The equations of ideal magnetohydrodynamics (MHD), which combines the Euler equations with the magnetic induction equation, are a central model in astrophysics. Compared with the Euler equations, it induces extra challenges in numerical computation, such as the control of the magnetic divergence error. If this error is not controlled well, a ficticious force proportional to the divergence error will be introduced into the momentum equation, thus&nbsp;the accuracy of the solver will be deteriorated. Furthermore, the increase of the divergence error will finally affect&nbsp;the stability of the solver.When using the discontinuous Galerkin or spectral difference method to discretize magnetohydrodynamic (MHD) equations, it is challenging to satisfy the divergence-free constraint for the magnetic field. To tackle this challenge, an unstaggered constrained transport method is integrated with the spectral difference scheme. In addition to solving the two-dimensional ideal MHD equations, one more equation describing the transport of the magnetic potential is introduced. For 2D problems, this additional equation is strongly hyperbolic, therefore discontinuous Galerkin and&nbsp;spectral difference method are both&nbsp;appropriate methods to solve it. The magnetic potential will be solved&nbsp;with respect to time, just like other conservative variables.&nbsp;After each step of time marching, the magnetic field will be updated by computing the curl of the magnetic potential. This strategy could effectively control the magnitude of the divergence error. Meanwhile, the additional computational cost is also small,&nbsp; approximately 20%&nbsp;more than the spectral difference solver without constrained transport. We successfully demonstrated that the divergence of the magnetic field is approximately zero in machine precision in the field loop advection test and the simulation of propagation of the Alfven wave. In the field loop advection test, when using&nbsp;the spectral difference solver without constrained transport, spurious stripes with high-frequency oscillations are witnessed in the contour of the magnetic energy. These stripes are&nbsp;aligned with the direction of the velocity field. When constrained transport is implemented, the contour is completely smooth. In the simulation of the propagation of the Alfven wave, when using&nbsp;the spectral difference solver without constrained transport, the divergence error will grow with time. When the divergence error reaches some threshold, the stability of the solver is lost. After constrained transport is added, the divergence error will not grow with time, and the simulation will be stable in long-time integration.&nbsp;Future works involve adding the shock-capturing features. The plan is to add artificial viscosity terms to provide local dissipation near the discontinuity so that the oscillation near shocks&nbsp;will not affect the stability of the solver.

A High-Order Spectral Difference Solver for 2D Ideal MHD Equations With Constrained Transport

Paper Proceedings: https://asmedigitalcollection.asme.org/IMECE/IMECE2021/volume/85543

ASME 2021 International Mechanical Engineering Congress and Exposition (IMECE2021)

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