×

zbMATH — the first resource for mathematics

Von Neumann stability analysis of globally divergence-free RKDG schemes for the induction equation using multidimensional Riemann solvers. (English) Zbl 1375.76212
Summary: In this paper, we focus on the numerical solution of the induction equation using Runge-Kutta Discontinuous Galerkin (RKDG)-like schemes that are globally divergence-free. The induction equation plays a role in numerical MHD and other systems like it. It ensures that the magnetic field evolves in a divergence-free fashion; and that same property is shared by the numerical schemes presented here. The algorithms presented here are based on a novel DG-like method as it applies to the magnetic field components in the faces of a mesh. (I.e., this is not a conventional DG algorithm for conservation laws.) The other two novel building blocks of the method include divergence-free reconstruction of the magnetic field and multidimensional Riemann solvers; both of which have been developed in recent years by the first author. Since the method is linear, a von Neumann stability analysis is carried out in two-dimensions to understand its stability properties. The von Neumann stability analysis that we develop in this paper relies on transcribing from a modal to a nodal DG formulation in order to develop discrete evolutionary equations for the nodal values. These are then coupled to a suitable Runge-Kutta timestepping strategy so that one can analyze the stability of the entire scheme which is suitably high order in space and time. We show that our scheme permits CFL numbers that are comparable to those of traditional RKDG schemes. We also analyze the wave propagation characteristics of the method and show that with increasing order of accuracy the wave propagation becomes more isotropic and free of dissipation for a larger range of long wavelength modes. This makes a strong case for investing in higher order methods. We also use the von Neumann stability analysis to show that the divergence-free reconstruction and multidimensional Riemann solvers are essential algorithmic ingredients of a globally divergence-free RKDG-like scheme. Numerical accuracy analyses of the RKDG-like schemes are presented and compared with the accuracy of PNPM schemes. It is found that PNPM retrieve much of the accuracy of the RKDG-like schemes while permitting a larger CFL number.

MSC:
76W05 Magnetohydrodynamics and electrohydrodynamics
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
Software:
RIEMANN
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Balsara, D. S., Linearized formulation of the Riemann problem for adiabatic and isothermal magnetohydrodynamics, Astrophys. J. Suppl. Ser., 116, 119, (1998)
[2] Balsara, D. S., Total variation diminishing algorithm for adiabatic and isothermal magnetohydrodynamics, Astrophys. J. Suppl. Ser., 116, 133, (1998)
[3] Balsara, D. S.; Spicer, D. S., Maintaining pressure positivity in magnetohydrodynamic simulations, J. Comput. Phys., 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, J. Comput. Phys., 149, 270-292, (1999) · Zbl 0936.76051
[5] Balsara, D. S., Divergence-free adaptive mesh refinement for magnetohydrodynamics, J. Comput. Phys., 174, 614-648, (2001) · Zbl 1157.76369
[6] Balsara, D. S., Second-order-accurate schemes for magnetohydrodynamics with divergence-free reconstruction, Astrophys. J. Suppl. Ser., 151, 149-184, (2004)
[7] Balsara, D. S., Divergence-free reconstruction of magnetic fields and WENO schemes for magnetohydrodynamics, J. Comput. Phys., 228, 5040-5056, (2009) · Zbl 1280.76030
[8] Balsara, D. S.; Kim, J. S., An intercomparison between divergence-cleaning and staggered mesh formulations for numerical magnetohydrodynamics, Astrophys. J., 602, 1079, (2004)
[9] Balsara, D. S.; Shu, C.-W., Monotonicity preserving weighted non-oscillatory schemes with increasingly high order of accuracy, J. Comput. Phys., 160, 405-452, (2000) · Zbl 0961.65078
[10] 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, J. Comput. Phys., 226, 586-620, (2007) · Zbl 1124.65072
[11] Balsara, D. S.; Rumpf, T.; Dumbser, M.; Munz, C.-D., Efficient, high accuracy ADER-WENO schemes for hydrodynamics and divergence-free magnetohydrodynamics, J. Comput. Phys., 228, 2480-2516, (2009) · Zbl 1275.76169
[12] Balsara, D. S.; Dumbser, M.; Meyer, C.; Du, H.; Xu, Z., Efficient implementation of ADER schemes for Euler and magnetohydrodynamic flow on structured meshes - comparison with Runge-Kutta methods, J. Comput. Phys., 235, 934-969, (2013) · Zbl 1291.76237
[13] Balsara, D. S., Multidimensional HLLE Riemann solver; application to Euler and magnetohydrodynamic flows, J. Comput. Phys., 229, 1970-1993, (2010) · Zbl 1303.76140
[14] Balsara, D. S., A two-dimensional HLLC Riemann solver for conservation laws: application to Euler and magnetohydrodynamic flows, J. Comput. Phys., 231, 7476-7503, (2012) · Zbl 1284.76261
[15] Balsara, D. S.; Dumbser, M.; Abgrall, R., Multidimensional HLL and HLLC Riemann solvers for unstructured meshes - with application to Euler and MHD flows, J. Comput. Phys., 261, 172-208, (2014) · Zbl 1349.76426
[16] Balsara, D. S., Multidimensional Riemann problem with self-similar internal structure - part I - application to hyperbolic conservation laws on structured meshes, J. Comput. Phys., 277, 163-200, (2014) · Zbl 1349.76303
[17] Balsara, D. S.; Dumbser, M., Multidimensional Riemann problem with self-similar internal structure - part II - application to hyperbolic conservation laws on unstructured meshes, J. Comput. Phys., 287, 269-292, (2015) · Zbl 1351.76091
[18] Balsara, D. S., Self-adjusting, positivity preserving high order schemes for hydrodynamics and magnetohydrodynamics, J. Comput. Phys., 231, 7504-7517, (2012)
[19] Balsara, D. S., Three dimensional HLL Riemann solver for structured meshes; application to Euler and MHD flow, J. Comput. Phys., 295, 1-23, (2015) · Zbl 1349.76584
[20] Balsara, D. S.; Dumbser, M., Divergence-free MHD on unstructured meshes using high order finite volume schemes based on multidimensional Riemann solvers, J. Comput. Phys., 299, 687-715, (2015) · Zbl 1351.76092
[21] Balsara, D. S.; Vides, J.; Gurski, K.; Nkonga, B.; Dumbser, M.; Garain, S.; Audit, E., A two-dimensional Riemann solver with self-similar sub-structure - alternative formulation based on least squares projection, J. Comput. Phys., 304, 138-161, (2016) · Zbl 1349.76157
[22] Balsara, D. S.; Nkonga, B., Formulating multidimensional Riemann solvers in similarity variables - part III: a multidimensional analogue of the HLLEM Riemann solver for conservative hyperbolic systems, J. Comput. Phys., (2017), submitted for publication
[23] Brackbill, J. U.; Barnes, D. C., The effect of nonzero \(\operatorname{\nabla} \cdot \mathbf{B}\) on the numerical solution of the magnetohydrodynamic equations, J. Comput. Phys., 35, 426-430, (1980) · Zbl 0429.76079
[24] Brackbill, J., Fluid modelling of magnetized plasmas, Space Sci. Rev., 42, 153, (1985)
[25] Cheng, Y.; Li, F.; Qiu, J.; Xu, L., Positivity-preserving DG and central DG methods for ideal MHD equations, J. Comput. Phys., 238, 255, (2013) · Zbl 1286.76162
[26] Christlieb, A. J.; Rossmanith, J. A.; Tang, Q., Finite difference weighted essentially non-oscillatory schemes with constrained transport for ideal magnetohydrodynamics, J. Comput. Phys., 268, 302-325, (2014) · Zbl 1349.76442
[27] Cockburn, B.; Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Math. Comput., 52, 411-435, (1989) · Zbl 0662.65083
[28] Cockburn, B.; Hou, S.; Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case, J. Comput. Phys., 54, 545-581, (1990) · Zbl 0695.65066
[29] Cockburn, B.; Shu, C.-W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J. Comput. Phys., 141, 199-224, (1998) · Zbl 0920.65059
[30] Cockburn, B.; Karniadakis, G.; Shu, C.-W., The development of discontinuous Galerkin methods, (Cockburn, B.; Karniadakis, G.; Shu, C.-W., Discontinuous Galerkin Methods: Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, vol. 11, (2000), Springer), 3-50 · Zbl 0989.76045
[31] Cockburn, B.; Shu, C.-W., Runge-Kutta discontinuous Galerkin methods for convection dominated problems, J. Sci. Comput., 16, 173-261, (2001) · Zbl 1065.76135
[32] Cockburn, B.; Li, F.; Shu, C.-W., Locally divergence-free discontinuous Galerkin method for the Maxwell equations, J. Comput. Phys., 141, 413-442, (2005) · Zbl 1123.76341
[33] Colella, P., Multidimensional upwind methods for conservation laws, J. Comput. Phys., 87, 171, (1990) · Zbl 0694.65041
[34] Dai, W.; Woodward, P. R., On the divergence-free condition and conservationlaws in numerical simulations for supersonic magnetohydrodynamic flows, Astrophys. J., 494, 317-335, (1998)
[35] 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, J. Comput. Phys., 227, 8209-8253, (2008) · Zbl 1147.65075
[36] Dumbser, M.; Zanotti, O.; Hidalgo, A.; Balsara, D. S., ADER-WENO finite volume schemes with space-time adaptive mesh refinement, J. Comput. Phys., 248, 257-286, (2013) · Zbl 1349.76325
[37] Gardiner, T.; Stone, J. M., An unsplit Godunov method for ideal MHD via constrained transport, J. Comput. Phys., 205, 509, (2005) · Zbl 1087.76536
[38] Gottlieb, S., On higher order strong stability preserving Runge-Kutta and multistep time discretizations, J. Sci. Comput., 25, 1/2, 105, (2005) · Zbl 1203.65166
[39] Gottlieb, S.; Shu, C.-W.; Tadmor, E., Strong stability-preserving higher order time discretization methods, SIAM Rev., 43, 1, 89-112, (2001) · Zbl 0967.65098
[40] Li, F.; Xu, L.; Yakovlev, S., Central discontinuous Galerkin methods for ideal MHD equations with the exactly divergence-free magnetic field, J. Comput. Phys., 230, 12, 4828-4847, (2011) · Zbl 1416.76117
[41] Liu, Y.; Shu, C.-W.; Tadmor, E.; Zhang, M., L2 stability analysis of the central discontinuous Galerkin method and comparison between the central and regular discontinuous Galerkin methods, Math. Model. Numer. Anal., 42, 593-607, (2008) · Zbl 1152.65095
[42] Londrillo, P.; DelZanna, L., On the divergence-free condition in Godunov-type schemes for ideal magnetohydrodynamics: the upwind constrained transport method, J. Comput. Phys., 195, 17-48, (2004) · Zbl 1087.76074
[43] Reed, W. H.; Hill, T. R., Triangular mesh methods for the neutron transport equation, (1973), Los Alamos Scientific Laboratory, Tech. Report LA-UR-73-479
[44] Ryu, D.; Miniati, F.; Jones, T. W.; Frank, A., A divergence-free upwind code for multidimensional magnetohydrodynamic flows, Astrophys. J., 509, 244-255, (1998)
[45] Shu, C.-W., Total variation-diminishing time discretizations, SIAM J. Sci. Stat. Comput., 9, 1073-1084, (1988) · Zbl 0662.65081
[46] Shu, C.-W.; Osher, S. J., Efficient implementation of essentially non-oscillatory shock capturing schemes, J. Comput. Phys., 77, 439-471, (1988) · Zbl 0653.65072
[47] Shu, C.-W.; Osher, S. J., Efficient implementation of essentially non-oscillatory shock capturing schemes II, J. Comput. Phys., 83, 32-78, (1989) · Zbl 0674.65061
[48] Spiteri, R. J.; Ruuth, S. J., A new class of optimal high-order strong-stability-preserving time-stepping schemes, SIAM J. Numer. Anal., 40, 469-491, (2002) · Zbl 1020.65064
[49] Taube, A.; Dumbser, M.; Balsara, D. S.; Munz, C. D., Arbitrary high order discontinuous Galerkin schemes for the MHD equations, SIAM J. Sci. Comput., 30, 3, 441-461, (2007) · Zbl 1176.76075
[50] Vides, J.; Nkonga, B.; Audit, E., A simple two-dimensional extension of the HLL Riemann solver for hyperbolic conservation laws, J. Comput. Phys., 280, 1, 643-675, (2015) · Zbl 1349.76403
[51] Xu, Z.; Balsara, D. S.; Du, H., Divergence-free WENO reconstruction-based finite volume scheme for solving ideal MHD equations on triangular meshes, Commun. Comput. Phys., 19, 4, 841-880, (2016) · Zbl 1373.76149
[52] Yang, H.; Li, F., Stability analysis and error estimates of an exactly divergence-free method for the magnetic induction equations, ESAIM: Math. Model. Numer. Anal., 50, 4, 965-993, (2016) · Zbl 1348.78028
[53] Yee, K. S., Numerical solution of initial boundary value problems involving Maxwell equation in an isotropic media, IEEE Trans. Antennas Propag., 14, 302, (1966) · Zbl 1155.78304
[54] Stroud, A. H., Approximate calculation of multiple integrals, (1971), Prentice-Hall Inc. Englewood Cliffs, NJ · Zbl 0379.65013
[55] Zhang, M.; Shu, C.-W., An analysis of and a comparison between the discontinuous Galerkin and the spectral finite volume methods, Comput. Fluids, 34, 581-592, (2005) · Zbl 1138.76391
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.