Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-26T15:11:48.592Z Has data issue: false hasContentIssue false
  • English
  • Français

An Efficient Implementation of the Divergence Free Constraint in a Discontinuous Galerkin Method for Magnetohydrodynamics on Unstructured Meshes

Part of: Hyperbolic equations and systems Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  07 February 2017

Christian Klingenberg ,
Frank Pörner  and
Yinhua Xia
Christian Klingenberg*
Affiliation:
Department of Mathematics, University of Würzburg, Emil-Fischer-Str. 40, 97074 Würzburg, Germany
Frank Pörner*
Affiliation:
Department of Mathematics, University of Würzburg, Emil-Fischer-Str. 40, 97074 Würzburg, Germany
Yinhua Xia*
Affiliation:
School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China
*
*Corresponding author.Email addresses:[email protected] (C. Klingenberg), [email protected] (F. Pörner), [email protected] (Y. Xia)
*Corresponding author.Email addresses:[email protected] (C. Klingenberg), [email protected] (F. Pörner), [email protected] (Y. Xia)
*Corresponding author.Email addresses:[email protected] (C. Klingenberg), [email protected] (F. Pörner), [email protected] (Y. Xia)
Get access

Abstract

In this paper we consider a discontinuous Galerkin discretization of the ideal magnetohydrodynamics (MHD) equations on unstructured meshes, and the divergence free constraint (∇·B=0) of its magnetic field B. We first present two approaches for maintaining the divergence free constraint, namely the approach of a locally divergence free projection inspired by locally divergence free elements [19], and another approach of the divergence cleaning technique given by Dedner et al. [15]. By combining these two approaches we obtain a efficient method at the almost same numerical cost. Finally, numerical experiments are performed to show the capacity and efficiency of the scheme.

Keywords

Ideal magnetohydrodynamics equationsdiscontinuous Galerkin methoddivergence free constraintlocally divergence free projectiondivergence free cleaning technique

MSC classification

Secondary: 65M12: Stability and convergence of numerical methods 65M20: Method of lines 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 35L65: Conservation laws
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 2 , February 2017 , pp. 423 - 442
Copyright
Copyright © Global-Science Press 2017 

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] Altmann, C.. Explicit Discontinuous Galerkin Methods for Magnetohydrodynamics. PhD thesis, University of Suttgart, 2012.Google Scholar
[2] Balsara, D.S. and 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.Google Scholar
[3] Bouchut, F., Klingenberg, C., and Waagan, K.. A multiwave approximate rieman solver for ideal mhd based on relaxation. i: theoretical framework. Numerische Mathematik, 108:7–42, 2007.Google Scholar
[4] Bouchut, F., Klingenberg, C., and Waagan, K.. A multiwave approximate rieman solver for ideal mhd based on relaxation. ii: numerical implementation with 3 and 5 waves. Numerische Mathematik, 115:647–679, 2010.Google Scholar
[5] Brackbill, J. U. and Barnes, D.C.. The effect of nonzero ∇·b on the numerical solution of the magnetohydrodynamic equations. Journal of Computational Physics, 35:426–430, 1980.Google Scholar
[6] Brezzi, F., Douglas, J., and Marini, L.D.. Two families ofmixed finite elements for second order elliptic problems. Numerical Mathematics, 47:217–235, 1985.Google Scholar
[7] Cheng, Y., Li, F., Qiu, J., and Xu, L.. Positivity-preserving dg and central dg methods for ideal mhd equations. Journal of Computational Physics, 238:255–280, 2013.Google Scholar
[8] Cockburn, B., Li, F., and Shu, C.-W.. Locally divergence-free discontinuous galerkin methods for the maxwell equations. Journal of Computational Physics, 194:588–610, 2003.Google Scholar
[9] Cockburn, B. and Shu, C.-W.. Tvb runge-kutta local projection discontinuous galerkin finite element method for conservation laws ii: general framework. Mathematics of Computation, 52:411–435, 1989.Google Scholar
[10] Cockburn, B. and Shu, C.-W.. Tvb runge-kutta local projection discontinuous galerkin finite elementmethod for conservation laws iii: One-dimensional systems. Journal of Computational Physics, 84:90–113, 1989.CrossRefGoogle Scholar
[11] Cockburn, B. and Shu, C.-W.. The runge-kutta local projection discontinuous galerkin finite element method for conservation laws iv: The multidimensional case. Mathematics of Computation, 54:545–581, 1990.Google Scholar
[12] Cockburn, B. and Shu, C.-W.. The runge-kutta local projection p 1-discontinuous-galerkin finite element method for scalar conservation laws. Mathematical Modelling and Numerical Analysis, 25:337–361, 1991.Google Scholar
[13] Cockburn, B. and Shu, C.-W.. The runge-kutta discontinuous galerkin method for conservation laws v: Multidimensional systems. Journal of Computational Physics, 141:199–224, 1997.Google Scholar
[14] Cockburn, B. and Shu, C.-W.. Runge-kutta discontinuous galerkin methods for convection-dominated problems. Journal of Scientific Computing, 16:173–261, 2001.Google Scholar
[15] Dedner, A., Kemm, F., Kroener, D., Schnitzer, T. Munz, C.-D., and Wesenberg, M.. Hyperbolic divergence cleaning for the mhd equations. Journal of Computational Physics, 175:645–673, 2002.CrossRefGoogle Scholar
[16] Gottlieb, S., Shu, C.-W., and Tadmor, E.. Strong stability preserving high order time discretization method. SIAM Rev., 43:89–112, 2001.Google Scholar
[17] Guzmán, J. and Rivière, B.. Sub-optimal convergence of non-symmetric discontinuous galerkin methods for odd polynomial approximations. J. Sci. Comp., 40:273–280, 2009.CrossRefGoogle Scholar
[18] Hesthaven, J. S. and Warburton, T.. Nodal Discontinuous Galerkin Methods. Springer Verlag, 2008.Google Scholar
[19] Li, F. and Shu, C.-W.. Locally divergence-free discontinuous galerkin methods for mhd equations. Journal of Scientific Computing, 22-23:413–442, 2005.Google Scholar
[20] Li, F. and Xu, L.. Arbitrary order exactly divergence-free central discontinuous galerkin methods for ideal mhd equations. Journal of Computational Physics, 231:2655–2675, 2012.Google Scholar
[21] Li, F., Xu, L., and Yakovlev, S.. Central discontinuous galerkin methods for ideal mhd equations with the exactly divergence-free magnetic field. Journal of Computational Physics, 230:4828–4847, 2011.Google Scholar
[22] Nédélec, J.C.. A new family of mixed finite elements in r3. Numerical Mathematics, 50:57–81, 1986.Google Scholar
[23] Oden, J. T., Babuska, I., and Baumann, C. E.. A discontinuous hp finite element method for diffusion problems. Journal of Computational Physics, 146:491–519, 1998.Google Scholar
[24] Orszag, S. and Tang, C.-M.. Small-scale structure of two-dimensional magnetohydrodynamic turbulence. Journal of Fluid Mechanics, 90:129–143, 1979.Google Scholar
[25] Powell, K. G.. An approximate riemann solver for magnetohydrodynamics (that works in more than one dimension). ICASE-Report, 94-24, 1994.Google Scholar
[26] Powell, K. G., Row, P. L., Myong, R. S., Gombosi, T., and De Zeeuw, D.. The symmetric form of magnetohydrodynamics equation. Numer. Methods Mech. Contin. Media, 1:26, 1994.Google Scholar
[27] Powell, K. G., Row, P. L., Myong, R. S., Gombosi, T., and De Zeeuw, D.. An upwind scheme for magnetohydrodynamics. Numerical Methods for Fluid Dynamics, V:163, 1994.Google Scholar
[28] Reed, W. H. and Hill, T. R.. Triangular mesh methods for the neutron transport equation. Tech. Report LA-UR-73-479, Los Alamos Scientific Laboratory, 1973.Google Scholar
[29] Shu, C.-W.. Tvb uniformly high-order schemes for conservation laws. Mathematics of Computation, 49:105–121, 1987.Google Scholar
[30] Shu, C.-W. and Osher, S.. Efficient implementation of essentially non-oscillatory shoch-capturing schemes. Journal of Computational Physics, 77:439–471, 1988.Google Scholar
[31] Shu, C.-W. and Zhang, X.. On positivity-preserving high order discontinuous galerkin schemes for compressible euler equations on rectangular meshes. Journal of Computational Physics, 229:8918–8934, 2010.Google Scholar
[32] Torrilhon, M.. Zur Numerik der idealen Magnetohydrodynamik. PhD thesis, Eidgenssische Technische Hochschule Zrich, 2003.Google Scholar
[33] Tóth, G.. The ∇·b=0 constraint in shock-capturing magnetohydrodynamics codes. Journal of Computational Physics, 161:605, 2000.CrossRefGoogle Scholar
[34] Waagan, K., Klingenberg, C., and Federrath, C.. A robust numerical scheme for highly compressible magnetohydrodynamics: Nonlinear stability, implementation and tests. Journal of Computational Physics, 230:3331–3351, 2011.Google Scholar
[35] Zhang, X., Xia, Y., and Shu, C.-W.. Maximum-principle-satisfying and positivity-preserving high order discontinuous galerkin schemes for conservation laws on triangular meshes. Journal of Scientific Computing, 50(1):29–62, 2012.CrossRefGoogle Scholar