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A Locally Divergence-Free Oscillation-Eliminating Discontinuous Galerkin Method for Ideal Magnetohydrodynamic Equations

Published 22 Apr 2024 in math.NA and cs.NA | (2404.14274v1)

Abstract: Numerical simulations of ideal compressible magnetohydrodynamic (MHD) equations are challenging, as the solutions are required to be magnetic divergence-free for general cases as well as oscillation-free for cases involving discontinuities. To overcome these difficulties, we develop a locally divergence-free oscillation-eliminating discontinuous Galerkin (LDF-OEDG) method for ideal compressible MHD equations. In the LDF-OEDG method, the numerical solution is advanced in time by using a strong stability preserving Runge-Kutta scheme. Following the solution update in each Runge-Kutta stage, an oscillation-eliminating (OE) procedure is performed to suppress spurious oscillations near discontinuities by damping the modal coefficients of the numerical solution. Subsequently, on each element, the magnetic filed of the oscillation-free DG solution is projected onto a local divergence-free space, to satisfy the divergence-free condition. The OE procedure and the LDF projection are fully decoupled from the Runge-Kutta stage update, and can be non-intrusively integrated into existing DG codes as independent modules. The damping equation of the OE procedure can be solved exactly, making the LDF-OEDG method remain stable under normal CFL conditions. These features enable a straightforward implementation of a high-order LDF-OEDG solver, which can be used to efficiently simulate the ideal compressible MHD equations. Numerical results for benchmark cases demonstrate the high-order accuracy, strong shock capturing capability and robustness of the LDF-OEDG method.

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Authors (2)

  1. Wei Zeng 
  2. Qian Wang 

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