro EF. On Riemann solvers for compressible liquids. International Journal for Numerical Methods in Fluids 1998; 28:395-418.
27. Guinot V. Numerical simulation of two-phase flow in pipes using Godunov method. International Journal for Numerical Methods in Engineering 2001; 50:1169-1189.
28. Jeffrey A, Taniuti T. Non-Linear Wave Propagation. Academic Press: New York, 1964.
29. Stoker JJR. Water Waves. Interscience: New York, 1957.