Logically rectangular finite volume methods with adaptive refinement on the sphere

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 367, Issue 1907, 2009, Pages 4483-4496

  Berger, Marsha J ^a   Calhoun, Donna A ^b   Helzel, Christiane ^c   LeVeque, Randall J ^d  

^a NEW YORK UNIVERSITY   (United States)

^b Centre de Saclay ^*  (France)

^c RUHR UNIVERSITY BOCHUM   (Germany)

^d UNIVERSITY OF WASHINGTON   (United States)

Author keywords

Adaptive mesh refinement; Bathymetry; Finite volume; Shallow water equations; Sphere; Well balanced schemes

Indexed keywords

BATHYMETRY; DIFFERENTIAL EQUATIONS; OCEANOGRAPHY;

ADAPTIVE MESH REFINEMENT; ADAPTIVE REFINEMENT; COURANT NUMBERS; EQUILIBRIUM SOLUTIONS; FINITE VOLUME; SHALLOW WATER EQUATIONS; SPHERICAL SHELL; THREE-DIMENSIONAL PROBLEMS; UNIFORM CELLS; WELL- BALANCED SCHEMES;

SPHERES;

EID: 73249117387     PISSN: 1364503X     EISSN: None     Source Type: Journal    
DOI: 10.1098/rsta.2009.0168     Document Type: Article

References (20)

1. Audusse, E., Bouchut, F., Bristeau, M. O., Klein, R. & Perthame, B. 2004 A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput.25, 2050-2065. (doi:10.1137/ S1064827503431090)
2. Bale, D., LeVeque, R. J., Mitran, S. &Rossmanith, J. A. 2002 A wave-propagation method for conservation laws and balance laws with spatially varying flux functions. SIAM J. Sci. Comput.24, 955-978. (doi:10.1137/ S106482750139738X)
3. Berger, M. J. & LeVeque, R. J. 1998 Adaptive mesh refinement using wave-propagation algorithms for hyperbolic systems. SIAM J. Numer. Anal. 35, 2298-2316. (doi:10.1137/S0036142997315974)
4. Bouchut, F. 2004 Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources. Boston, MA: Birkḧuser.
5. Calhoun, D.A. Helzel, C. In press. A finite-volume method for solving parabolic equations on logically Cartesian curved surface meshesSIAM J. Sci. Comput. Seehttp://www.amath. washington.edu/∼calhoun/Surfaces/.
6. Calhoun, D. A., Helzel, C. & LeVeque, R. J. 2008a A finite volume grid for solving hyperbolic problems on the sphere. In Hyperbolic problems: theory, numerics, applications. Proc. 11th Int. Conf. on Hyperbolic Problems, Lyon, France, 17-21 July 2006, pp. 355-362. Berlin, Germany: Springer.
7. Calhoun, D. A., Helzel, C. & LeVeque, R. J. 2008b Logically rectangular finite volume grids and methods for 'circular' and 'spherical' domains. SIAM Rev. 50, 723-752. (doi:10.1137/060664094)
8. Castro, M., Gallardo, J. M., Lpez, J. A. & Paŕs, C. 2008 Well-balanced high order extensions of Godunov's method for semilinear balance laws. SIAM J. Numer. Anal. 46, 1012-1039. (doi:10.1137/060674879)
9. George, D. L. 2008 Augmented Riemann solvers for the shallow water equations over variable topography with steady states and inundation. J. Comput. Phys. 227, 3089-3113.(doi:10.1016/ j.jcp.2007.10.027)
10. George, D. L. & LeVeque, R. J. 2006 Finite volume methods and adaptive refinement for global tsunami propagation and local inundation. Sci. Tsunami Hazards 24, 319-328.
11. George, D. L. & LeVeque, R. J. 2008 High resolution methods and adaptive refinement for tsunami propagation and inundation. In Hyperbolic problems: theory, numerics, applications. Proc. 11th Int. Conf. on Hyperbolic Problems, Lyon, France, 17-21 July 2006, pp. 541-550. Berlin, Germany: Springer.
12. Giraldo, F. X., Hesthaven, J. S. & Warburton, T. 2002 Nodal high-order discontinuous Galerkin methods for the spherical shallow water equations. J. Comput. Phys. 181, 499-525. (doi:10.1006/ jcph.2002.7139)
13. Gosse, L. 2000 A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms. Comput. Math. Appl. 39, 135-159. (doi:10.1142/S021820250100088X)
14. Gosse, L. 2001 A well-balanced scheme using non-conservative products designed for hyperbolic systems of conservation laws with source terms. Math. Models Meth. Appl. Sci. 11, 339-365. (doi:10.1142/S021820250100088X)Gosse, L. 2001 A well-balanced scheme using non-conservative products designed for hyperbolic systems of conservation laws with source terms. Math. Models Meth. Appl. Sci. 11, 339-365. (doi:10.1142/S021820250100088X)
15. Hubbard, M. E. & Nikiforakis, N. 2003 A three-dimensional, adaptive, Godunov-type model for global atmospheric flows. Mon. Wea. Rev. 131, 1848-1864. (doi:10.1175//2568.1) (Pubitemid 37124278)
16. LeVeque, R. J. 1998 Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comput. Phys. 146, 346-365. (doi:10.1006/jcph.1998.6058) (Pubitemid 128347128)
17. LeVeque, R. J. 2002 Finite volume methods for hyperbolic problems. Cambridge, UK: Cambridge University Press.
18. LeVeque, R. J. & George, D. L. 2007 High-resolution finite volume methods for the shallow water equations with bathymetry and dry states. In Advanced numerical models for simulating tsunami waves and runup (eds P. L.-F. Liu, H. Yeh &C. Synolakis). Advances in Coastal and Ocean Engineering, vol. 10, pp. 43-73. Singapore: World Scientific.
19. Swarztrauber, P. N., Williamson, D. L. & Drake, J. B. 1997 The Cartesian method for solving partial differential equations in spherical geometry. Dyn. Atmos. Oceans 27, 679-706. (doi:10.1016/S0377-0265(97)00038-9)
20. Williamson, D. L., Drake, J. B., Hack, J. J., Jakob, R. & Swarztrauber, P. N. 1992 A standard test set for numerical approximations to the shallow water equations in spherical geometry. J. Comput. Phys. 102, 211-224. (doi:10.1016/S0021-9991(05)80016-6)