Performance of numerical methods on the non-unique solution to the Riemann problem for the shallow water equations

International Journal for Numerical Methods in Fluids

Volumn 47, Issue 8-9, 2005, Pages 825-831

  Andrianov, Nikolai ^a  

^a UNIVERSITÉ BORDEAUX 1   (France)

Author keywords

Non strictly hyperbolic non conservative system; Riemann problem; Shallow water equations; Test cases

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; PROBLEM SOLVING; WATER;

EXACT SOLUTION; RIEMANN PROBLEMS; SHALLOW WATER EQUATIONS;

NUMERICAL METHODS;

MATHEMATICAL METHOD; SHALLOW-WATER EQUATION;

EID: 15744377636     PISSN: 02712091     EISSN: None     Source Type: Journal    
DOI: 10.1002/fld.846     Document Type: Article

References (16)

1. Stoker JJ. Water Waves. Interscience: New York, 1957.
2. Smoller J. Shock Waves and Reaction-Diffusion Equations. Springer: New York, 1983.
3. Bouchut F. Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws, and Well-balanced Schemes for Sources. Frontiers in Math. Series. Birkhäuser: Basel, 2004.
4. Audusse E, Bouchut F, Bristeau MO, Klein R, Perthame B. A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM Journal on Scientific Computing 2004; 25 6):2050-2065.
5. Gallouët T, Hérard JM, Seguin N. Some approximate Godunov schemes to compute shallow-water equations with topography. Computers and Fluids 2003; 32(4):479-513.
6. Perthame B, Simeoni C. A kinetic scheme for the Saint-Venant system with a source term. Calcolo 2001; 38(4):201-231.
7. Toro EF. Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer: Berlin, 1999.
8. Andrianov N. CONSTRUCT: a collection of MATLAB routines for constructing the exact solution to the Riemann problem for the shallow water equations. Available at http://www-ian.math.uni-magdeburg.de/home/ andriano/CONSTRUCT.
9. Chinnayya A, LeRoux AY, Seguin N. A well-balanced numerical scheme for the approximation of the shallow-water equations with topography: the resonance phenomenon. Preprint, 2003. Available at http:// seguin.univ-tln.fr.
10. Goatin P, LeFloch PG. The Riemann problem for a class of resonant hyperbolic systems of balance laws. Preprint, 2003. Available at http:// www.math.ntnu.no/conservation/2003/008.html.
11. LeFloch PG, Thanh MD. The Riemann problem for fluid flows in a nozzle with discontinuous cross-section. Preprint, 2003. Available at http:// www.cmap.polytechnique.fr/~lefloch.
12. Isaacson E, Temple B. Nonlinear resonance in systems of conservation laws. SIAM Journal on Applied Mathematics 1992; 52:1260-1278.
13. Isaacson E, Temple B. Convergence of the 2 × 2 Godunov method for a general resonant nonlinear balance law. SIAM Journal on Applied Mathematics 1995; 55:625-640.
14. Andrianov N, Warnecke G. On the solution to the Riemann problem for the compressible duct flow. SIAM Journal on Applied Mathematics 2004; 64(3):878-901.
15. Andrianov N. Testing numerical schemes for the shallow water equations. Preprint, 2004. Available at http://www-ian.math.uni-magdeburg.de/home/ andriano/CONSTRUCT.
16. Tzavaras AE. Materials with internal variables and relaxation to conservation laws. Archive for Rational Mechanics and Analysis 1999; 146(2):129-155.