Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-16T09:22:25.016Z Has data issue: false hasContentIssue false
  • English
  • Français

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:[email protected]
Email:[email protected]
Email:[email protected]
Email:[email protected]
Email:[email protected]
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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