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Approximate Riemann Solvers and Robust High-Order Finite Volume Schemes for Multi-Dimensional Ideal MHD Equations

Published online by Cambridge University Press:  20 August 2015

Franz Georg Fuchs ,
Andrew D. McMurry ,
Siddhartha Mishra ,
Nils Henrik Risebro  and
Knut Waagan
Franz Georg Fuchs*
Affiliation:
Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern N-0316 Oslo, Norway
Andrew D. McMurry*
Affiliation:
Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern N-0316 Oslo, Norway
Siddhartha Mishra*
Affiliation:
Seminar for Applied Mathematics, D-Math, ETH Zürich, HG G. 57.2, Rämistrasse 101, Zürich-8092, Switzerland
Nils Henrik Risebro*
Affiliation:
Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern N-0316 Oslo, Norway
Knut Waagan*
Affiliation:
Center for Scientific Computation and Mathematical Modeling, The University of Maryland, CSCAMM 4146, CSIC Building #406, Paint Branch Drive College Park, MD 20742-3289, USA
*
Corresponding author.Email:franzf@math.uio.no
Email:a.d.mcmurry@ifi.uio.no
Email:smishra@sam.math.ethz.ch
Email:nilshr@math.uio.no
Email:knutwa@ucar.edu
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Abstract

We design stable and high-order accurate finite volume schemes for the ideal MHD equations in multi-dimensions. We obtain excellent numerical stability due to some new elements in the algorithm. The schemes are based on three- and five-wave approximate Riemann solvers of the HLL-type, with the novelty that we allow a varying normal magnetic field. This is achieved by considering the semi-conservative Godunov-Powell form of the MHD equations. We show that it is important to discretize the Godunov-Powell source term in the right way, and that the HLL-type solvers naturally provide a stable upwind discretization. Second-order versions of the ENO- and WENO-type reconstructions are proposed, together with precise modifications necessary to preserve positive pressure and density. Extending the discrete source term to second order while maintaining stability requires non-standard techniques, which we present. The first- and second-order schemes are tested on a suite of numerical experiments demonstrating impressive numerical resolution as well as stability, even on very fine meshes.

Keywords

35L6574S1065M12Conservation lawsMHDdivergence constraintGodunov-Powell source termsupwinded source termshigh-order schemes
Type
Research Article
Information
Communications in Computational Physics , Volume 9 , Issue 2 , February 2011 , pp. 324 - 362
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Audusse, E., Bouchut, F., Bristeau, M. O., Klien, R. and Perthame, B., A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIA M. J. Sci. Comp., 25(6) (2004), 2050–2065.Google Scholar
[2]Balsara, D. S. and Spicer, D., A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations, J. Comp. Phys., 149(2) (1999), 270–292.Google Scholar
[3]Balsara, D. S., Total variation diminishing algorithm for adiabatic and isothermal magneto-hydrodynamics, Astrophys. J. Supp. Ser., 116 (1998), 133–153.CrossRefGoogle Scholar
[4]Balsara, D. S., Divergence-free reconstructions of magnetic fields and WENO schemes for magnetohydrodynamics, J. Comp. Phys., 228(14) (2009), 5040–5056.CrossRefGoogle Scholar
[5]Balsara, D. S., Rumpf, T., Dumbser, M. and Munz, C. D., Efficient high accuracy ADER-WENO schemes for hydrodynamics and divergence-free MHD, J. Comp. Phys., 228(7) (2009), 2480–2516.Google Scholar
[6]Barth, T. J., Numerical methods for gas dynamics systems, in: An introduction to Recent Developments in Theory and Numerics for Conservation Laws, Kröner, D., Ohlberger, M., Rohde, C. (Eds.), Springer, 1999.Google Scholar
[7]Berthon, C., Why the MUSCL-Hancock scheme is L 1-stable, Numer. Math., 104 (2006), 27–46.Google Scholar
[8]Bouchut, F., Klingenberg, C. and Waagan, K., A multi-wave HLL approximate Riemann solver for ideal MHD based on relaxation I-theoretical framework, Numer. Math., 108(1) (2007), 7–42.CrossRefGoogle Scholar
[9]Bouchut, F., Klingenberg, C. and Waagan, K., A multi-wave HLL approximate Riemann solver for ideal MHD based on relaxation II-numerical experiments, Preprint, 2008.Google Scholar
[10]Brackbill, J. U. and Barnes, D. C., The effect of nonzero div B on the numerical solution of the magnetohydrodynamic equations, J. Comp. Phys., 35 (1980), 426–430.Google Scholar
[11]Brio, M. and Wu, C. C., An upwind differencing scheme for the equations of ideal MHD, J. Comp. Phys., 75(2) (1988), 400–422.CrossRefGoogle Scholar
[12]Cargo, P. and Gallice, G., Roe matrices for ideal MHD and systematic construction of Roe matrices for systems of conservation laws, J. Comp. Phys., 136(2) (1997), 446–466.CrossRefGoogle Scholar
[13]Dai, W. and Woodward, P. R., A simple finite difference scheme for multi-dimensional mag-netohydrodynamic equations, J. Comp. Phys., 142(2) (1998), 331–369.Google Scholar
[14]Dedner, A., Kemm, F., Kröner, D., Munz, C. D., Schnitzer, T. and Wesenberg, M., Hyperbolic divergence cleaning for the MHD equations, J. Comp. Phys., 175 (2002), 645–673.Google Scholar
[15]Einfeldt, B., On the Godunov type methods for gas dynamics, SIA M. J. Num. Anal., 25(2) (1988), 294–318.Google Scholar
[16]Evans, C. and Hawley, J. F., Simulation of magnetohydrodynamic flow: a constrained transport method, Astrophys. J., 332 (1998), 659–677.Google Scholar
[17]Fuchs, F., Karlsen, K. H., Mishra, S. and Risebro, N. H., Stable upwind schemes for the magnetic induction equation, Math. Model. Num. Anal., 43(5) (2009), 825–852.Google Scholar
[18]Fuchs, F., Mishra, S. and Risebro, N. H., Splitting based finite volume schemes for the ideal MHD equations, J. Comp. Phys., 228(3) (2009), 641–660.Google Scholar
[19]Fuchs, F., McMurry, A. D., Mishra, S. and Risebro, N. H., Finite volume methods for wave propagation in stratified magneto-atmospheres, Commun. Comput. Phys., to appear, 2010.Google Scholar
[20]Godunov, S. K., The symmetric form of magnetohydrodynamics equation, Num. Meth. Mech. Cont. Med., 1 (1972), 26–34.Google Scholar
[21]Gurski, K. F., An HLLC-type approximate Riemann solver for ideal Magneto-hydro dynamics, SIA M. J. Sci. Comp., 25(6) (2004), 2165–2187.Google Scholar
[22]Gottlieb, S., Shu, C. W. and Tadmor, E., High order time discretizations with strong stability property, SIAM. Rev., 43 (2001), 89–112.Google Scholar
[23]Harten, A., Engquist, B., Osher, S. and Chakravarty, S. R., Uniformly high order accurate essentially non-oscillatory schemes, J. Comp. Phys., 71 (1987), 231–303.Google Scholar
[24]Harten, A., Lax, P. D. and Van Leer, B., On upstream differencing and Godunov type schemes for hyperbolic conservation laws, SIAM. Rev., 25(1) (1983), 35–61.Google Scholar
[25]Honkkila, V. and Janhunen, P., HLLC solver for ideal relativistic MHD, J. Comp. Phys., 223 (2007), 643–656.Google Scholar
[26]Kolgan, V. L., Application of the minimum-derivative principle in the construction of finite-difference schemes for the numerical analysis of discontinuous solutions in gas dynamics, Transactions of the Central Aerohydrodynamics Institute, 3(6) (1972), 68–77.Google Scholar
[27]Kolgan, V. L., Finite-difference schemes for computation of three dimensional solutions of gas dynamics and calculation of a flow over a body under an angle of attack, Transactions of the Central Aerohydrodynamics Institute, 6(2) (1975), 1–6.Google Scholar
[28]LeVeque, R. J., Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, 2002.Google Scholar
[29]LeVeque, R. J., Wave propagation algorithms for multi-dimensional hyperbolic systems, J. Comp. Phys., 131 (1997), 327–353.CrossRefGoogle Scholar
[30]Li, S., An HLLC Riemann solver for magneto-hydrodynamics, J. Comp. Phys., 203 (2005), 344–357.CrossRefGoogle Scholar
[31]Linde, T. J., A Three Adaptive Multi Fluid MHD Model for the Heliosphere, Ph.D thesis, University of Michigan, Ann-Arbor, 1998.Google Scholar
[32]Londrillo, P. and Zanna, L. del, On the divergence-free condition in Godunov-type schemes for ideal magnetohydrodynamics: the upwind constrained transport method, J. Comp. Phys., 195(1) (2004), 17–48.Google Scholar
[33]Mishra, S. and Tadmor, E., Constraint preserving schemes using potential-based fluxes: III, genuinely multi-dimensional central schemes for MHD equations, Preprint, 2009.Google Scholar
[34]Miyoshi, T. and Kusano, K., A multi-state HLL approximate Riemann solver for ideal magneto hydro dynamics, J. Comp. Phys., 208(1) (2005), 315–344.Google Scholar
[35]Powell, K. G., An approximate Riemann solver for magneto-hydro dynamics (that works in more than one space dimension), Technical Report, 94-24, ICASE, Langley, VA, 1994.Google Scholar
[36]Powell, K. G., Roe, P. L., Linde, T. J., Gombosi, T. I. and De zeeuw, D. L., A solution adaptive upwind scheme for ideal MHD, J. Comp. Phys., 154(2) (1999), 284–309.Google Scholar
[37]Roe, P. L. and Balsara, D. S., Notes on the eigensystem of magnetohydrodynamics, SIA M. J. Appl. Math., 56(1) (1996), 57–67.Google Scholar
[38]Rossmanith, J., A Wave Propagation Method with Constrained Transport for Shallow Water and Ideal Magnetohydrodynamics, Ph.D thesis, University of Washington, Seattle, 2002.Google Scholar
[39]Ryu, D. S., Miniati, F., Jones, T. W. and Frank, A., A divergence free upwind code for multidimensional magnetohydrodynamic flows, Astrophys. J., 509(1) (1988), 244–255.Google Scholar
[40]Shu, C. W. and Osher, S., Efficient implementation of essentially non-oscillatory schemes-II, J. Comp. Phys., 83 (1989), 32–78.CrossRefGoogle Scholar
[41]Stone, J. M., Gardiner, T.A., Thomas, A., Teuben, P., Hawley, J. F. and Simon, J. B., Athena: a new code for astrophysical MHD, Astrophys. J. Supp. Ser., 178(1) (2008), 137–177.Google Scholar
[42]Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer Verlag, third edition, 2009.Google Scholar
[43]Toro, E. F., Spruce, M. and Speares, W., Restoration of the contact surface in the HLL Riemann solver, Tech. Report COA-9204, College of Aeronautics, Cranfield Institute of Technology, U.K., 1992.Google Scholar
[44]Toro, E. F., Spruce, M. and Speares, W., Restoration of the contact surface in the Harten-Lax-van Leer Riemann solver, Shock. Waves., 4 (1994), 25–34.CrossRefGoogle Scholar
[45]Torrilhon, M. and Fey, M., Constraint-preserving upwind methods for multidimensional ad-vection equations, SIA M. J. Num. Anal., 42(4) (2004), 1694–1728.CrossRefGoogle Scholar
[46]Torrilhon, M., Locally divergence preserving upwind finite volume schemes for magnetohy-rodynamic equations, SIA M. J. Sci. Comp., 26(4) (2005), 1166–1191.Google Scholar
[47]Torrilhon, M., Uniqueness conditions for Riemann problems of ideal Magneto-hydrodynamics, J. Plasma. Phys., 69(3) (2003), 253–276.Google Scholar
[48]Torrilhon, M. and Balsara, D. S., High-order WENO schemes: investigations on non-uniform convergence for MHD Riemann problems, J. Comp. Phys., 201(2) (2004), 586–600.Google Scholar
[49]Toth, G.. The divB=0 constraint in shock capturing magnetohydrodynamics codes, J. Comp. Phys., 161 (2000), 605–652.Google Scholar
[50]Van Leer, B., Towards the ultimate conservative difference scheme. V, J. Comp. Phys., 32 (1979), 101–136.CrossRefGoogle Scholar
[51]Waagan, K., A positive MUSCL-Hancock scheme for ideal MHD, J. Comp. Phys., to appear.Google Scholar