Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-27T09:43:33.866Z Has data issue: false hasContentIssue false
  • English
  • Français

Constraint Preserving Schemes Using Potential-Based Fluxes I. Multidimensional Transport Equations

Published online by Cambridge University Press:  20 August 2015

Siddhartha Mishra  and
Eitan Tadmor
Siddhartha Mishra*
Affiliation:
Centre of Mathematics for Applications (CMA), University of Oslo, P.O. Box 1053, Blindern, N-0316 Oslo, Norway
Eitan Tadmor*
Affiliation:
Center of Scientific Computation and Mathematical Modeling (CSCAMM), and Department of Mathematics, and Institute for Physical Sciences and Technology (IPST), University of Maryland, MD 20742-4015, USA
*
Email:[email protected]
Corresponding author.Email:[email protected]
Get access

Abstract

We consider constraint preserving multidimensional evolution equations. A prototypical example is provided by the magnetic induction equation of plasma physics. The constraint of interest is the divergence of the magnetic field. We design finite volume schemes which approximate these equations in a stable manner and preserve a discrete version of the constraint. The schemes are based on reformulating standard edge centered finite volume fluxes in terms of vertex centered potentials. The potential-based approach provides a general framework for faithful discretizations of constraint transport and we apply it to both divergence preserving as well as curl preserving equations. We present benchmark numerical tests which confirm that our potential-based schemes achieve high resolution, while being constraint preserving.

Keywords

65M0635L65Multidimensional evolution equationsmagnetohydrodynamicsfinite-differencefinite volume schemesconstraint transportpotential-based schemes
Type
Research Article
Information
Communications in Computational Physics , Volume 9 , Issue 3 , March 2011 , pp. 688 - 710
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]Artebrant, R. and Torrilhon, M., Increasing the accuracy of local divergence preserving schemes for MHD, J. Comp. Phys., 227(6) (2008), 3405–3427.Google Scholar
[2]Bálbas, J., Tadmor, E., Non-oscillatory central schemes for one- and two-dimensional MHD equations. II: high-order semi-discrete schemes, SIAM J. Sci. Comput., 28 (2006), 533–560.CrossRefGoogle Scholar
[3]Bálbas, J., Tadmor, E. and Wu, C. C., Non-oscillatory central schemes for one and two-dimensional magnetohydrodynamics I, J. Comp. Phys., 201(1) (2004), 261–285.CrossRefGoogle Scholar
[4]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
[5]Bell, J. B., Colella, P. and Glaz, H. M., A second-order projection method for the incompressible Navier-Stokes equations, J. Comp. Phys., 85 (1989), 257–283.Google Scholar
[6]Brackbill, J. U. and Barnes, D. C., The effect of nonzero DivB on the numerical solution of the magnetohydrodynamic equations, J. Comp. Phys., 35 (1980), 426–430.Google Scholar
[7]Chorin, A. J., Numerical solutions of the Navier-Stokes equations, Math. Comp., 22 (1968), 745–762.CrossRefGoogle Scholar
[8]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.CrossRefGoogle Scholar
[9]Evans, C. and Hawley, J. F., Simulation of magnetohydrodynamic flow: a constrained transport method, Astrophys. J., 332 (1998), 659–677.Google Scholar
[10]Font, A. J., Numerical hydrodynamics in general relativity, Living Rev. Relativ., 6 (2003), http://relativity.livingreviews.org/Articles/lrr-2003-4/.Google Scholar
[11]Fuchs, F., Karlsen, K. H., Mishra, S. and Risebro, N. H., Stable upwind schemes for the magnetic induction equation, Math. Model. Num. Anal., v43 (2009), 825–852.Google Scholar
[12]Fuchs, F., McMurry, A. D., Mishra, S., Risebro, N. H. and Waagan, K., Approximate Riemann solver based high order finite volume schemes for the Godunov-Powell form of the ideal MHD equations in multi dimensions, preprint, submitted for publication, 2009.Google Scholar
[13]Godunov, S. K., The symmetric form of magnetohydrodynamics equation, Num. Meth. Mech. Cont. Media., 1 (1972), 26–34.Google Scholar
[14]Gottlieb, S., Shu, C. W. and Tadmor, E., High order time discretizations with strong stability property, SIAM. Rev., 43 (2001), 89–112.Google Scholar
[15]Harten, A., Engquist, B., Osher, S. and Chakravarty, S. R., Uniformly high order accurate essentially non-oscillatory schemes, J. Comput. Phys., 71 (1987), 231–303.Google Scholar
[16]Jeltsch, R. and Torrilhon, M., On curl preserving finite volume discretizations of the shallow water equations, BIT, 46(6) (2006), 35–53.CrossRefGoogle Scholar
[17]Kurganov, A. and Tadmor, E., New high resolution central schemes for non-linear conservation laws and convection-diffusion equations, J. Comput. Phys., 160(1) (2000), 241–282.Google Scholar
[18]LeVeque, R. J., Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, 2002.CrossRefGoogle Scholar
[19]Mishra, S. and Tadmor, E., Constraint preserving schemes using potential-based fluxes II. genuinely multi-dimensional central schemes for systems of conservation laws, preprint.Google Scholar
[20]Mishra, S. and Tadmor, E., Constraint preserving schemes using potential-based fluxes III. genuinely multi-dimensional schemes for MHD equations, preprint.Google Scholar
[21]Morton, K. W. and Roe, P. L., Vorticity preserving Lax-Wendroff type schemes for the system wave equation, SIA M. J. Sci. Comput., 23(1) (2001), 170–192.CrossRefGoogle Scholar
[22]Powell, K. G., An approximate Riemann solver for magneto-hydro dynamics (that works in more than one space dimension), Technical report, ICASE, Langley, VA, 1994.Google Scholar
[23]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
[24]Ryu, D. S., Miniati, F., Jones, T. W. and Frank, A., A divergence free upwind code for multidimensional magnetohydrodynamic flows, Astrophys. J., 509(1) (1998), 244–255.Google Scholar
[25]Shu, C. W. and Osher, S., Efficient implementation of essentially non-oscillatory schemes–II, J. Comput. Phys., 83 (1989), 32–78.Google Scholar
[26]Tadmor, E., Numerical viscosity and entropy conditions for conservative difference schemes, Math. Comp., 43(168) (1984), 369–381.Google Scholar
[27]Tadmor, E., From semi-discrete to fully discrete: stability of Runge-Kutta schemes by the energy method. II, in “Collected Lectures on the Preservation of Stability under Discretization”, Proc. in Applied Mathematics (Estep, D. and Tavener, S., eds.) 109, SIAM 2002, 25–49.Google Scholar
[28]Tadmor, E., Approximate solutions of nonlinear conservation laws, Advanced Numerical Approximations of Nonlinear Hyperbolic Equations, Lecture notes in Mathematics, Springer Verlag (1998), 1–149.Google Scholar
[29]Torrilhon, M., Locally divergence preserving upwind finite volume schemes for magneto-hydro dynamics, SIAM J. Sci. Comp., 26(4) (2005), 1166–1191.Google Scholar
[30]Torrilhon, M. and Fey, M., Constraint-preserving upwind methods for multidimensional ad-vection equations, SIAM J. Num. Anal., 42(4) (2004), 1694–1728.CrossRefGoogle Scholar
[31]Toth, G., The Div B=0 constraint in shock capturing magnetohydrodynamics codes, J. Comp. Phys., 161 (2000), 605–652.Google Scholar