×

zbMATH — the first resource for mathematics

Efficient implementation of ADER schemes for Euler and magnetohydrodynamical flows on structured meshes – speed comparisons with Runge-Kutta methods. (English) Zbl 1291.76237
Summary: ADER (Arbitrary DERivative in space and time) methods for the time-evolution of hyperbolic conservation laws have recently generated a fair bit of interest. The ADER time update can be carried out in a single step, which is desirable in many applications. However, prior papers have focused on the theory while downplaying implementation details. The purpose of the present paper is to make ADER schemes accessible by providing two useful formulations of the method as well as their implementation details on three-dimensional structured meshes. We therefore provide a detailed formulation of ADER schemes for conservation laws with non-stiff source terms in nodal as well as modal space along with useful implementation-related details. A good implementation of ADER requires a fast method for transcribing from nodal to modal space and vice versa and we provide innovative transcription strategies that are computationally efficient. We also provide details for the efficient use of ADER schemes in obtaining the numerical flux for conservation laws as well as electric fields for divergence-free magnetohydrodynamics (MHD). An efficient WENO-based strategy for obtaining zone-averaged magnetic fields from face-centered magnetic fields in MHD is also presented. Several explicit formulae have been provided in all instances for ADER schemes spanning second to fourth orders.
The schemes catalogued here have been implemented in the first author’s RIEMANN code. The speed of ADER schemes is shown to be almost twice as fast as that of strong stability preserving Runge-Kutta time stepping schemes for all the orders of accuracy that we tested. The modal and nodal ADER schemes have speeds that are within ten percent of each other. When a linearized Riemann solver is used, the third order ADER schemes are half as fast as the second order ADER schemes and the fourth order ADER schemes are a third as fast as the third order ADER schemes. The third order ADER scheme, either with an HLL or linearized Riemann solver, represents an excellent upgrade path for scientists and engineers who are working with a second order Runge-Kutta based total variation diminishing (TVD) scheme. Several stringent test problems have been catalogued.

MSC:
76M25 Other numerical methods (fluid mechanics) (MSC2010)
76W05 Magnetohydrodynamics and electrohydrodynamics
65Y20 Complexity and performance of numerical algorithms
65L05 Numerical methods for initial value problems involving ordinary differential equations
Software:
HLLE; RIEMANN
PDF BibTeX Cite
Full Text: DOI
References:
[1] Balsara, D. S., Linearized formulation of the Riemann problem for adiabatic and isothermal magnetohydrodynamics, Astrophysical Journal Supplement, 116, 119, (1998)
[2] Balsara, D. S., Total variation diminishing algorithm for adiabatic and isothermal magnetohydrodynamics, Astrophysical Journal Supplement, 116, 133, (1998)
[3] Balsara, D. S.; Spicer, D. S., Maintaining pressure positivity in magnetohydrodynamic simulations, Journal of Computational Physics, 148, 133-148, (1999) · Zbl 0930.76050
[4] Balsara, D. S.; Spicer, D. S., A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations, Journal of Computational Physics, 149, 270-292, (1999) · Zbl 0936.76051
[5] Balsara, D. S.; Shu, C.-W., Monotonicity preserving weighted non-oscillatory schemes with increasingly high order of accuracy, Journal of Computational Physics, 160, 405-452, (2000) · Zbl 0961.65078
[6] Balsara, D. S., Divergence-free adaptive mesh refinement for magnetohydrodynamics, Journal of Computational Physics, 174, 614-648, (2001) · Zbl 1157.76369
[7] Balsara, D. S., Total variation diminishing scheme for relativistic magneto-hydrodynamics, Astrophysical Journal Supplement, 132, 83, (2001)
[8] Balsara, D. S., Second-order-accurate schemes for magnetohydrodynamics with divergence-free reconstruction, Astrophysical Journal Supplement, 151, 149-184, (2004)
[9] Balsara, D. S.; Altmann, C.; Munz, C. D.; Dumbser, M., A sub-cell based indicator for troubled zones in RKDG schemes and a novel class oh hybrid RKDG+HWENO schemes, Journal of Computational Physics, 226, 586-620, (2007) · Zbl 1124.65072
[10] Balsara, D. S., Divergence-free reconstruction of magnetic fields and WENO schemes for magnetohydrodynamics, Journal of Computational Physics, 228, 5040, (2009) · Zbl 1280.76030
[11] Balsara, D. S.; Rumpf, T.; Dumbser, M.; Munz, C.-D., Efficient, high-accuracy ADER-WENO schemes for hydrodynamics and divergence-free magnetohydrodynamics, Journal Computational Physics, 228, 2480, (2009) · Zbl 1275.76169
[12] Balsara, D. S., Multidimensional extension of the HLL Riemann solver; application to Euler and magnetohydrodynamical flows, Journal of Computational Physics, 229, 1970-1993, (2010) · Zbl 1303.76140
[13] D.S. Balsara, A two-dimensional HLLC Riemann solver: application to Euler and MHD flows, Journal Computational Physics, in press. · Zbl 1284.76261
[14] D.S. Balsara, Self-adjusting, positivity preserving schemes for hydrodynamics and MHD, Journal Computational Physics, in press.
[15] T.J.Barth, P.O. Frederickson, Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction, in: AIAA Paper no. 90-0013, 28th Aerospace Sciences Meeting, January, 1990.
[16] Ben-Artzi, M.; Falcovitz, J., A second-order Godunov-type scheme for compressible fluid dynamics, Journal of Computational Physics, 55, 1-32, (1984) · Zbl 0535.76070
[17] Berger, M.; Colella, P., Local adaptive mesh refinement for shock hydrodynamics, Journal of Computational Physics, 82, 64-84, (1989) · Zbl 0665.76070
[18] Brackbill, J. U.; Barnes, D. C., The effect of nonzero ∇·B on the numerical solution of the magnetohydrodynamic equations, Journal of Computational Physics, 35, 426-430, (1980) · Zbl 0429.76079
[19] Brecht, S. H.; Lyon, J. G.; Fedder, J. A.; Hain, K., A simulation study of east-west IMF effects on the magnetosphere, Geophysical Reserach Letters, 8, 397, (1981)
[20] Brio, M.; Wu, C.-C., An upwind differencing scheme for the equations of MHD, Journal of Computational Physics, 75, 400, (1988) · Zbl 0637.76125
[21] Castro, C. E.; Toro, E. F., Solvers for the high-order Riemann problem for hyperbolic balance laws, Journal of Computational Physics, 227, 2481-2513, (2008) · Zbl 1148.65066
[22] Cockburn, B.; Shu, C.-W., The runge – kutta discontinuous Galerkin method for conservation laws V, Journal of Computational Physics, 141, 199-224, (1998) · Zbl 0920.65059
[23] Colella, P.; Sekora, M. D., A limiter for PPM that preserves accuracy at smooth extrema, Journal of Computational Physics, 227, 7069, (2008) · Zbl 1152.65090
[24] Crockett, R. K.; Colella, P.; Fisher, R. T.; Klein, R. I.; McKee, C. F., An unsplit cell-centered Godunov method for ideal MHD, Journal of Computational Physics, 203, 422, (2005) · Zbl 1143.76599
[25] Dai, W.; Woodward, P. R., An approximate Riemann solver for ideal magnetohydrodynamics, Journal of Computational Physics, 111, 354-372, (1994) · Zbl 0797.76052
[26] Dai, W.; Woodward, P. R., On the divergence-free condition and conservation laws in numerical simulations for supersonic magnetohydrodynamic flows, Astrophysical Journal, 494, 317-335, (1998)
[27] Dedner, A.; Kemm, F.; Kröner, D.; Munz, C.-D.; Schnitzer, T.; Wesenberg, M., Hyperbolic divergence cleaning for MHD equations, Journal of Computational Physics, 175, 645-673, (2002) · Zbl 1059.76040
[28] DeVore, C. R., Flux-corrected transport techniques for multidimensional compressible magnetohydrodynamics, Journal of Computational Physics, 92, 142-160, (1991) · Zbl 0716.76056
[29] Dubiner, M., Spectral methods on triangles and other domains, Journal of Scientific Computing, 6, 345-390, (1991) · Zbl 0742.76059
[30] Dumbser, M.; Käser, M., Arbitary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems, Journal of Computational Physics, 221, 693-723, (2007) · Zbl 1110.65077
[31] Dumbser, M.; Käser, M.; Titarev, V. A.; Toro, E. F., Quadrature-free non- oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems, Journal of Computational Physics, 226, 204-243, (2007) · Zbl 1124.65074
[32] Dumbser, M.; Balsara, D. S.; Toro, E. F.; Munz, C. D., A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes, Journal of Computational Physics, 227, 8209-8253, (2008) · Zbl 1147.65075
[33] Dumbser, M.; Enaux, C.; Toro, E. F., Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws, Journal of Computational Physics, 227, 3971-4001, (2008) · Zbl 1142.65070
[34] Dumbser, M.; Balsara, D. S., High-order unstructured one-step PNPM schemes for the viscous and resistive MHD equations, Computer Modeling for Engineers and Scientists, 54, 3, 301-334, (2010) · Zbl 1231.76345
[35] Dumbser, M.; Schwartzkopff, T.; Munz, C.-D., Arbitrary high order finite volume schemes for linear wave propagation, (Krause, E.; Shokin, Y.; Resch, M.; Shokina, N., Computational Science and High Performance Computing II, Series: Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM), (2006), Springer Verlag Berlin Heidelberg), 129-144
[36] Evans, C. R.; Hawley, J. F., Simulation of magnetohydrodynamic flows: a constrained transport method, Astrophysics Journal, 332, 659, (1989)
[37] Einfeldt, B.; Munz, C.-D.; Roe, P. L.; Sjogreen, B., On Godunov type methods near low densities, Journal of Computational Physics, 92, 273-295, (1991) · Zbl 0709.76102
[38] Falle, S. A.E. G.; Komissarov, S. S.; Joarder, P., A multidimensional upwind scheme for magnetohydrodynamics, Monthly Notices of the Royal Astronomical Society, 297, 265-277, (1998)
[39] Fuchs, F.; Mishra, S.; Risebro, N. H., Splitting based finite volume schemes for the ideal MHD equations, Journal of Computational Physics, 228, 3, 641-660, (2009) · Zbl 1259.76021
[40] Gardiner, T.; Stone, J. M., An unsplit Godunov method for ideal MHD via constrained transport, Journal of Computational Physics, 205, 509, (2005) · Zbl 1087.76536
[41] Gottlieb, S.; Shu, C., Total-variation-diminishing runge – kutta schemes, Mathematics of Computation, 67, 73-85, (1998) · Zbl 0897.65058
[42] Harten, A.; Lax, P. D.; van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Review, 25, 289-315, (1983) · Zbl 0565.65051
[43] Harten, A.; Hyman, J., Self-adjusting grid methods for one-dimensional hyperbolic conservation laws, Journal of Computational Physics, 50, 297-322, (1983) · Zbl 0565.65049
[44] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S., Uniformly high order essentially non-oscillatory schemes III, Journal of Computational Physics, 71, 231-303, (1987) · Zbl 0652.65067
[45] Horn, M. K., Fourth and fifth order, scaled runge – kutta algorithms for treating dense output, SIAM Journal of Numerical Analysis, 20, 3, 588, (1983) · Zbl 0511.65048
[46] Jeffrey, A.; Taniuti, T., Non-linear wave propagation, (1964), Academic Press · Zbl 0117.21103
[47] Jiang, G.-S.; Shu, C.-W., Efficient implementation of weighted ENO schemes, Journal of Computational Physics, 126, 202-228, (1996) · Zbl 0877.65065
[48] Liu, X.-D.; Osher, S.; Chan, T., Weighted essentially non-oscillatory schemes, Journal of Computational Physics, 115, 200-212, (1994) · Zbl 0811.65076
[49] Londrillo, P.; DelZanna, L., On the divergence-free condition in Godunov-type schemes for ideal magnetohydrodynamics: the upwind constrained transport method, Journal of Computational Physics, 195, 17-48, (2004) · Zbl 1087.76074
[50] K.G. Powell, An approximate Riemann solver for MHD (That actually works in more than one dimension), ICASE Report 94-24.
[51] Qiu, J.; Shu, C.-W., Hermite WENO schemes and their application as limiters for runge – kutta discontinuous Galerkin schemes: the one dimensional case, Journal of Computational Physics, 193, 115, (2004) · Zbl 1039.65068
[52] Roe, P. L., Approximate Riemann solvers, parameter vectors, and difference schemes, Journal of Computational Physics, 43, 357-372, (1981) · Zbl 0474.65066
[53] Roe, P. L.; Balsara, D. S., Notes on the eigensystem of magnetohydrodynamics, SIAM Journal of applied Mathematics, 56, 57, (1996) · Zbl 0845.35092
[54] Ryu, D.; Jones, T. W., Numerical MHD in astrophysics: algorithm and tests for one-dimensional flow, Astrophysical Journal, 442, 228, (1995)
[55] Ryu, D.; Miniati, F.; Jones, T. W.; Frank, A., A divergence-free upwind code for multidimensional magnetohydrodynamic flows, Astrophysical Journal, 509, 244-255, (1998)
[56] Schwartzkopff, T.; Dumbser, M.; Munz, C.-D., Fast high order ADER schemes for linear hyperbolic equations, Journal of Computational Physics, 197, 532, (2004) · Zbl 1052.65078
[57] Shu, C.-W.; Osher, S. J., Efficient implementation of essentially non-oscillatory shock capturing schemes, Journal of Computational Physics, 77, 439-471, (1988) · Zbl 0653.65072
[58] Shu, C.-W.; Osher, S. J., Efficient implementation of essentially non-oscillatory shock capturing schemes II, Journal of Computational Physics, 83, 32-78, (1989) · Zbl 0674.65061
[59] Spiteri, R. J.; Ruuth, S. J., A new class of optimal high-order strong-stability- preserving time-stepping schemes, SIAM Journal of Numerical Analysis, 40, 469-491, (2002) · Zbl 1020.65064
[60] Spiteri, R. J.; Ruuth, S. J., Non-linear evolution using optimal fourth-order strong- stability-preserving runge – kutta methods, Mathematics and Computers in Simulation, 62, 125-135, (2003) · Zbl 1015.65031
[61] Suresh, A.; Huynh, H. T., Accurate monotonicity preserving scheme with runge – kutta time-stepping, Journal of Computational Physics, 136, 83-99, (1997) · Zbl 0886.65099
[62] Taube, A.; Dumbser, M.; Balsara, D. S.; Munz, C. D., Arbitrary high order discontinuous Galerkin schemes for the magnetohydrodynamic equations, Journal of Scientific Computing, 30, 441-464, (2007) · Zbl 1176.76075
[63] Titarev, V. A.; Toro, E. F., ADER: arbitrary high order Godunov approach, Journal of Scientific Computing, 17, 1-4, 609-618, (2002) · Zbl 1024.76028
[64] Titarev, V. A.; Toro, E. F., ADER schemes for three-dimensional nonlinear hyperbolic systems, Journal of Computational Physics, 204, 715-736, (2005) · Zbl 1060.65641
[65] Toro, E. F.; Millington, R. C.; M Nejad, L. A., Towards very high-order Godunov schemes, (Toro, E. F., Godunov Methods: Theory and Applications, (2001), Kluwer Academic/Plenum Publishers), 905-938 · Zbl 0989.65094
[66] Toro, E. F.; Titarev, V. A., Solution of the generalized Riemann problem for advection-reaction equations, Proceedings of the Royal Society of London, Series A, 458, 271-281, (2002) · Zbl 1019.35061
[67] Toro, E. F.; Titarev, V. A., Derivative Riemann solvers for systems of conservation laws and ADER methods, Journal of Computational Physics, 212, 1, 150-165, (2006) · Zbl 1087.65590
[68] Ustyugov, S. D.; Popov, M. V.; Kritsuk, A. G.; Norman, M. L., Piecewise parabolic method on a local stencil for supersonic turbulence simulations, Journal of Computational Physics, 228, 7614, (2009) · Zbl 1391.76583
[69] van Leer, B., Towards the ultimate conservative difference scheme V. A second order sequel to godunov’s method, Journal of Computational Physics, 32, 101-136, (1979) · Zbl 1364.65223
[70] Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, Journal of Computational Physics, 54, 115-173, (1984) · Zbl 0573.76057
[71] Xu, Z.; Liu, Y.; Shu, C.-W., Hierarchical reconstruction for discontinuous Galerkin methods on unstructured grids with a WENO-type linear reconstruction and partial neighboring cells, Journal of Computational Physics, 228, 2194, (2009) · Zbl 1165.65392
[72] Xu, Z.; Liu, Y.; Shu, C.-W., Hierarchical reconstruction for spectral volume method on unstructured grids, Journal of Computational Physics, 228, 5787, (2009) · Zbl 1169.65339
[73] Yee, K. S., Numerical solution of initial boundary value problems involving Maxwell equation in an isotropic media, IEEE Transactions on Antenna Propagation, 14, 302, (1966) · Zbl 1155.78304
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.