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A wave propagation algorithm for hyperbolic systems on curved manifolds. (English) Zbl 1126.76350

Summary: An extension of the wave propagation algorithm first introduced by R. J. LeVeque [J. Comput. Phys. 131, No. 2, 327–353 (1997; Zbl 0872.76075)] is developed for hyperbolic systems on a general curved manifold. This extension is important in a variety of applications, including the propagation of sound waves on a curved surface, shallow water flow on the surface of the Earth, shallow water magnetohydrodynamics in the solar tachocline, and relativistic hydrodynamics in the presence of compact objects such as neutron stars and black holes. As is the case for the Cartesian wave propagation algorithm, this new approach is second order accurate for smooth flows and high-resolution shock-capturing. The algorithm is formulated such that scalar variables are numerically conserved and vector variables have a geometric source term that is naturally incorporated into a modified Riemann solver. Furthermore, all necessary one-dimensional Riemann problems are solved in a locally valid orthonormal basis. This orthonormalization allows one to solve Cartesian Riemann problems that are devoid of geometric terms. The new method is tested via application to the linear wave equation on a curved manifold as well as the shallow water equations on part of a sphere. The proposed algorithm has been implemented in the software package CLAWPACK and is freely available on the

MSC:

76M25 Other numerical methods (fluid mechanics) (MSC2010)
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L60 First-order nonlinear hyperbolic equations
86-08 Computational methods for problems pertaining to geophysics

Citations:

Zbl 0872.76075

Software:

CLAWPACK; chammp
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Full Text: DOI Link

References:

[1] D.S. Bale, Wave propagation algorithms on curved manifolds with applications to relativistic hydrodynamics, Ph.D. Thesis, University of Washington, Seattle, WA, 2002; D.S. Bale, Wave propagation algorithms on curved manifolds with applications to relativistic hydrodynamics, Ph.D. Thesis, University of Washington, Seattle, WA, 2002
[2] 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. Comput., 24, 955-978 (2003) · Zbl 1034.65068
[3] 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)
[4] Cho, J. Y.-K.; Polvani, L. M., The emergence of jets and vortices in freely evolving shallow-water turbulence on a sphere, Phys. Fluids, 8, 1531-1552 (1995) · Zbl 1087.76057
[5] Coté, J., A Lagrange multiplier approach for the metric terms of semi-Langrangian models on the sphere, Q. J. R. Meteorol. Soc., 114, 1347-1352 (1988)
[6] Cushman-Roisin, B., Introduction to Geophysical Fluid Dynamics (1994), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ
[7] Donat, R.; Marquina, A., Capturing shock reflections: an improved flux formula, J. Comput. Phys., 125, 42-58 (1996) · Zbl 0847.76049
[8] T. Fogarty, Finite volume methods for acoustics and elasto-plasticity with damage in a heterogeneous medium, Ph.D. Thesis, University of Washington, Seattle, WA, 2002; T. Fogarty, Finite volume methods for acoustics and elasto-plasticity with damage in a heterogeneous medium, Ph.D. Thesis, University of Washington, Seattle, WA, 2002
[9] T.R. Fogarty, High-resolution finite volume methods for acoustics in a rapidly-varying heterogeneous medium, Master’s Thesis, University of Washington, Seattle, WA, 1997; T.R. Fogarty, High-resolution finite volume methods for acoustics in a rapidly-varying heterogeneous medium, Master’s Thesis, University of Washington, Seattle, WA, 1997
[10] 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)
[11] Font, J. A., Numerical hydrodynamics in general relativity, Living Rev. Rel. (2000) · Zbl 0944.83007
[12] Gilman, P. A., Magnetohydrodynamic “shallow water” equations for the solar tachocline, Astrophys. J., 544, L79-L82 (2000)
[13] Harten, A.; Hyman, J. M., Self-adjusting grid methods for one-dimensional hyperbolic conservation laws, J. Comput. Phys., 50, 235-269 (1983) · Zbl 0565.65049
[14] Heikes, R.; Randall, D. A., Numerical integration integration of the shallow water equations on a twisted icosahedral grid. Part I: basic design and results of tests, Month. Weather Rev., 123, 1862-1880 (1995)
[15] Heikes, R.; Randall, D. A., Numerical integration 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, Month. Weather Rev., 123, 1881-1887 (1995)
[16] 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; 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
[17] Helzel, C.; LeVeque, R. J.; Warnecke, G., A modified fractional step method for the accurate approximation of detonation waves, SIAM J. Sci. Comput., 22, 1489-1510 (2000) · Zbl 0983.65105
[18] Hern, S. D.; Stewart, J. M., The Gowdy T-3 cosmologies revisited, Classical Quant. Grav., 15, 1581-1593 (1998) · Zbl 0933.83043
[19] Iacono, R.; Struglia, M. V.; Ronchi, C., Spontaneous formation of equatorial jets in freely decaying shallow water turbulence, Phys. Fluids, 11, 1272-1274 (1999) · Zbl 1147.76422
[20] Kevorkian, J., Partial Differential Equations: Analytic Solution Techniques (2000), Springer: Springer New York · Zbl 0937.35001
[21] 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
[22] LeVeque, R. J.; Yong, D. H., Solitary waves in layered nonlinear media, SIAM J. Appl. Math., 63, 1539-1560 (2003) · Zbl 1075.74047
[23] LeVeque, R. J., High-resolution conservative algorithms for advection in incompressible flow, SIAM J. Numer. Anal., 33, 627-665 (1996) · Zbl 0852.76057
[24] LeVeque, R. J., Wave propagation algorithms for multi-dimensional hyperbolic systems, J. Comput. Phys., 131, 327-335 (1997)
[25] LeVeque, R. J., Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm, J. Comput. Phys., 146, 346-365 (1998) · Zbl 0931.76059
[26] 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; 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
[27] LeVeque, R. J., Finite Volume Methods for Hyperbolic Problems (2002), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1010.65040
[28] LeVeque, R. J.; Pelanti, M., A class of approximate Riemann solvers and their relation to relaxation schemes, J. Comput. Phys., 172, 572-591 (2001) · Zbl 0988.65072
[29] Martı́, J. M.; Müller, E., Numerical hydrodynamics in special relativity, Living Rev. Rel. (1999) · Zbl 0944.83006
[30] McGregor, J., Semi-Lagrangian advection on conformal cubic grids, Month. Weather Rev., 124, 1311-1322 (1996)
[31] Misner, C. W.; Thorne, K. S.; Wheeler, J. A., Gravitation (1973), Freeman: Freeman San Fransisco
[32] Pons, J. A.; Font, J. A.; Ibá ñez, J. M.; Martı́, J. M.; Miralles, J. A., General relativistic hydrodynamics with special relativistic Riemann solvers, Astron. Astrophys., 339, 638-642 (1998)
[33] Rancic, M.; Purser, R. J.; Messinger, F., A global shallow-water model using an expanded spherical cube: gnonomic versus conformal coordinates, Q. J. R. Meteorol. Soc., 122, 959-982 (1996)
[34] 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
[35] J.A. Rossmanith, A wave propagation method with constrained transport for ideal and shallow water magnetohydrodynamics, Ph.D. Thesis, University of Washington, Seattle, WA, 2002; J.A. Rossmanith, A wave propagation method with constrained transport for ideal and shallow water magnetohydrodynamics, Ph.D. Thesis, University of Washington, Seattle, WA, 2002
[36] Sadourny, R., Conservative finite-difference approximations of the primitive equations on quasi-uniform spherical grids, Month. Weather Rev., 100, 211-224 (1972)
[37] Schecter, D. A.; Boyd, J. F.; Gilman, P. A., “Shallow-water” magneto-hydrodynamic waves in the solar tachocline, Astrophys. J., 551, L185-L188 (2001)
[38] H. De Sterck, Multi-dimensional upwind constrained transport on unstructured grids for “shallow water” magnetohydrodynamics, in: AIAA 2001-2623, June, 2001; H. De Sterck, Multi-dimensional upwind constrained transport on unstructured grids for “shallow water” magnetohydrodynamics, in: AIAA 2001-2623, June, 2001
[39] Swarztrauber, P. N.; Williamson, D. L.; Drake, J. B., The Cartesian grid method for solving partial differential equations in spherical geometry, Dyn. Atmos. Oceans, 27, 679-706 (1998)
[40] Thompson, J. F.; Soni, B.; Weatherrill, N. P., Handbook of Grid Generation (1998), CRC Press: CRC Press Boca Raton
[41] Tsukahara, Y.; Nakaso, N.; Cho, H.; Yamanaka, K., Observation of diffraction-free propagation of surface acoustic waves around a homogeneous isotropic solid sphere, Appl. Phys. Lett., 77, 2926-2928 (2000)
[42] Vinokur, M., Conservation equations of gasdynamics in curvilinear coordinates, J. Comput. Phys., 14, 105-125 (1974) · Zbl 0277.76061
[43] Williamson, D. L.; Drake, J. B.; Hack, J. J.; Jakob, R.; Swartztrauber, 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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