A simplified adaptive Cartesian grid system for solving the 2D shallow water equations

International Journal for Numerical Methods in Fluids

Volumn 69, Issue 2, 2012, Pages 442-458

  Liang, Qiuhua ^a  

^a NEWCASTLE UNIVERSITY   (United Kingdom)

Author keywords

Adaptive grid refinement; Cartesian grid; Dynamic grid adaptation; Godunov type scheme; Riemann solver; Shallow water equations; Shock capturing

Indexed keywords

ADAPTIVE GRID REFINEMENT; CARTESIAN GRID; DYNAMIC GRID ADAPTATIONS; GODUNOV-TYPE SCHEMES; RIEMANN SOLVER; SHALLOW WATER EQUATIONS; SHOCK-CAPTURING;

COMPUTATIONAL FLUID DYNAMICS; DATA STRUCTURES;

EQUATIONS OF MOTION;

EID: 84859768622     PISSN: 02712091     EISSN: 10970363     Source Type: Journal    
DOI: 10.1002/fld.2568     Document Type: Article

References (37)

1. Baeza A, Mulet P. Adaptive mesh refinement techniques for high-order shock capturing schemes for multi-dimensional hydrodynamic simulations. International Journal for Numerical Methods in Fluids 2006; 52:455-471.
2. Berger MJ, Colella P. Local adaptive mesh refinement for shock hydrodynamics. Journal of Computational Physics 1989; 82:64-84.
3. Berger MJ, Oliger J. Adaptive mesh refinement for hyperbolic partial differential equations. Journal of Computational Physics 1984; 53:484-512.
4. Rai MM, Anderson D. Grid evolution in time asymptotic problems. Journal of Computational Physics 1981; 43:327-344.
5. Brackbill JU, Saltzman JS. Adaptive zoning for singular problems in two dimensions. Journal of Computational Physics 1982; 46(3):342-368.
6. Ivanenko SA, Muratova GV. Adaptive grid shallow water modeling. Applied Numerical Mathematics 2000; 32:447-482.
7. Tang H. Solution of the shallow-water equations using an adaptive moving mesh method. International Journal for Numerical Methods in Fluids 2004; 44:789-810.
8. Skoula ZD, Borthwick AGL, Moutzouris CI. Godunov-type solution of the shallow water equations on adaptive unstructured triangular grids. International Journal of Computational Fluid Dynamics 2006; 20(9):621-636.
9. Sleigh PA, Gaskell PH, Berzina M, Wright NG. An unstructured finite-volume algorithm for predicting flow in rivers and estuaries. Computers and Fluids 1998; 27(4):479-508.
10. Ji H, Lien F-S, Yee E. A new adaptive mesh refinement data structure with an application to detonation. Journal of Computational Physics 2010; 229(23):8981-8993.
11. Bell J, Berger M, Saltzman J, Welcome M. Three-dimensional adaptive mesh refinement for hyperbolic conservation laws. SIAM Journal on Scientific Computing 1994; 15:127-138.
12. Berger M, Jameson A. Automatic adaptive grid refinement for the Euler equations. AIAA Journal 1985; 23:561-568.
13. Berger MJ, LeVeque RJ. Adaptive mesh refinement using wave-propagation algorithms for hyperbolic systems. SIAM Journal on Numerical Analysis 1998; 35:2298-2316.
14. George DL, LeVeque RJ. Finite volume methods and adaptive refinement for global tsunami propagation and local inundation. Science of Tsunami Hazards 2006; 24:319-328.
15. Rogers BD, Fujihara M, Borthwick AGL. Adaptive Q-tree Godunov-type scheme for shallow water equations. International Journal for Numerical Methods in Fluids 2001; 35:247-280.
16. Liang Q, Borthwick AGL, Stelling G. Simulation of dam and dyke-break hydrodynamics on dynamically adaptive quadtree grids. International Journal for Numerical Methods in Fluids 2004; 46:127-162.
17. Liang Q, Zang J, Borthwick AGL, Taylor PH. Shallow flow simulation on dynamically adaptive cut-cell quadtree grids. International Journal for Numerical Methods in Fluids 2007; 53(12):1777-1799.
18. Krámer T, Józsa J. Solution-adaptivity in modelling complex shallow flows. Computers and Fluids 2007; 36(3):562-577.
19. Anderson JD. Computational Fluid Dynamics: The Basics with Applications. McGraw-Hill: New York, 1995.
20. Greenberg JM, LeRoux A-Y. A well-balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM Journal on Numerical Analysis 1996; 33:1-16.
21. 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.
22. Liang Q, Marche F. Numerical resolution of well-balanced shallow water equations with complex source terms. Advances in Water Resources 2009; 32(6):873-884.
23. van Leer B. Towards the ultimate conservative difference scheme, V-A second-order sequel to Godunov's method. Journal of Computational Physics 1979; 32:101-136.
24. Harten A, Engquist B, Osher S, Chakravarthy SR. Uniformly high order accurate essentially non-oscillatory schemes. Journal of Computational Physics 1987; 71:231-303.
25. Hirsch C. Numerical computation of internal and external flows. Computational Methods for Inviscid and Viscous Flows, vol. 2. Wiley: New York, 1990.
26. Toro EF, Spruce M, Speares W. Restoration of the contact surface in the HLL Riemann solver. Shock Waves 1994; 4(1):25-34.
27. Toro EF. Shock-capturing Methods for Free-surface Shallow Flows. Wiley: Chichester, 2001.
28. Batten P, Clarke N, Lambert C, Causon DM. On the choice of wavespeeds for the HLLC Riemann solver. SIAM Journal on Scientific Computing 1997; 18(6):1553-1570.
29. Fraccarollo L, Toro EF. Experimental and numerical assessment of the shallow water model for two-dimension Dam-Break type problems. Journal of Hydraulic Research 1995; 33(6):843-863.
30. Liang Q, Borthwick AGL. Simple treatment of non-aligned boundaries in a Cartesian grid shallow flow model. International Journal for Numerical Methods in Fluids 2008; 56(11):2091-2110.
31. Alcrudo F, García-Navarro P. A high-resolution Godunov-type scheme in finite volumes for the 2D shallow-water equations. International Journal for Numerical Methods in Fluids 1993; 16:489-505.
32. Anastasiou K, Chan CT. Solution of the 2D shallow water equations using the finite volume method on unstructured triangular meshes. International Journal for Numerical Methods in Fluids 1997; 24:1225-1245.
33. Lamby P, Müller S, Stiriba Y. Solution of shallow water equations using fully adaptive multiscale schemes. International Journal for Numerical Methods in Fluids 2005; 49:417-437.
34. Duan Y-L, Liu R-X. Lattice Boltzmann simulations of triangular cavity flow and free-surface problems. Journal of Hydrodynamics Series B 2007; 19(2):127-134.
35. Causon DM, Ingram DM, Mingham CG, Yang G, Pearson RV. Calculation of shallow water flows using a Cartesian cut cell approach. Advances in Water Resources 2000; 23(5):545-562.
36. Krivodonova L. An efficient local time-stepping scheme for solution of nonlinear conservation laws. Journal of Computational Physics 2010; 229(22):8537-8551.
37. 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.