Advertisement

A Runge–Kutta Discontinuous Galerkin Scheme for the Ideal Magnetohydrodynamical Model

  • Praveen Chandrashekar
  • Juan Pablo Gallego-Valencia
  • Christian KlingenbergEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 236)

Abstract

A numerical scheme for solving the system of ideal Magnetohydrodynamics (MHD) model, using an explicit high-order Runge–Kutta Discontinuous Galerkin method (RKDG) is proposed. An entropy stable numerical flux introduced in the context of Finite Volume (FV) method in Chandrashekar and Klingenberg (SIAM J Numer Anal, 2016, [4]) is used in the RKDG scheme. To illustrate the usefulness of the implementation, some specific test cases for the ideal Magnetohydrodynamics model (MHD equations) are shown.

Keywords

Ideal MHD Discontinuous Galerkin Entropy stability 

Notes

Acknowledgements

J.P. G.-V. thanks the GRK 1147 and the DAAD STIBET fellowship at the University of Würzburg for their support.

References

  1. 1.
    W. Bangerth, D. Davydov, T. Heister, L. Heltai, G. Kanschat, M. Kronbichler, M. Maier, B. Turcksin, D. Wells, The deal.II library, version 8.4. J. Numer. Math. 24 (2016)Google Scholar
  2. 2.
    T.J. Barth, Numerical methods for gasdynamic systems on unstructured meshes, An Introduction to Recent Developments in Theory and Numerics for Conservation Laws (Springer, Berlin, 1999), pp. 195–285CrossRefGoogle Scholar
  3. 3.
    M. Brio, C.C. Wu, An upwind differencing scheme for the equations of ideal magnetohydrodynamics. J. Comput. Phys. 75(2), 400–422 (1988)MathSciNetCrossRefGoogle Scholar
  4. 4.
    P. Chandrashekar, C. Klingenberg, Entropy stable finite volume scheme for ideal compressible mhd on 2-d cartesian meshes. SIAM J. Numer. Anal. (2016)Google Scholar
  5. 5.
    B. Cockburn, C.W. Shu, The Runge-Kutta discontinuous Galerkin method for conservation laws v: multidimensional systems. J. Comput. Phys. 141(2), 199–224 (1998)MathSciNetCrossRefGoogle Scholar
  6. 6.
    J.E. Flaherty, L. Krivodonova, J.F. Remacle, M.S. Shephard, Aspects of discontinuous galerkin methods for hyperbolic conservation laws. Finite Elem. Anal. Des. 38(10), 889–908 (2002): Robert J. Melosh Medal Compet. (2001)MathSciNetCrossRefGoogle Scholar
  7. 7.
    J.P. Gallego-Valencia, C. Klingenberg, P. Chandrashekar, On limiting for higher order discontinuous galerkin method for 2d euler equations. Bull. Braz. Math. Soc. New Ser. 47(1), 335–345 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    T.A. Gardiner, J.M. Stone, An unsplit godunov method for ideal mhd via constrained transport. J. Comput. Phys. 205(2), 509–539 (2005)MathSciNetCrossRefGoogle Scholar
  9. 9.
    G.S. Jiang, C.c. Wu, A high-order weno finite difference scheme for the equations of ideal magnetohydrodynamics. J. Comput. Phys. 150(2), 561–594 (1999)MathSciNetCrossRefGoogle Scholar
  10. 10.
    L. Krivodonova, J. Xin, J.F. Remacle, N. Chevaugeon, J. Flaherty, Shock detection and limiting with discontinuous galerkin methods for hyperbolic conservation laws. Appl. Numer. Math. 48(3–4), 323–338 (2004). (Workshop on Innovative Time Integrators for PDEs)MathSciNetCrossRefGoogle Scholar
  11. 11.
    S.A. Orszag, C.M. Tang, Small-scale structure of two-dimensional magnetohydrodynamic turbulence. J. Fluid Mech. 90(01), 129–143 (1979)CrossRefGoogle Scholar
  12. 12.
    K.G. Powell, An approximate riemann solver for magnetohydrodynamics (that works in more than one dimension). Technical report (1994)Google Scholar
  13. 13.
    G. Tóth, The \(\nabla \cdot b= 0\) constraint in shock-capturing magnetohydrodynamics codes. J. Comput. Phys. 161(2), 605–652 (2000)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Praveen Chandrashekar
    • 1
  • Juan Pablo Gallego-Valencia
    • 2
  • Christian Klingenberg
    • 1
    Email author
  1. 1.Tata Institute for Fundamental ResearchBangaloreIndia
  2. 2.University of WürzburgWürzburgGermany

Personalised recommendations