Well-balanced finite volume evolution Galerkin methods for the shallow water equations

Journal of Computational Physics

Volumn 221, Issue 1, 2007, Pages 122-147

  Lukacova Medvid'ova M ^a   Noelle, S ^b   Kraft, M ^a  

^a HAMBURG UNIVERSITY OF TECHNOLOGY   (Germany)

^b RWTH AACHEN UNIVERSITY   (Germany)

Author keywords

Evolution Galerkin schemes; Geostrophic balance; Shallow water equations; Steady states; Systems of hyperbolic balance laws; Well balanced schemes

Indexed keywords

EQUATIONS OF MOTION; EQUATIONS OF STATE; GALERKIN METHODS; INTEGRAL EQUATIONS; WAVE PROPAGATION;

BALANCE LAW; EVOLUTION GALERKIN SCHEMES; FINITE-VOLUME; GEOSTROPHIC BALANCE; SHALLOW WATER EQUATIONS; STEADY JET; STEADY STATE; SYSTEM OF HYPERBOLIC BALANCE LAW; WELL BALANCED; WELL- BALANCED SCHEMES;

FINITE VOLUME METHOD;

EID: 33846137646     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2006.06.015     Document Type: Article

References (34)

1. Abgrall R., and Mezine M. Construction of second order accurate monotone and stable residual distribution schemes for unsteady flow problems. J. Comput. Phys. 188 1 (2003) 16-55
2. Audusse E., Bouchut F., Bristeau M.-O., Klein R., and Perthame B. A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comp. 25 (2004) 205-206
3. Billet S., and Toro E.F. On WAF-type schemes for multidimensional hyperbolic conservation laws. J. Comput. Phys. 130 (1997) 1-24
4. Botta N., Klein R., Langenberg S., and Lützenkirchen S. Well balanced finite volume methods for nearly hydrostatic flows. J. Comput. Phys. 196 (2004) 539-565
5. Bouchut F., LeSommer J., and Zeitlin V. Frontal geostrophic adjustment and nonlinear-wave phenomena in one dimensional rotating shallow water. Part 2: high-resolution numerical simulations. J. Fluid Mech. 514 (2004) 35-63
6. Colella P. Multidimensional upwind methods for hyperbolic conservation laws. J. Comput. Phys. 87 (1990) 171-200
7. Csík A., Ricchiuto M., and Deconinck H. A conservative formulation of the multidimensional upwind residual distribution schemes for general nonlinear conservation laws. J. Comput. Phys. 179 1 (2002) 286-312
8. Deconinck H., Roe P., and Struijs R. A multidimensional generalization of Roe's flux difference splitter for the Euler equations. Comput. Fluids 22 2-3 (1993) 215-222
9. Fey M. Multidimensional upwinding, Part II Decomposition of the Euler equations into advection equations. J. Comput. Phys. 143 (1998) 181-199
10. Gallouët T., Hérard J.-M., and Seguin N. Some approximate Godunov schemes to compute shallow-water equations with topography. Comput. Fluids 32 (2003) 479-513
11. Gjevik B., Moe H., and Ommundsen A. A high resolution tidal model for the area around The Lofoten Islands, northern Norway. Continental Shelf Res. 22 1 (2002) 485-504
12. Greenberg J.M., and LeRoux A.-Y. A well-balanced scheme for numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33 (1996) 1-16
13. 2AN, Math. Model. Numer. Anal. 35 4 (2001) 631-646
14. Klein R. An applied mathematical view of meteorological modelling, in: J.M. Hill, R. Moore (Eds.), Proceedings ICAM 2003, 2003, pp. 227-270.
15. Kröger T., and Lukáčová-Medvid'ová M. An evolution Galerkin scheme for the shallow water magneto-hydro-dynamic (SMHD) equations in two space dimensions. J. Comput. Phys. 206 (2005) 122-149
16. Kröger T., and Noelle S. On the connection between some Riemann-solver free approaches to the approximation of multi-dimensional systems of hyperbolic conservation laws. Math. Model. Numer. Anal. 38 (2004) 989-1009
17. 2AN, Math. Model. Numer. Anal. 36 3 (2002) 397-425
18. A. Kurganov, G. Petrova, B. Popov, Adaptive semi-discrete central-upwind schemes for nonconvex hyperbolic conservation laws, SIAM J. Sci. Comp. (accepted).
19. LeVeque R.J. Wave propagation algorithms for multi-dimensional hyperbolic systems. J. Comput. Phys. 131 (1997) 327-353
20. LeVeque R.J. Balancing source terms and flux gradients in high-resolution Godunov methods: The quasi-steady wave propagation algorithm. J. Comput. Phys. 146 (1998) 346-365
21. Lie K.-A., and Noelle S. On the artificial compression method for second order non-oscillatory central difference schemes for systems of conservation laws. SIAM J. Sci. Comp. 24 4 (2003) 1157-1174
22. M. Lukáčová-Medvid'ová, Multidimensional bicharacteristics finite volume methods for the shallow water equations, in: R. Hérbin, D. Kröner (Eds.), Finite Volumes for Complex Applications, Hermes, 2002, pp. 389-397.
23. Lukáčová-Medvid'ová M., Morton K.W., and Warnecke G. Evolution Galerkin methods for hyperbolic systems in two space dimensions. MathComp. 69 (2000) 1355-1384
24. M. Lukáčová-Medvid'ová, K.W. Morton, G. Warnecke, On high-resolution finite volume evolution Galerkin schemes for genuinely multidimensional hyperbolic conservation laws, in: Onate et al. (Eds.), European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2000, CIMNE 2000, pp. 1-14.
25. Lukáčová-Medvid'ová M., Morton K.W., and Warnecke G. Finite volume evolution Galerkin methods for Euler equations of gas dynamics. Int. J. Numer. Meth. Fluids 40 3-4 (2002) 425-434
26. Lukáčová-Medvid'ová M., Saibertová J., and Warnecke G. Finite volume evolution Galerkin methods for nonlinear hyperbolic systems. J. Comput. Phys. 183 (2002) 533-562
27. Lukáčová-Medvid'ová M., Morton K.W., and Warnecke G. Finite volume evolution Galerkin (FVEG) methods for hyperbolic problems. SIAM J. Sci. Comput. 26 1 (2004) 1-30
28. M. Lukáčová-Medvid'ová, G. Warnecke, Y. Zahaykah, On the stability of the evolution Galerkin schemes applied to a two-dimensional wave equation system, SIAM J. Numer. Anal. (accepted).
29. M. Lukáčová-Medvid'ová, G. Warnecke, Y. Zahaykah, Finite volume evolution Galerkin schemes for three-dimensional wave equation system (submitted).
30. Noelle S. The MOT-ICE: a new high-resolution wave-propagation algorithm for multi-dimensional systems of conservative laws based on Fey's method of transport. J. Comput. Phys. 164 (2000) 283-334
31. Noelle S., Pankratz N., Puppo G., and Natvig J. Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. J. Comput. Phys. 213 (2006) 474-499
32. Richtmyer R., and Morton K.W. Difference Methods for Initial Value Problems. second ed. (1967), Interscience, New York
33. Sweby P.K. High-resolution schemes using flux limiters for hyperbolic conservation-laws. SIAM J. Numer. Anal. 21 (1984) 995-1011
34. Xing Y., and Shu C.W. High order finite difference WENO schemes with the exact conservation property for the shallow water equations. J. Comput. Phys. 208 (2005) 206-227