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A discontinuous Galerkin method for the shallow water equations in spherical triangular coordinates. (English) Zbl 1218.76028
Summary: A global model of the atmosphere is presented governed by the shallow water equations and discretized by a Runge-Kutta discontinuous Galerkin method on an unstructured triangular grid. The shallow water equations on the sphere, a two-dimensional surface in \(\mathbb R^3\), are locally represented in terms of spherical triangular coordinates, the appropriate local coordinate mappings on triangles. On every triangular grid element, this leads to a two-dimensional representation of tangential momentum and therefore only two discrete momentum equations.
The discontinuous Galerkin method consists of an integral formulation which requires both area (elements) and line (element faces) integrals. Here, we use a Rusanov numerical flux to resolve the discontinuous fluxes at the element faces. A strong stability-preserving third-order Runge-Kutta method is applied for the time discretization. The polynomial space of order \(k\) on each curved triangle of the grid is characterized by a Lagrange basis and requires high-order quadrature rules for the integration over elements and element faces. For the presented method no mass matrix inversion is necessary, except in a preprocessing step.
The validation of the atmospheric model has been done considering standard tests from Williamson et al. [D. L. Williamson, J. B. Drake, J. J. Hack, R. Jakob, P. N. Swarztrauber, J. Comput. Phys. 102, No. 1, 211–224 (1992; Zbl 0756.76060)], unsteady analytical solutions of the nonlinear shallow water equations and a barotropic instability caused by an initial perturbation of a jet stream. A convergence rate of \(O(\Delta x^{k+1})\) was observed in the model experiments. Furthermore, a numerical experiment is presented, for which the third-order time-integration method limits the model error. Thus, the time step \(\Delta t\) is restricted by both the CFL-condition and accuracy demands. Conservation of mass was shown up to machine precision and energy conservation converges for both increasing grid resolution and increasing polynomial order \(k\).

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76B07 Free-surface potential flows for incompressible inviscid fluids
Software:
chammp; AMATOS
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[1] Baumgardner, J.R.; Frederickson, P.O., Icosahedral discretization of the two-sphere, SIAM J. numer. anal., 22, 1107-1115, (1985) · Zbl 0601.65084
[2] Behrens, J.; Rakowsky, N.; Hiller, W.; Handorf, D.; Läuter, M.; Päpke, J.; Dethloff, K., Amatos: parallel adaptive mesh generator for atmospheric and oceanic simulation, Ocean model., 10, 171-183, (2005)
[3] Ciarlet, P.G.; Raviart, P.-A., Interpolation theory over curved elements, with applications to finite element methods, Comput. methods appl. mech. engrg., 1, 217-249, (1972) · Zbl 0261.65079
[4] Cockburn, B.; Shu, C.-W., The runge – kutta local projection \(p^1\)-discontinuous-Galerkin finite element method for scalar conservation laws, Math. mod. num. anal., 25, 337-361, (1991) · Zbl 0732.65094
[5] Cockburn, B.; Shu, C.-W., Runge – kutta discontinuous Galerkin methods for convection-dominated problems, J. sci. comput., 16, 173-261, (2001) · Zbl 1065.76135
[6] Cools, R., Monomial cubature rules Since stroud: a compilation – part 2, J. comput. appl. math., 112, 21-27, (1999) · Zbl 0954.65021
[7] Cools, R.; Rabinowitz, P., Monomial cubature rules Since stroud: a compilation, J. comput. appl. math., 48, 309-326, (1993) · Zbl 0799.65027
[8] Côté, J., A Lagrange multiplier approach for the metric terms of semi-Lagrangian models on the sphere, Q.J.R. meteorol. soc., 114, 1347-1352, (1988)
[9] Dennis, J.; Fournier, A.; Spotz, W.F.; St-Cyr, A.; Taylor, M.A.; Thomas, S.J.; Tufo, H., High-resolution mesh convergence porperties and parallel efficiency of a spectral element atmospheric dynamical core, Int. J. high perfor. comput. appl., 19, 225-235, (2005)
[10] Dolejsi, V.; Feistauer, M., A semi-implicit discontinuous Galerkin finite element method for the numerical solution of inviscid compressible flow, J. comput. phys., 198, 727-746, (2004) · Zbl 1116.76386
[11] 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. wea. rev., 132, 726-748, (2004)
[12] Galewsky, J.; Scott, R.K.; Polvani, L.M., An initial-value problem for testing numerical models of the global shallow water equations, Tellus A, 56, 429-440, (2004)
[13] Giraldo, F.X., Lagrange-Galerkin methods on spherical geodesic grids, J. comput. phys., 136, 197-213, (1997) · Zbl 0909.65066
[14] Giraldo, F.X., Semi-implicit time-integrators for a scalable spectral element atmospheric model, Q.J.R. meteorol. soc., 131, 2431-2454, (2005)
[15] Giraldo, F.X., High-order triangle-based discontinuous Galerkin methods for hyperbolic equations on a rotating sphere, J. comput. phys., 214, 447-465, (2006) · Zbl 1089.65096
[16] 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
[17] Giraldo, F.X.; Warburton, T., A nodal triangle-based spectral element method for the shallow water equations on the sphere, J. comput. phys., 207, 129-150, (2005) · Zbl 1177.86002
[18] Giraldo, F.X.; Warburton, T., A high-order triangular discontinuous Galerkin oceanic shallow water model, Int. J. numer. meth. fluids, 56, 899-925, (2008) · Zbl 1290.86002
[19] Gottlieb, S.; Shu, C.-W.; Tadmor, E., Strong stability-preserving high-order time discretization methods, SIAM rev., 43, 89-112, (2001) · Zbl 0967.65098
[20] 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. wea. rev., 123, 1862-1880, (1995)
[21] Heinze, T.; Hense, A., The shallow water equations on the sphere and their Lagrange-Galerkin solution, Meteorol. atmos. phys., 81, 129-137, (2002)
[22] Hesthaven, J.S., From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex, SIAM J. numer. anal., 35, 655-676, (1998) · Zbl 0933.41004
[23] Krivodonova, L.; Xin, J.; Remacle, J.-F.; Chevaugeon, N.; Flaherty, J.E., Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws, Appl. numer. math., 48, 323-338, (2004) · Zbl 1038.65096
[24] Läuter, M.; Handorf, D.; Dethloff, K., Unsteady analytical solutions of the spherical shallow water equations, J. comput. phys., 210, 535-553, (2005) · Zbl 1078.86001
[25] Läuter, M.; Handorf, D.; Rakowsky, N.; Behrens, J.; Frickenhaus, S.; Best, M.; Dethloff, K.; Hiller, W., A parallel adaptive barotropic model of the atmosphere, J. comput. phys., 223, 609-628, (2007) · Zbl 1162.35437
[26] LeSaint, P.; Raviart, P.A., On a finite element method for solving the neutron transport equation, (), 89-145
[27] Lin, S.-J.; Rood, R.B., An explicit flux-form semi-Lagrangian shallow-water model on the sphere, Q.J.R. meteorol. soc., 123, 2477-2498, (1997)
[28] Lyness, J.; Cools, R., A survey of numerical cubature over triangles, Proc. symp. appl. math., 48, 127-150, (1994) · Zbl 0820.41026
[29] Müller, Detlev, Angular-momentum budget of shallow waters on a rotating sphere, Phys. rev. A, 45, 8, 5545-5555, (1992)
[30] Nair, R.D.; Thomas, S.J.; Loft, R.D., A discontinuous Galerkin global shallow water model, Mon. wea. rev., 133, 876-888, (2005)
[31] Rancic, M.; Purser, R.J.; Mesinger, F., A global shallow water model using an expanded spherical cube: gnomonic versus conformal coordinates, Q.J.R. meteorol. soc., 122, 959-982, (1996)
[32] M. Restelli, F.X. Giraldo, A conservative discontinuous Galerkin sem-implicit formulation for the Navier-Stokes equations in nonhydrostatic mesoscale modeling, SIAM J. Sci. Comput., in review. · Zbl 1405.65127
[33] 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
[34] Rossmanith, J.A., A wave propagation method for hyperbolic systems on the sphere, J. comput. phys., 213, 629-658, (2006) · Zbl 1089.65088
[35] Rossmanith, J.A.; Bale, D.S.; LeVeque, R.J., A wave propagation algorithm for hyperbolic systems on curved manifolds, J. comput. phys., 199, 631-662, (2004) · Zbl 1126.76350
[36] Ruuth, S.J.; Spiteri, R.J., Two barriers on strong-stability preserving time discretization methods, J. sci. comput., 17, 211-220, (2002) · Zbl 1003.65107
[37] Sadourny, R., Conservative finite difference approximations of the primitive equations on quasi-uniform spherical grids, Mon. wea. rev., 100, 136-144, (1972)
[38] Sadourny, R.; Arakawa, A.; Mintz, Y., Integration of nondivergent barotropic vorticity equation with an icosahedral-hexagonal grid for sphere, Mon. wea. rev., 6, 351, (1968)
[39] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. comput. phys., 77, 439-471, (1988) · Zbl 0653.65072
[40] A. St-Cyr, C. Jablonowski, J.M. Dennis, H.M. Tufo, S.J. Thomas, A comparison of two shallow water models with non-conforming adaptive grids, Mon. Wea. Rev., in press.
[41] Stroud, A.H., Approximate calculation of multiple integrals, (1971), Prentice-Hall London · Zbl 0379.65013
[42] Stuhne, G.R.; Peltier, W.R., New icosahedral grid-point discretizations of the shallow water equations on the sphere, J. comput. phys., 148, 23-58, (1999) · Zbl 0930.76067
[43] Swarztrauber, P.N., The approximation of vector functions and their derivatives on the sphere, SIAM J. numer. anal., 18, 191-210, (1981) · Zbl 0492.65010
[44] Swarztrauber, P.N.; Williamson, D.L.; Drake, J.B., The Cartesian method for solving partial differential equations in spherical geometry, Dyn. atmos. oceans, 27, 679-706, (1997)
[45] Taylor, M.; Tribbia, J.; Iskandarani, M., The spectral element method for the shallow water equations on the sphere, J. comput. phys., 130, 92-108, (1997) · Zbl 0868.76072
[46] Taylor, M.A.; Wingate, B.A.; Vicent, R.E., An algorithm for computing Fekete points in the triangle, SIAM J. numer. anal., 38, 5, 1707-1720, (2000) · Zbl 0986.65017
[47] Thuburn, J., Pv-based shallow water model on a hexagonal-icosahedral grid, Mon. wea. rev., 125, 2328-2347, (1997)
[48] 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
[49] Williamson, D.L., Integration of barotropic flow on a spherical geodesic grid, Bull. am. meteor. soc., 48, 589, (1967)
[50] 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, (1992) · Zbl 0756.76060
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