ADER scheme on unstructured meshes for shallow water: Simulation of tsunami waves

Geophysical Journal International

Volumn 189, Issue 3, 2012, Pages 1505-1520

  Castro, Cristóbal E ^a   Toro, Eleuterio F ^b   Käser, Martin ^c  

^a UNIVERSITY OF HAMBURG   (Germany)

^b UNIVERSITY OF TRENTO   (Italy)

^c UNIVERSITY OF MUNICH   (Germany)

Author keywords

Non linear differential equations; Numerical solutions; Tsunamis

Indexed keywords

ADER SCHEMES; ARBITRARY ACCURACY; CARTESIAN GRID; DISCRETE LEVELS; FINITE VOLUME; FINITE VOLUME SCHEMES; HIGH ORDER; HIGH-ORDER ACCURACY; MATHEMATICAL FORMULATION; NONLINEAR DIFFERENTIAL EQUATION; NUMERICAL SOLUTION; PHYSICAL PROCESS; QUADRATURE RULES; SHALLOW WATER EQUATIONS; SHALLOW WATERS; STEADY STATE SOLUTION; STEADY-STATE CONDITION; STEADY-STATE FLOWS; TSUNAMI WAVES; UNSTRUCTURED MESHES;

BATHYMETRY; EQUATIONS OF MOTION; TSUNAMIS;

NUMERICAL METHODS;

ACCURACY ASSESSMENT; BATHYMETRY; COMPUTER SIMULATION; FINITE ELEMENT METHOD; NUMERICAL METHOD; SHALLOW WATER; STEADY-STATE EQUILIBRIUM; TSUNAMI; WAVE PROPAGATION;

EID: 84860833642     PISSN: 0956540X     EISSN: 1365246X     Source Type: Journal    
DOI: 10.1111/j.1365-246X.2012.05471.x     Document Type: Article

References (50)

1. Audusse, E., Bouchut, F., Bristeau, M., Klein, R. & Perthame, B., 2004. A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J. Sci. Comput., 25(6), 2050-2065.
2. Benkhaldoun, F., Elmahi, I. & Seaïd, M., 2007. Well-balanced finite volume schemes for pollutant transport by shallow water equations on unstructured meshes, J. Comput. Phys., 226, 180-203.
3. Berger, M.J., George, D.L., LeVeque, R.J. & Mandli, K.T., 2011. The geoclaw software for depth-averaged flows with adaptive refinement, Adv. Water Resour., 34(9), 1195-1206.
4. Bermúdez, A. & Vázquez, M., 1994. Upwind methods for hyperbolic conservation laws with source terms, Comput. Fluids, 23(8), 1049-1071.
5. Bernetti, R., Titarev, V. & Toro, E., 2008. Exact solution of the Riemann problem for the shallow water equation with discontinuous bottom geometry, J. Comput. Phys., 227, 3212-3243.
6. Caleffi, V., Valiani, A. & Bernini, A., 2006. Fourth-order balanced source term treatment in central WENO schemes for shallow water equations, J. Comput. Phys., 218, 228-245.
7. Castro, C.E., 2007. High order ADER FV/DG numerical methods for hyperbolic equations, Monographs of the School of Doctoral Studies in Environmental Engineering 10, University of Trento, Trento.
8. Castro, C.E. & Toro, E.F., 2008. Solvers for the high-order Riemann problem for hyperbolic balance laws, J. Comput. Phys., 227(4), 2481-2513.
9. Dumbser, M., 2005. Arbitrary high order schemes for the solution of hyperbolic conservation laws in complex domains, PhD thesis, Universität Stuttgart, Stuttgart
10. Dumbser, M. & Käser, M., 2007. Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems, J. Comput. Phys., 221, 693-723.
11. Dumbser, M. & Munz, C., 2006. Building blocks for arbitrary high order discontinuous Galerkin schemes, J. Sci. Comput., 27, 215-230.
12. Dumbser, M., Käser, M. & Toro, E.F., 2007. An arbitrary high order discontinuous Galerkin method for elastic waves on unstructured meshes v: local time stepping and p-adaptivity, Geophys. J. Int., 171, 695-717.
13. Dumbser, M., Enaux, C. & Toro, E., 2008. Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws, J. Comput. Phys., 227(8), 3971-4001.
14. Ern, A., Piperno, S. & Djadel, K., 2008. A well-balanced Runge-Kutta discontinuous Galerkin method for the shallow-water equations with flooding and drying, Int. J. Numer. Meth. Fl., 58, 1-25.
15. ETOPO2v2, 2006. ETOPO2v2 Global Gridded 2-minute Database, National Geophysical Data Center, (last accessed 2009 June).
16. GEBCO, 2009. The GEBCO_08 Grid, version 20091120, (last accessed 2010 June).
17. George, D.L., 2008. Augmented Riemann solvers for the shallow water equation over variable topography with steady state and inundation, J. Comput. Phys., 227, 3089-3113.
18. Godunov, S., 1959. A finite difference method for the computation of discontinuous solutions of the equations of fluid dynamics, Mat. Sbornik., 47, 357-393.
19. Harig, S., Chaeroni, Pranowo, W.S. & Behrens, J., 2008. Tsunami simulations on several scales, Ocean Dyn., 58, 429-440.
20. Imamura, F., 1996. Review of tsunami simulation with a finite difference method, in Long-Wave Run-Up Models, pp. 25-42, eds Yeh, H., Liu, P. L.-F. & Synolakis, C. E., World Scientific, Singapore
21. Karypis, G. & Kumar, V., 1998. Multilevel k-way partitioning scheme for irregular graphs, J. Parallel Distrib. Comput., 48, 96-129.
22. Käser, M. & Iske, A., 2005. ADER schemes on adaptive triangular meshes for scalar conservation laws, J. Comput. Phys., 205, 486-508.
23. Kawahara, M., Takeuchi, N. & Yoshida, T., 1978. Two step explicit finite element method for tsunami wave propagation analysis, Int. J. Numer. Methods Eng., 12, 331-351.
24. LeVeque, R., 1998. Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm, J. Comput. Phys., 146, 346-365.
25. Liu, P.L.-F., Woo, S.-B. & Chorin, Y.-S., 1998. Computer programs for tsunami propagation and inundation, Tech. rep., Cornell University.
26. Mader, C.L., 1974. Numerical simulation of tsunamis, J. Phys. Oceanogr., 4, 74-82.
27. NDBC, 2010. National data buoy center, NOAA, (last accessed 2010 June).
28. Noelle, S., Pankratz, N., Puppo, G. & Natvig, J., 2006. Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows, J. Comput. Phys., 213, 474-499.
29. Noelle, S., Xing, Y. & Shu, C., 2007. High-order well-balanced finite volume WENO schemes for shallow water equation with moving water, J. Comput. Phys., 226, 29-58.
30. Oeser, J., Bunge, H.-P. & Mohr, M., 2006. Cluster design in the earth sciences: TETHYS, in HPCC 2006: High Performance Computing and Communications-Second International Conference, Lecture Notes in Computer Science, Vol. 4208, pp. 31-40 eds Gerndt, M. & Kranzlmüller, D., Springer, Dordrecht, doi:.
31. Olabarrieta, M., Medina, R., Gonzalez, M. & Otero, L., 2011. C3: a finite volume-finite difference hybrid model for tsunami propagation and runup, Comput. Geosci., 37, 1003-1014.
32. Pranowo, W.S., 2010. Adaptive mesh refinement applied to tsunami modeling: TsunaFLASH, PhD thesis, Universität Bremen, Bremen
33. Saito, T. & Furumura, T., 2009. Three-dimensional simulation of tsunami generation and propagation: application to intraplate events, J. Geophys. Res., 114, B02307, doi:.
34. Schwartzkopff, T., Munz, C. & Toro, E., 2002. ADER-2D: a high-order approach for linear hyperbolic systems in 2D, J. Sci. Comput., 17, 231-240.
35. Schwartzkopff, T., Dumbser, M. & Munz, C., 2004. Fast high order ADER schemes for linear hyperbolic equations, J. Comput. Phys., 197(2), 532-539.
36. Sladen, A. et al., 2010. A coseismic distributed slip model for the 2010 Mw8.8 Maule (Chile) earthquake, in AGU Chapman Conference on Giant Earthquakes and Their Tsunamis, 2010 May 16-24, Valparaíso, Viña del Mar.
37. Takakura, Y. & Toro, E., 2002. Arbitrarily accurate non-oscillatory schemes for a nonlinear conservation law, J. Comput. Fluid Dyn., 11(1), 7-18.
38. Titarev, V.A. & Toro, E.F., 2002. ADER: arbitrary high order Godunov approach., J. Sci. Comput., 17(1-4), 609-618.
39. Titarev, V.A. & Toro, F.E., 2005. ADER schemes for three-dimensional non-linear hyperbolic systems, J. Comput. Phys., 204, 715-736.
40. Titov, V. & Gonzalez, F., 1997. Implementation and testing of the method of splitting tsunami (most) model, Tech. rep., NOAA/Pacific Marine Environmental Laboratory, NOAA Technical Memorandum ERL PMEL-112. Contribution No. 1927.
41. Toro, E., 2009. Riemann Solvers and Numerical Methods for Fluid Dynamics, 3rd edn, Springer-Verlag, Berlin Heidelberg
42. Toro, E.F., 2001. Shock-Capturing Methods for Free-Surface Shallow Flows, John Wiley and Sons Ltd, Chichester.
43. Toro, E.F. & Titarev, V.A., 2002. Solution of the generalized Riemann problem for advection-reaction equations., Proc. R. Soc. Lond., A, 458, 271-281.
44. Toro, E.F. & Titarev, V.A., 2005. TVD fluxes for the high-order ADER schemes, J. Sci. Comput., 24(3), 285-309.
45. Toro, E.F., Millington, R.C. & Nejad, L.A.M., 2001. Towards very high order Godunov schemes., in Godunov Methods. Theory and Applications, pp. 907-940, ed. Toro, E. F., Kluwer/Plenum Academic Publishers, New York, NY.
46. USGS, 2010. Magnitude 8.8-OFFSHORE MAULE, CHILE, (last accessed 2010 June).
47. Vázquez-Cendón, M., 1999. Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry, J. Comput. Phys., 148, 497-526.
48. Vignoli, G., Titarev, V. & Toro, E., 2008. ADER schemes for the shallow water equations in channel with irregular bottom elevation, J. Comput. Phys., 227, 2463-2480.
49. Xing, Y. & Shu, C., 2005. High order finite difference WENO schemes with the exact conservation property for the shallow water equations, J. Comput. Phys., 208, 206-227.
50. Xing, Y. & Shu, C., 2006. High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms, J. Comput. Phys., 214, 567-598.