Well-balancing issues related to the RKDG2 scheme for the shallow water equations

International Journal for Numerical Methods in Fluids

Volumn 62, Issue 4, 2010, Pages 428-448

  Kesserwani, Georges ^a,b   Liang, Qiuhua ^a   Vazquez, José ^b   Mosé, Robert ^b  

^a NEWCASTLE UNIVERSITY   (United Kingdom)

^b UNIVERSITÉ DE STRASBOURG   (France)

Author keywords

C property; Discontinuous galerkin finite element method; Godunov type scheme; Irregular topography; Local linear approximations; Saint venant equations; Slope limiting; Source terms

Indexed keywords

DISCONTINUOUS GALERKIN FINITE-ELEMENT METHOD; GODUNOV-TYPE SCHEMES; IRREGULAR TOPOGRAPHY; LOCAL LINEAR APPROXIMATION; SAINT VENANT EQUATION; SOURCE TERMS;

COMPUTATIONAL FLUID DYNAMICS; FLOW OF WATER; FLOW SIMULATION; GALERKIN METHODS; HYDRAULICS; RUNGE KUTTA METHODS; SURFACE TOPOGRAPHY; THREE TERM CONTROL SYSTEMS;

FINITE ELEMENT METHOD;

EID: 77950207731     PISSN: 02712091     EISSN: 10970363     Source Type: Journal    
DOI: 10.1002/fld.2027     Document Type: Article

References (40)

1. Guinot V. Godunov-type Schemes: An Introduction for Engineers. Elsevier: Amsterdam, 2003.
2. Hirsch C. Numerical Computation of Internal and External Flows, Computational Methods for Inviscid and Viscous Flows. Wiley: Chichester, England, New York, 1990.
3. Liang Q, Borthwick AGL. Adaptive quadtree simulation of shallow flows with wet-dry fronts over complex topography. Computers and Fluids 2009; 38(2):221-234.
4. Schwanenberg D. Die Runge-Kutta-discontinuous-Galerkin-Methode zur Lösung konvektionsdominierter tiefengemittelter Flachwasserprobleme. Ph.D. Thesis, Fakultät für Bauingenieurwesen, 2003.
5. Toro EF. Shock-capturing Methods for Free-surface Shallow Flows. Wiley: New York, 2001.
6. Toro EF, García-Navarro P. Godunov-type methods for free-surface shallow flows: a review. Journal of Hydraulic Research 2007; 45(6):737-751.
7. Lhomme J, Guinot V. A general approximate-state Riemann solver for hyperbolic systems of conservation laws with source terms. International Journal for Numerical Methods in Fluids 2007; 55(9):1509-1540.
8. Kesserwani G, Mosé R, Vazquez J, Ghenaim A. A practical implementation of high-order RKDG models for the 1D open-channel flow equations. International Journal for Numerical Methods in Fluids 2009; DOI: 10.1002/fld.1879.
9. Shu CW, Osher S. Efficient implementation of essentially non-oscillatory shock capturing schemes. Journal of Computational Physics 1988; 77(2):439-471.
10. Cockburn B, Shu CW. Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. Journal of Scientific Computing 2001; 16(3):173-261.
11. Cockburn B, Lin S, Shu CW. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one dimensional systems. Journal of Computational Physics 1989; 84(1):90-113.
12. Cockburn B, Shu CW. The Runge-Kutta local projection P1-discontinuous Galerkin method for scalar conservation laws. ESAIM: Mathematical Modelling and Numerical Analysis 1991; 25(3):337-361.
13. Cockburn B, Shu CW. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws II: general framework. Mathematics of Computation 1989; 52(186):411-435.
14. Cockburn B. Discontinuous Galerkin methods. Zeitschrift für Angewandte Mathematik und Mechanik 2003; 83(11):731-754.
15. Kurganov A, Petrova G. A second-order well-balanced positivity preserving central-upwind scheme for the Saint-Venant system. Communications in Mathematical Sciences 2007; 5(1):133-160.
16. LeVeque RJ. Balancing source terms and flux gradients in high resolution Godunov methods. Journal of Computational Physics 1998; 146(1):346-365.
17. Mohammadian A, Le Roux DY. Simulation of shallow flows over variable topographies using unstructured grids. International Journal for Numerical Methods in Fluids 2006; 52(5):473-498.
18. Noelle S, Pankratz N, Puppo G, Natvig JR. Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. Journal of Computational Physics 2006; 213(2):474-499.
19. Vukovic S, Sopta L. ENO and WENO schemes with the exact conservation property for one-dimensional shallow water equations. Journal of Computational Physics 2002; 179(2):593-621.
20. Roe PL. Upwind differencing schemes for hyperbolic conservation laws with source terms. In Proceedings of Nonlinear Hyperbolic Problems, Carasso C, Raviart P, Serre D (eds). Lecture Notes in Mathematics, vol. 1270. Springer: Berlin, 1986; 41-51.
21. Bermúdez A, Vázquez-Cendón ME. Upwind methods for hyperbolic conservation laws with source terms. Computers and Fluids 1994; 23(8):1049-1071.
22. Črnjarić-Žic N, Vuković S, Sopta L. Balanced finite volume WENO and central WENO schemes for the shallow water and the open-channel flow equations. Journal of Computational Physics 2004; 200(2):512-548.
23. Crossley AJ, Wright NG. Time accurate local time stepping for the unsteady shallow water equations. International Journal for Numerical Methods in Fluids 2005; 48(7):775-799.
24. García-Navarro P, Vázquez-Cendón ME. On numerical treatment of the source terms in shallow water equations. Computers and Fluids 2000; 29(8):951-979.
25. Vázquez-Cendón ME. Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry. Journal of Computational Physics 1999; 148(2):497-526.
26. Greenberg JM, LeRoux AY. A well-balanced scheme for numerical processing of source terms in hyperbolic equations. SIAM Journal on Numerical Analysis 1996; 33(1):1-16.
27. 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.
28. Aureli F, Maranzoni A, Mignosa P, Ziveri C. A weighted surface-depth gradient method for the numerical integration of the 2D shallow water equations with topography. Advances in Water Resources 2008; 31(7):962-974.
29. Hubbard ME, García-Navarro P. Flux difference splitting and the balancing of source terms and flux gradients. Journal of Computational Physics 2000; 65(1):89-125.
30. Zhou JG, Causon DM, Ingram DM, Mingham CG. Numerical solutions of the shallow water equations with discontinuous bed topography. International Journal for Numerical Methods in Fluids 2002; 38(8):769-788.
31. Zhou JG, Causon DM, Ingram DM, Mingham CG. The surface gradient method for the treatment of source terms in the shallow-water equations. Journal of Computational Physics 2001; 168(1):1-25.
32. Rogers BD, Borthwick AGL, Taylor PH. Mathematical balancing of flux gradient and source terms prior to using Roe's approximate Riemann solver. Journal of Computational Physics 2003; 192(2):422-451.
33. Tseng MH. Improved Treatment of Source Terms in TVD Scheme for Shallow Water Equations. Advances in Water Resources 2004; 27(6):617-629.
34. Tseng MH. The improved surface gradient method for flows simulation in variable bed topography channel using TVD-MacCormack scheme. International Journal for Numerical Methods in Fluids 2003; 43(1):71-91.
35. Caleffi V, Valiani A, Bernini A. Fourth-order balanced source term treatment in central WENO schemes for shallow water equations. Journal of Computational Physics 2006; 218(1):228-245.
36. Xing Y, Shu CW. High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms. Journal of Computational Physics 2006; 214(2):567-598.
37. Xing Y, Shu CW. A new approach of high order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms. Communications in Computational Physics 2006; 1(1):101-135.
38. Kesserwani G, Ghostine R, Vazquez J, Ghenaim A, Mosé R. Riemann solvers with Runge-Kutta discontinuous Galerkin schemes for the 1D shallow water equations. Journal of Hydraulic Engineering (ASCE) 2008; 134(2):243-255.
39. Kesserwani G, Ghostine R, Vazquez J, Ghenaim A, Mosé R. Application of a second order Runge-Kutta discontinuous Galerkin scheme for the shallow water equations with source terms. International Journal for Numerical Methods in Fluids 2008; 56(7):805-821.
40. Goutal N, Maurel F. Proceedings of the 2nd Workshop on Dam Break Wave Simulation, Technical Report IIE-13/97/016A, Electricité de France,Départment Laboratoire National d'Hydraulique, Groupe Hydraulique Fluviale, 1997.