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Efficient, high accuracy ADER-WENO schemes for hydrodynamics and divergence-free magnetohydrodynamics. (English) Zbl 1275.76169
Summary: The present paper introduces a class of finite volume schemes of increasing order of accuracy in space and time for hyperbolic systems that are in conservation form. The methods are specially suited for efficient implementation on structured meshes. The hyperbolic system is required to be non-stiff. This paper specifically focuses on Euler system that is used for modeling the flow of neutral fluids and the divergence-free, ideal magnetohydrodynamics (MHD) system that is used for large scale modeling of ionized plasmas.
Efficient techniques for weighted essentially non-oscillatory (WENO) interpolation have been developed for finite volume reconstruction on structured meshes. We have shown that the most elegant and compact formulation of WENO reconstruction obtains when the interpolating functions are expressed in modal space. Explicit formulae have been provided for schemes having up to fourth order of spatial accuracy. Divergence-free evolution of magnetic fields requires the magnetic field components and their moments to be defined in the zone faces. We draw on a reconstruction strategy developed recently by the first author to show that a high order specification of the magnetic field components in zone-faces naturally furnishes an appropriately high order representation of the magnetic field within the zone.
We also present a new formulation of the ADER (for Arbitrary Derivative Riemann Problem) schemes that relies on a local continuous space-time Galerkin formulation instead of the usual Cauchy-Kovalewski procedure. We call such schemes ADER-CG and show that a very elegant and compact formulation results when the scheme is formulated in modal space. Explicit formulae have been provided on structured meshes for ADER-CG schemes in three dimensions for all orders of accuracy that extend up to fourth order. Such ADER schemes have been used to temporally evolve the WENO-based spatial reconstruction. The resulting ADER-WENO schemes provide temporal accuracy that matches the spatial accuracy of the underlying WENO reconstruction.
In this paper we have also provided a point-wise description of ADER-WENO schemes for divergence-free MHD in a fashion that facilitates computer implementation. The schemes reported here have all been implemented in the RIEMANN framework for computational astrophysics. All the methods presented have a one-step update, making them low-storage alternatives to the usual Runge-Kutta time-discretization. Their one-step update also makes them suitable building blocks for adaptive mesh refinement (AMR) calculations.
We demonstrate that the ADER-WENO meet their design accuracies. Several stringent test problems of Euler flows and MHD flows are presented in one, two and three dimensions. Many of our test problems involve near infinite shocks in multiple dimensions and the higher order schemes are shown to perform very robustly and accurately under all conditions. It is shown that the increasing computational complexity with increasing order is handily offset by the increased accuracy of the scheme. The resulting ADER-WENO schemes are, therefore, very worthy alternatives to the standard second order schemes for compressible Euler and MHD flow.

76M25 Other numerical methods (fluid mechanics) (MSC2010)
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
76W05 Magnetohydrodynamics and electrohydrodynamics
Full Text: DOI
[1] Balsara, D.S., Linearized formulation of the Riemann problem for adiabatic and isothermal magnetohydrodynamics, Astrophysical journal, Suppl. 116, 119, (1998)
[2] Balsara, D.S., Total variation diminishing algorithm for adiabatic and isothermal magnetohydrodynamics, Astrophysical journal, Suppl. 116, 133, (1998)
[3] Balsara, D.S., Divergence-free adaptive mesh refinement for magnetohydrodynamics, Journal of computational physics, 174, 614-648, (2001) · Zbl 1157.76369
[4] Balsara, D.S., Total variation diminishing scheme for relativistic magneto-hydrodynamics, Astrophysical journal, Suppl. 132, 83, (2001)
[5] Balsara, D.S., Second-order-accurate schemes for magnetohydrodynamics with divergence-free reconstruction, Astrophysical journal, Suppl. 151, 149-184, (2004)
[6] Balsara, D.S., Divergence-free reconstruction of magnetic fields and WENO schemes for magnetohydrodynamics submitted, Journal of computational physics, (2008)
[7] 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
[8] Balsara, D.S.; Dumbser, M.; Tilley, D.A., ADER schemes for problems with stiff source terms – method and application to two-fluid magnetohydrodynamics, in preparation, Journal computational physics, (2008)
[9] Balsara, D.S.; Shu, C.-W., Monotonicity preserving weighted non-oscillatory schemes wirh increasingly high order of accuracy, Journal of computational physics, 160, 405-452, (2000) · Zbl 0961.65078
[10] Balsara, D.S.; Spicer, D.S., Maintaining pressure positivity in magnetohydrodynamic simulations, Journal of computational physics, 148, 133-148, (1999) · Zbl 0930.76050
[11] 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
[12] T.J.Barth, P.O. Frederickson, Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction, AIAA Paper no. 90-0013, 28th Aerospace Sciences Meeting, January 1990.
[13] Berger, M.; Colella, P., Local adaptive mesh refinement for shock hydrodynamics, Journal of computational physics, 82, 64-84, (1989) · Zbl 0665.76070
[14] Brackbill, J.U.; Barnes, D.C., The effect of nonzero δB on the numerical solution of the magnetohydrodynamic equations, Journal of computational physics, 35, 426-430, (1980) · Zbl 0429.76079
[15] Brecht, S.H.; Lyon, J.G.; Fedder, J.A.; Hain, K., A simulation study of east-west IMF effects on the magnetosphere, Geophysical reserach letter, 8, 397, (1981)
[16] 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
[17] Colella, P., Multidimensional upwind methods for hyperbolic conservation laws, Journal of computational physics, 87, 171-200, (1990) · Zbl 0694.65041
[18] Colella, P.; Woodward, P., The piecewise parabolic method (PPM) for gas-dynamical simulations, Journal of computational physics, 54, 174-201, (1984) · Zbl 0531.76082
[19] 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
[20] Dai, W.; Woodward, P.R., An approximate Riemann solver for ideal magnetohydrodynamics, Journal of computational physics, 111, 354-372, (1994) · Zbl 0797.76052
[21] 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)
[22] Dedner, A.; Kemm, F.; Kröner, D.; Munz, C.-D.; Schnitzer, T.; Wesenberg, M., Hyperbolic divergence cleaning for MHD equations, Journsl of computational physics, 175, 645-673, (2002) · Zbl 1059.76040
[23] Del Zanna, L.; Velli, M.; Londrillo, P., Parametric decay of circularly polarized Alfvén waves: multidimensional simulations in periodic and open domains, Astronomy and astrophysics, 367, 705-718, (2001)
[24] DeVore, C.R., Flux-corrected transport techniques for multidimensional compressible magnetohydrodynamics, Journal of computational physics, 92, 142-160, (1991) · Zbl 0716.76056
[25] Dubiner, M., Spectral methods on triangles and other domains, Journal of scientific computing, 6, 345-390, (1991) · Zbl 0742.76059
[26] Dumbser, M.; Balsara, D.; 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
[27] 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
[28] Dumbser, M.; Käser, M., Arbitary high order non-oscillatory finite volume schmes on unstuructured meshes for linear hyperbolic systems, Journal of computational physcis, 221, 693-723, (2007) · Zbl 1110.65077
[29] 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
[30] M. Dumbser, D. Balsara, J.M. Powers, ADER-DG schemes for stiff source terms: applications to reactive flow, Journal of Computational Physics, in preparation.
[31] 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
[32] 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)
[33] Friedrich, O., Weighted essentially non-oscillatory schemes for the interpolation of Mean values on unstructured grids, Journal of computational physics, 144, 194-212, (1998) · Zbl 1392.76048
[34] Goldstein, M.L., An instability of finite amplitude circularly polarized alfven waves, Astrophysical journal, 219, 700, (1978)
[35] Hanawa, T.; Mikami, H.; Matsumoto, T., Improving shock irregularities based on the characteristics of the MHD equations, Journal of computational physics, 227, 7952-7976, (2008) · Zbl 1268.76038
[36] 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
[37] Hu, C.; Shu, C.W., Weighted essentially non-oscillatory schemes on triangular meshes, Journal of computational physics, 150, 97-127, (1999) · Zbl 0926.65090
[38] Jayanti, V.; Hollweg, J.V., On the dispersion relations for parametric instabilities of parallel-propagating Alfvén waves, Journal of geophysical research, 98, 13247-13252, (1993)
[39] Jeffrey, A.; Taniuti, T., Non-linear wave propagation, (1964), Academic Press · Zbl 0117.21103
[40] Jiang, G.-S.; Shu, C.-W., Efficient implementation of weighted ENO schemes, Journal of computational physics, 126, 202-228, (1996) · Zbl 0877.65065
[41] Kim, J.S.; Ryu, D.; Jones, T.W.; Hong, S.S., A multidimensional code for isothermal magnetohydrodynamic flows in astrophysics, Astrophysical journal, 514, 506-519, (1999)
[42] Liu, X.-D.; Osher, S.; Chan, T., Weighted essentially non-oscillatory schemes, Journal of computational physics, 115, 200-212, (1994) · Zbl 0811.65076
[43] 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
[44] Pandolfini, M.; D’Ambrosio, D., Numerical instabilities in upwind methods: analysis and cures for the “carbuncle” phenomenon, J journal of computational physics, 166, 271, (2001) · Zbl 0990.76051
[45] K.G. Powell, An approximate Riemann solver for MHD (That actually works in more than one dimension, ICASE Report 94-24.
[46] Quirk, J.J., A contribution to the great Riemann solver debate, International journal numerical methods in fluids, 18, 555, (1994) · Zbl 0794.76061
[47] Roe, P.L.; Balsara, D.S., Notes on the eigensystem of magnetohydrodynamics, SIAM journal of applied mathematics, 56, 57, (1996) · Zbl 0845.35092
[48] Ryu, D.; Jones, T.W., Numerical MHD in astrophysics: algorithm and tests for one-dimensional flow, Astrophysical journal, 442, 228, (1995)
[49] Ryu, D.; Miniati, F.; Jones, T.W.; Frank, A., A divergence-free upwind code for multidimensional magnetohydrodynamic flows, Astrophysical journal, 509, 244-255, (1998)
[50] 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
[51] 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
[52] 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
[53] 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
[54] Taube, A.; Dumbser, M.; Balsara, D.S.; Munz, C.D., Arbitrary high order discontinuous Galerkin schemes fort the magnetohydrodynamic equations, Journal of scientific computing, 30, 441-464, (2007) · Zbl 1176.76075
[55] 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
[56] 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
[57] 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
[58] 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
[59] 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
[60] Zhang, Y.-T.; Shu, C.-W., High order WENO schemes for hamilton – jacobi equations on triangular meshes, SIAM journal on scientific computing, 24, 1005-1030, (2003) · Zbl 1034.65051
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