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Locally Divergence-Free Discontinuous Galerkin Methods for MHD Equations

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Abstract

In this paper, we continue our investigation of the locally divergence-free discontinuous Galerkin method, originally developed for the linear Maxwell equations (J. Comput. Phys. 194 588–610 (2004)), to solve the nonlinear ideal magnetohydrodynamics (MHD) equations. The distinctive feature of such method is the use of approximate solutions that are exactly divergence-free inside each element for the magnetic field. As a consequence, this method has a smaller computational cost than the traditional discontinuous Galerkin method with standard piecewise polynomial spaces. We formulate the locally divergence-free discontinuous Galerkin method for the MHD equations and perform extensive one and two-dimensional numerical experiments for both smooth solutions and solutions with discontinuities. Our computational results demonstrate that the locally divergence-free discontinuous Galerkin method, with a reduced cost comparing to the traditional discontinuous Galerkin method, can maintain the same accuracy for smooth solutions and can enhance the numerical stability of the scheme and reduce certain nonphysical features in some of the test cases.

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References

  • N. Aslan T. Kammash (1997) ArticleTitleDeveloping numerical fluxes with new sonic fix for MHD equations J.Comput.Phys. 133 43–55 Occurrence Handle10.1006/jcph.1997.5644

    Article  Google Scholar 

  • G.A. Baker W.N. Jureidini O.A. Karakashian (1990) ArticleTitlePiecewise solenoidal vector fields and the Stokes problem SIAMJ.Numer.Anal. 27 1466–1485 Occurrence Handle10.1137/0727085

    Article  Google Scholar 

  • D. S. Balsara (2001) ArticleTitleDivergence-free adaptive mesh refinement for magnetohydrodynamics J.Comput.Phys. 174 614–648 Occurrence Handle10.1006/jcph.2001.6917

    Article  Google Scholar 

  • D.S. Balsara (2004) ArticleTitleSecond order accurate schemes for magnetohydrodynamics with divergence-free reconstruction Astrophys.J.Suppl.Ser. 151 149–184 Occurrence Handle10.1086/381377

    Article  Google Scholar 

  • D.S. Balsara D.S. Spicer (1999) ArticleTitleA staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations J.Comput.Phys. 149 270–292 Occurrence Handle10.1006/jcph.1998.6153

    Article  Google Scholar 

  • J.U. Brackbill D.C. Barnes (1980) ArticleTitleThe effect of nonzero ∇.B B on the numerical solution of the magnetohydrodynamic equations J.Comput.Phys. 35 426–430 Occurrence Handle10.1016/0021-9991(80)90079-0

    Article  Google Scholar 

  • S.H. Brecht J.G. Lyon J.A. Fedder K. Hain (1981) ArticleTitleA simulation study of east-west IMF effects on the magnetosphere Geophys.Res.Lett. 8 397–400

    Google Scholar 

  • M. Brio C.C. Wu (1988) ArticleTitleAn upwind differencing scheme for the equations of ideal magnetohydrodynamics J.Comput.Phys. 75 400–422 Occurrence Handle10.1016/0021-9991(88)90120-9

    Article  Google Scholar 

  • B Cockburn S Hou CW Shu (1990) ArticleTitleThe Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case Math.Comput 54 545–581

    Google Scholar 

  • B Cockburn SY Lin CW Shu (1989) ArticleTitleTVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One dimensional systems J.Comput.Phys 84 90–113 Occurrence Handle10.1016/0021-9991(89)90183-6

    Article  Google Scholar 

  • B. Cockburn F. Li C.-W. Shu (2004) ArticleTitleLocally divergence-free discontinuous Galerkin methods for the Maxwell equations J.Comput.Phys. 194 588–610 Occurrence Handle10.1016/j.jcp.2003.09.007

    Article  Google Scholar 

  • B Cockburn CW Shu (1988) ArticleTitleThe Runge–Kutta discontinuous Galerkin method for conservation laws V: Multidimensional systems J.Comput.Phys 141 199–224 Occurrence Handle10.1006/jcph.1998.5892

    Article  Google Scholar 

  • B Cockburn CW Shu (2001) ArticleTitleRunge–Kutta Discontinuous Galerkin methods for convection-dominated problems J.Sci.Comput 16 173 Occurrence Handle10.1023/A:1012873910884

    Article  Google Scholar 

  • W. Dai P.R. Woodward (1998) ArticleTitleA simple finite difference scheme for multidimensional magnetohydrodynamic equations J.Comput.Phys. 142 331–369 Occurrence Handle10.1006/jcph.1998.5944 Occurrence HandleMR1624220

    Article  MathSciNet  Google Scholar 

  • A. Dedner F. Kemm D. Kröner C.-D. Munz T. Schnitner M. Wesenberg (2002) ArticleTitleHyperbolic divergence cleaning for the MHD equations J.Comput.Phys. 175 645–673 Occurrence Handle10.1006/jcph.2001.6961

    Article  Google Scholar 

  • C.R. Evans J.F. Hawley (1988) ArticleTitleSimulation of magnetohydrodynamic flows: A constrained transport method Astrophys.J. 332 659–677 Occurrence Handle10.1086/166684

    Article  Google Scholar 

  • B.-N. Jiang J. Wu L.A Povinelli (1996) ArticleTitleThe origin of spurious solutions in computational electromagnetics J.Comput.Phys. 125 104–123 Occurrence Handle10.1006/jcph.1996.0082

    Article  Google Scholar 

  • G.-S. Jiang C.-C. Wu (1999) ArticleTitleA high-order WENO finite difference scheme for the equations of ideal magnetohydrodynamics J.Comput.Phys. 150 561–594 Occurrence Handle10.1006/jcph.1999.6207

    Article  Google Scholar 

  • OA Karakashian WN Jureidini (1998) ArticleTitleA nonconforming finite element method for the stationary Navier–Stokes equations SIAM J.Numer.Anal 35 93–120 Occurrence Handle10.1137/S0036142996297199

    Article  Google Scholar 

  • C.-D. Munz P. Omnes R. Schneider E. Sonnendrücker U. Voβ (2000) ArticleTitleDivergence correction techniques for Maxwell solvers based on a hyperbolic model J. Comput. Phys. 161 484–511 Occurrence Handle10.1006/jcph.2000.6507

    Article  Google Scholar 

  • S.A Orszag CM Tang (1979) ArticleTitleSmall-scale structure of two-dimensional magnetohydrodynamic turbulence J. Fluid. Mech. 90 129–143

    Google Scholar 

  • Powell K. G. (1994). An Approximate Riemann solver for Magnetohydrodynamics (that works in more than one dimension), ICASE report No. 94–24, Langley, VA.

  • Qiu, J., and Shu, C.-W. Runge–Kutta discontinuous Galerkin method using WENO limiters. SIAM J. Sci. Comput., to appear.

  • J Qiu CW Shu (2003) ArticleTitleHermite WENO schemes and their application as limiters for Runge–Kutta discontinuous Galerkin method: One dimensional case J. Comput. Phys 193 115–135 Occurrence Handle10.1016/j.jcp.2003.07.026

    Article  Google Scholar 

  • D. Ryu F. Miniati T. W. Jones A. Frank (1998) ArticleTitleA divergence-free upwind code for multi-dimensional magnetohydrodynamic flows Astrophys. J. 509 244–255 Occurrence Handle10.1086/306481

    Article  Google Scholar 

  • C.-W. Shu (1987) ArticleTitleTVB uniformly high-order schemes for conservation laws Math. Comput. 49 105–121

    Google Scholar 

  • C.-W. Shu (1998) Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws B Cockburn C Johnson CW Shu E Tadmor A Quarteroni (Eds) Advanced Numerical Approximation of Nonlinear Hyperbolic Equations. Lecture Notes in Mathematics Springer Berlin 325–432

    Google Scholar 

  • C.-W. Shu S. Osher (1988) ArticleTitleEfficient implementation of essentially non-oscillatory shock-capturing schemes J. Comput. Phys. 77 439–471 Occurrence Handle10.1016/0021-9991(88)90177-5

    Article  Google Scholar 

  • J. M. Stone M. L. Norman (1992) ArticleTitleZEUS-2D: A radiation magnetohydrodynamics code for astrophysical flows in two space dimensions II The magnetohydrodynamic algorithms and tests. Astrophys. J. Suppl. Ser. 80 791–818

    Google Scholar 

  • G. Tóth (2000) ArticleTitleThe ∇. B B = 0 constraint in shock-capturing magnetohydrodynamics codes J. Comput. Phys. 161 605–652 Occurrence Handle10.1006/jcph.2000.6519

    Article  Google Scholar 

  • CC Wu (1986) ArticleTitleAn Kelvin–Helmholtz instability at the magnetopause boundary J. Geophys. Res 91 3042–3060

    Google Scholar 

  • KS Yee (1966) ArticleTitleNumerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media IEEE Trans. on Antenna Propagation AP-14 302–307

    Google Scholar 

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Authors and Affiliations

  1. Division of Applied Mathematics, Brown University, Providence, RI, 02912

    Fengyan Li & Chi-Wang Shu

  2. Department of Mathematics, University of South Carolina, Columbia, SC, 29308

    Chi-Wang Shu

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  1. Fengyan Li
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  2. Chi-Wang Shu
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Correspondence to Chi-Wang Shu.

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Li, F., Shu, CW. Locally Divergence-Free Discontinuous Galerkin Methods for MHD Equations. J Sci Comput 22, 413–442 (2005). https://doi.org/10.1007/s10915-004-4146-4

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  • DOI: https://doi.org/10.1007/s10915-004-4146-4

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