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A wave propagation method for hyperbolic systems on the sphere. (English) Zbl 1089.65088
Summary: The author presents an explicit finite volume method for solving general hyperbolic systems on the surface of a sphere. Applications where such systems arise include passive tracer advection in the atmosphere, shallow water models of the ocean and atmosphere, and shallow water magnetohydrodynamic models of the solar tachocline. The method is based on the curved manifold wave propagation algorithm of J. A. Rossmanith, D. S. Bale, and R. J. LeVeque [J. Comput. Phys. 199, No. 2, 631–662 (2004; Zbl 1126.76350)], which makes use of parallel transport to approximate geometric source terms and orthonormal Riemann solvers to carry out characteristic decompositions. This approach employs TVD wave limiters, which allows the method to be accurate for both smooth solutions and solutions in which large gradients or discontinuities can occur in the form of material interfaces or shock waves.
The numerical grid used in this work is the cubed sphere grid of C. Ronchi, R. Iacono, and P. S. Paolucci [ibid. 124, 93–114 (1996; Zbl 0849.76049)], which covers the sphere with nearly uniform resolution using six identical grid patches with grid lines lying on great circles. Boundary conditions across grid patches are applied either through direct copying from neighboring grid cells in the case of scalar equations or 1D interpolation along great circles in the case of more complicated systems. The resulting numerical method is applied to several test problems for the advection equation, the shallow water equations, and the shallow water magnetohydrodynamic (SMHD) equations. For the SMHD equations, we make use of an unstaggered constrained transport method to maintain a discrete divergence-free magnetic field.

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L45 Initial value problems for first-order hyperbolic systems
76M12 Finite volume methods applied to problems in fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76W05 Magnetohydrodynamics and electrohydrodynamics
Full Text: DOI
[1] Abgrall, R.; Karni, S., Computations of compressible multifluids, J. comput. phys., 169, 594-623, (2001) · Zbl 1033.76029
[2] D.S. Bale, Wave Propagation Algorithms on Curved Manifolds with Applications to Relativistic Hydrodynamics, Ph.D. Thesis, University of Washington, Seattle, Washington, 2002.
[3] Bale, D.S.; LeVeque, R.J.; Mitran, S.; Rossmanith, J.A., A wave propagation method for conservation laws and balance laws with spatially varying flux functions, SIAM J. sci. comp., 24, 955-978, (2003) · Zbl 1034.65068
[4] Bardeen, J.M.; Buchman, L.T., Numerical tests of evolution systems, gauge conditions, and boundary conditions for 1d colliding gravitational plane waves, Phys. rev. D, 65, (2002)
[5] Behrens, J., An adaptive semi-Lagrangian advection scheme and its parallelization, Mon. weather rev., 124, 2386-2395, (1996)
[6] Berger, M.J.; LeVeque, R.J., Adaptive mesh refinement using wave propagation algorithms for hyperbolic systems, SIAM J. numer. anal., 35, 2298-2316, (1998) · Zbl 0921.65070
[7] Berger, M.J.; Oliger, J., Adaptive mesh refinement for hyperbolic partial differential equations, J. comput. phys., 53, 484-512, (1984) · Zbl 0536.65071
[8] R. Blikberg, Nested Parellelism in Open MP with Application to Adaptive Mesh Refinement, Ph.D. Thesis, University of Bergen, Norway, 2003.
[9] Bourke, W., An efficient, one-level, primitive-equation spectral model, Mon. weather rev., 100, 683-698, (1972)
[10] Brackbill, J.U.; Barnes, D.C., The effect of nonzero ∇·b on the numerical solution of the magnetohydrodynamic equations, J. comput. phys., 35, 426-430, (1980) · Zbl 0429.76079
[11] Côté, J.; Staniforth, A., An accurate and efficient finite-element global model of the shallow-water primitive equations, Mon. weather rev., 118, 2707-2717, (1990)
[12] Chesshire, G.; Henshaw, W.D., Composite overlapping meshes for the solution of partial-differential equations, J. comput. phys., 90, 1-64, (1990) · Zbl 0709.65090
[13] de Saint-Venant, A.J.C., Théorie du mouvement non-permanent des eaux, avec crues des riviàre et à l’introduction des marées dans leur lit, CR acad. sci. Paris, 73, 147-154, (1871) · JFM 03.0482.04
[14] Evans, C.; Hawley, J.F., Simulation of magnetohydrodynamic flow: a constrained transport method, Astrophys. J., 332, 659, (1988)
[15] T. Fogarty, Finite Volume Methods for Acoustics and Elasto-plasticity with Damage in a Heterogeneous Medium, Ph.D. Thesis, University of Washington, Seattle, Washington, 2002.
[16] T.R. Fogarty, High-resolution Finite Volume Methods for Acoustics in a Rapidly-varying Heterogeneous Medium, Master’s Thesis, University of Washington, Seattle, Washington, 1997.
[17] Fogarty, T.R.; LeVeque, R.J., High-resolution finite volume methods for acoustics in periodic and random media, J. acoust. soc. am., 106, 1, 1-12, (1999)
[18] Fournier, A.; Taylor, M.A.; Tribbia, J.J., The spectral element atmosphere model (SEAM): high-resolution parallel computation and localized resolution of regional dynamics, Mon. weather rev., 132, 726-748, (2004)
[19] Galewsky, J.; Scott, R.K.; Polvani, L.M., An initial-value problem for testing numerical models of the global shallow-water equations, Tellus, 56A, 429-440, (2004)
[20] Gilman, P.A., Magnetohydrodynamic “shallow water” equations for the solar tachocline, Astrophys. J., 544, L79-L82, (2000)
[21] Giraldo, F.X., Lagrange-Galerkin methods on spherical geodesic grids: the shallow water equations, J. comput. phys., 160, 336-368, (2000) · Zbl 0977.76045
[22] Giraldo, F.X., A spectral element shallow water model spherical geodesic grids, Int. J. numer. meth. fluid, 35, 869-901, (2001) · Zbl 1030.76045
[23] Giraldo, F.X.; Hesthaven, J.S.; Warburton, T., Nodal high-order discontinuous Galerkin methods for the spherical shallow water equations, J. comput. phys., 181, 499-525, (2002) · Zbl 1178.76268
[24] Giraldo, F.X.; Rosmond, T.E., A scalable spectral element Eulerian atmospheric model (SEE-AM) for NWP: dynamical core tests, Mon. weather rev., 132, 133-153, (2004)
[25] Heikes, R.; Randall, D.A., Numerical integration of the shallow water equations on a twisted icosahedral grid. part I: basic design and results of tests, Mon. weather rev., 123, 1862-1880, (1995)
[26] Heikes, R.; Randall, D.A., Numerical integration of the shallow water equations on a twisted icosahedral grid. part II: A detailed description of the grid and an analysis of numerical accuracy, Mon. weather rev., 123, 1881-1887, (1995)
[27] C. Helzel, Numerical Approximation of Conservation Laws with Stiff Source Terms for the Modelling of Detonation Waves, Ph.D. Thesis, Otto-von-Guericke-Universität Magdeburg, Magdeburg, Germany, 2000.
[28] Helzel, C.; LeVeque, R.J.; Warnecke, G., A modified fractional step method for the accurate approximation of detonation waves, SIAM J. sci. comp., 22, 1489-1510, (2000) · Zbl 0983.65105
[29] Hern, S.D.; Stewart, J.M., The gowdy T-3 cosmologies revisited, Classic. quant. grav., 15, 1581-1593, (1998) · Zbl 0933.83043
[30] Hubbard, M.E.; Nikiforakis, N., A three-dimensional, adaptive, Godunov-type model for global atmospheric flows, Mon. weather rev., 131, 1848-1864, (2003)
[31] C. Jablonowski, Adaptive Grids in Weather and Climate Modeling, Ph.D. Thesis, University of Michigan, Ann Arbor, 2004.
[32] Jackson, J.D., Classical electrodynamics, (1999), Wiley New York · Zbl 0114.42903
[33] Langseth, J.O.; LeVeque, R.J., A wave propagation method for three-dimensional hyperbolic conservation laws, J. comput. phys., 165, 126-166, (2000) · Zbl 0967.65095
[34] Lanser, D.; Blom, J.G.; Verwer, J.G., Time integration of the shallow water equations in spherical geometry, J. comput. phys., 171, 373-393, (2001) · Zbl 1051.76047
[35] LeVeque, R.J.; Yong, D.H., Solitary waves in layered nonlinear media, SIAM J. appl. math., 63, 1539-1560, (2003) · Zbl 1075.74047
[36] LeVeque, R.J., High-resolution conservative algorithms for advection in incompressible flow, SIAM J. numer. anal., 33, 627-665, (1996) · Zbl 0852.76057
[37] LeVeque, R.J., Wave propagation algorithms for multi-dimensional hyperbolic systems, J. comput. phys., 131, 327-335, (1997)
[38] R.J. LeVeque, Finite Volume Methods for Nonlinear Elasticity in Heterogeneous Media, in: Proceedings of the ICFD Conference on Numerical Methods for Fluid Dynamics, 2001.
[39] LeVeque, R.J., Finite volume methods for hyperbolic problems, (2002), Cambridge University Press Cambridge · Zbl 1010.65040
[40] Liska, R.; Wendroff, B., Shallow water conservation laws on a sphere, (), 673-682
[41] McGregor, J., Semi-Lagrangian advection on conformal cubic grids, Mon. weather rev., 124, 1311-1322, (1996)
[42] Misner, C.W.; Thorne, K.S.; Wheeler, J.A., Gravitation, (1973), W.H. Freeman San Fransisco
[43] Nair, R.D.; Thomas, S.J.; Loft, R.D., A discontinuous Galerkin global shallow water model, Mon. weather rev., 133, 876-888, (2005)
[44] Nair, R.D.; Thomas, S.J.; Loft, R.D., A discontinuous Galerkin transport scheme on the cubed sphere, Mon. weather rev., 133, 814-828, (2005)
[45] Nihei, T.; Ishii, K., A fast solver for the shallow water equations on a sphere using a combined compact difference scheme, J. comput. phys., 187, 639-659, (2003) · Zbl 1061.76513
[46] Pedlosky, J., Geophysical fluid dynamics, (1987), Springer Berlin · Zbl 0713.76005
[47] Purser, R.J.; Rancic, M., Smooth quasi-homogeneous gridding of the sphere, J. comput. phys., 124, 637-647, (1998)
[48] Rancic, M.; Purser, R.J.; Mesinger, F., A global shallow-water model using an expanded spherical cube: gnomonic versus conformal coordinates, Q. J. roy. meteor. soc., 122, 959-982, (1996)
[49] Ronchi, C.; Iacono, R.; Paolucci, P.S., The ‘cubed sphere’: a new method for the solution of partial differential equations in spherical geometry, J. comput. phys., 124, 93-114, (1996) · Zbl 0849.76049
[50] J.A. Rossmanith, An unstaggered, high-resolution constrained transport method for magnetohydrodynamic flows, SIAM J. Sci. Comp. (submitted). Available from: <http://www.math.lsa.umich.edu/ rossmani/preprints/mhdpaper.pdf>. · Zbl 1344.76092
[51] Rossmanith, J.A.; Bale, D.S.; LeVeque, R.J., A wave propagation algorithm for hypberbolic systems on curved manifolds, J. comput. phys., 199, 631-662, (2004) · Zbl 1126.76350
[52] Sadourny, R., Conservative finite-difference approximations of the primitive equations on quasi-uniform spherical grids, Mon. weather rev., 100, 211-224, (1972)
[53] Spotz, W.F.; Taylor, M.A.; Swarztrauber, P.N., Fast shallow-water equation solvers in latitude-longitude coordinates, J. comput. phys., 145, 432-444, (1998) · Zbl 0928.76078
[54] De Sterck, H., Hyperbolic theory of the “shallow water” magnetohydrodynamics equations, Phys. plasmas, 8, 7, 3293-3304, (2001)
[55] Tóth, G., The ∇·B=0 constraint in shock-capturing magnetohydrodynamics codes, J. comput. phys., 161, 605-652, (2000) · Zbl 0980.76051
[56] Thuburn, J., A PV-based shallow-water model on a hexagonal-icosahedral grid, Mon. weather rev., 125, 2328-2347, (1997)
[57] Thuburn, J.; Li, Y., Numerical simulations of Rossby-haurwitz waves, Tellus, 52A, 181-189, (2000)
[58] Tolstykh, M.A., Vorticity-divergence semi-Lagrangian shallow-water model on the sphere based on compact finite differences, J. comput. phys., 179, 180-200, (2002) · Zbl 1060.76086
[59] Tomita, H.; Tsugawa, M.; Satoh, M.; Goto, K., Shallow water model on a modified icosahedral geodesic grid by using spring dynamics, J. comput. phys., 174, 579-613, (2001) · Zbl 1056.76058
[60] van Leer, B., Towards the ultimate conservative difference scheme II. monotonicity and conservation combined in a second order scheme, J. comput. phys., 14, 361-370, (1974) · Zbl 0276.65055
[61] Williamson, D.L.; Drake, J.B.; Hack, J.J.; Jakob, R.; Swarztrauber, P.N., A standard test set for numerical approximations to the shallow water equations in spherical geometry, J. comput. phys., 102, 211-224, (1994) · Zbl 0756.76060
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