An Explicit Divergence-Free DG Method for Incompressible Magnetohydrodynamics

Abstract

We extend the recently introduced explicit divergence-free DG scheme for incompressible hydrodynamics (Fu in Comput Methods Appl Mech Eng 345:502–517, 2019) to the incompressible magnetohydrodynamics. A globally divergence-free finite element space is used for both the velocity and the magnetic field. Highlights of the scheme include global and local conservation properties, high-order accuracy, energy-stability, and pressure-robustness. When forward Euler time stepping is used, we need two symmetric positive definite hybrid-mixed Poisson solvers (one for velocity and one for magnetic field) to advance the solution to the next time level. Since we treat both viscosity in the momentum equation and resistivity in the magnetic induction equation explicitly, the method shall be best suited for inviscid or high-Reynolds number, low resistivity flows so that the CFL constraint is not too restrictive.

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Acknowledgements

The author would like to thank Prof. Chi-Wang Shu for suggesting to work on the problem, and for many helpful discussions concerning the subject. Part of this research was conducted using computational resources and services at the Center for Computation and Visualization, Brown University.

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Correspondence to Guosheng Fu.

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Fu, G. An Explicit Divergence-Free DG Method for Incompressible Magnetohydrodynamics. J Sci Comput 79, 1737–1752 (2019). https://doi.org/10.1007/s10915-019-00909-2

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Keywords

  • Incompressible MHD
  • Exactly divergece-free
  • Discontinuous Galerkin

Mathematics Subject Classification

  • 65N30
  • 65N12
  • 76S05
  • 76D07