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GPU Accelerated Discontinuous Galerkin Methods for Shallow Water Equations

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Hyperbolic equations and systems Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  03 July 2015

Rajesh Gandham ,
David Medina  and
Timothy Warburton
Rajesh Gandham*
Affiliation:
Department of Computational and Applied Mathematics, Rice University, 6100 Main Street, MS-134, Houston, TX-77005, USA
David Medina
Affiliation:
Department of Computational and Applied Mathematics, Rice University, 6100 Main Street, MS-134, Houston, TX-77005, USA
Timothy Warburton
Affiliation:
Department of Computational and Applied Mathematics, Rice University, 6100 Main Street, MS-134, Houston, TX-77005, USA
*
*Corresponding author. Email addresses: Rajesh.Gandham@rice.edu (R. Gandham), David.S.Medina@rice.edu (D. Medina), timwar@rice.edu (T. Warburton)
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Abstract

We discuss the development, verification, and performance of a GPU accelerated discontinuous Galerkin method for the solutions of two dimensional nonlinear shallow water equations. The shallow water equations are hyperbolic partial differential equations and are widely used in the simulation of tsunami wave propagations. Our algorithms are tailored to take advantage of the single instruction multiple data (SIMD) architecture of graphic processing units. The time integration is accelerated by local time stepping based on a multi-rate Adams-Bashforthscheme. A total variational bounded limiter is adopted for nonlinear stability of the numerical scheme. This limiter is coupled with a mass and momentum conserving positivity preserving limiter for the special treatment of a dry or partially wet element in the triangulation. Accuracy, robustness and performance are demonstrated with the aid of test cases. Furthermore, we developed a unified multi-threading model OCCA. The kernels expressed in OCCA model can be cross-compiled with multi-threading models OpenCL, CUDA, and OpenMP. We compare the performance of the OCCA kernels when cross-compiled with these models.

Keywords

Shallow water equationsdiscontinuous Galerkinpositivity preservingslope limitingGPUsacceleratorsmulti-threading

MSC classification

Secondary: 35Q35: PDEs in connection with fluid mechanics 35L60: Nonlinear first-order hyperbolic equations 35L65: Conservation laws 65Z05: Applications to physics 65M99: None of the above, but in this section
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 1 , July 2015 , pp. 37 - 64
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Aizinger, V. and Dawson, C., A discontinuous Galerkin method for two-dimensional flow and transport in shallow water, Advances in Water Resources, 25 (2002), pp. 67–84.CrossRefGoogle Scholar
[2]Berger, M., George, D., LeVeque, R., and Mandli, K., The GeoClaw software for depth-averaged flows with adaptive refinement, Advances in Water Resources, 34 (2011), pp. 1195–1206.Google Scholar
[3]Bunya, S., Kubatko, E., Westerink, J., and Dawson, C., A wetting and drying treatment for the Runge-Kutta discontinuous Galerkin solution to the shallow water equations, Computer Methods in Applied Mechanics and Engineering, 198 (2009), pp. 1548–1562.Google Scholar
[4]Casulli, V., Semi-implicit finite difference methods for the two-dimensional shallow water equations, Journal of Computational Physics, 86 (1990), pp. 56–74.Google Scholar
[5]Cockburn, B. and Shu, C.-W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, Journal of Computational Physics, 141 (1998), pp. 199–224.CrossRefGoogle Scholar
[6]Cools, R., Monomial cubature rules since “Stroud”: a compilation-part 2, (1999).Google Scholar
[7]Ern, A., Piperno, S., and Djadel, K., A well-balanced Runge-Kutta discontinuous Galerkin method for the shallow-water equations with flooding and drying, International journal for numerical methods in fluids, 58 (2008), pp. 1–25.Google Scholar
[8]Eskilsson, C. and Sherwin, S., A triangular spectral/hp discontinuous Galerkin method for modelling 2D shallow water equations, International Journal for Numerical Methods in Fluids, 45 (2004), pp. 605–623.Google Scholar
[9]Gallardo, J. M., Parés, C., and Castro, M., On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas, Journal of Computational Physics, 227 (2007), pp. 574–601.Google Scholar
[10]Gear, C. W. and Wells, D., Multirate linear multistep methods, BIT Numerical Mathematics, 24 (1984), pp. 484–502.Google Scholar
[11]Giraldo, F. and Warburton, T., A high-order triangular discontinuous Galerkin oceanic shallow water model, International journal for numerical methods in fluids, 56 (2008), pp. 899–925.Google Scholar
[12]Godel, N., Schomann, S., Warburton, T., and Clemens, M., GPU accelerated Adams-Bashforth multirate discontinuous Galerkin FEM simulation of high-frequency electromagnetic fields, Magnetics, IEEE Transactions on, 46 (2010), pp. 2735–2738.Google Scholar
[13]Hesthaven, J. and Warburton, T., Nodal discontinuous Galerkin methods: algorithms, analysis, and applications, vol. 54, Springer Verlag, 2008.Google Scholar
[14]Johnson, C. and Pitkäranta, J., An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation., Math. Comput., 46 (1986), pp. 1–26.Google Scholar
[15]Klockner, A., High-performance high-order simulation of wave and plasma phenomena, PhD thesis, Brown University, 2010.Google Scholar
[16]Klockner, A., Warburton, T., Bridge, J., and Hesthaven, J., Nodal discontinuous Galerkin methods on graphics processors, Journal of Computational Physics, 228 (2009), pp. 7863–7882.CrossRefGoogle Scholar
[17]Kubatko, E., Westerink, J., and Dawson, C., hp Discontinuous Galerkin methods for advection dominated problems in shallow water flow, Computer Methods in Applied Mechanics and Engineering, 196 (2006), pp. 437–451.Google Scholar
[18]LeVeque, R., George, D., and Berger, M., Tsunami modelling with adaptively refined finite volume methods, Acta Numerica, 20 (2011), pp. 211–289.Google Scholar
[19]Medina, D. and Warburton, T., OCCA: A unified approach to multi-threading languages, white paper, (2014).Google Scholar
[20]Medina, D., Wilcox, L., and Warburton, T., OCCA 2.0: Extensible multi-threading programming API, 2014.Google Scholar
[21]Merrill, D., Finite difference and pseudospectral methods applied to the shallow water equations in spherical coordinates, PhD thesis, University of Colorado, 1997.Google Scholar
[22]Navon, I., A review of finite-element methods for solving the shallow-water equations, Schrefler, B. and Zienkiewicz, OC, ed., Computer Modelling in Ocean Engineering, Balkeman, Rotterdam, (1988), pp. 273–278.Google Scholar
[23]Shu, C.-W., TVB uniformly high-order schemes for conservation laws, Mathematics of Computation, 49 (1987), pp. 105–121.Google Scholar
[24]Thacker, W. C., Some exact solutions to the nonlinear shallow-water wave equations, Journal of Fluid Mechanics, 107 (1981), pp. 499–508.CrossRefGoogle Scholar
[25]Wen-mei, W. H., GPU Computing GEMs Jade Edition, Morgan Kaufmann, 2011.Google Scholar
[26]Xing, Y. and Shu, C., High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms, Journal of Computational Physics, 214 (2006), pp. 567–598.Google Scholar
[27]Xing, Y., Zhang, X., and Shu, C., Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations, Advances in Water Resources, 33 (2010), pp. 1476–1493.Google Scholar
[28]Zhang, X. and Shu, C., On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes, Journal of Computational Physics, 229 (2010), pp. 8918–8934.Google Scholar