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

Core Concepts

A locally divergence-free oscillation-eliminating discontinuous Galerkin (LDF-OEDG) method is developed to efficiently simulate ideal compressible magnetohydrodynamic (MHD) equations, which overcomes the challenges of preserving the magnetic divergence-free constraint and eliminating spurious oscillations near discontinuities.

Abstract

The authors develop a locally divergence-free oscillation-eliminating discontinuous Galerkin (LDF-OEDG) method for solving the ideal compressible magnetohydrodynamic (MHD) equations. The key features of the LDF-OEDG method are:

The numerical solution is advanced in time using a strong stability preserving Runge-Kutta scheme.

After each Runge-Kutta stage update, an oscillation-eliminating (OE) procedure is performed to suppress spurious oscillations near discontinuities by damping the modal coefficients of the numerical solution. The damping equation can be solved exactly, making the LDF-OEDG method stable under normal CFL conditions.

Subsequently, the magnetic field of the oscillation-free discontinuous Galerkin (DG) solution is projected onto a local divergence-free space to satisfy the divergence-free constraint.

The OE procedure and the local divergence-free (LDF) projection are fully decoupled from the Runge-Kutta stage update, enabling a non-intrusive implementation of the LDF-OEDG method by integrating them as independent modules into existing DG codes.

The authors demonstrate the high-order accuracy, strong shock capturing capability and robustness of the LDF-OEDG method through numerical results for several benchmark ideal MHD test cases.

Stats

The density, thermal pressure, Mach number, and magnetic pressure are used to support the key insights.

Quotes

"The OE procedure and the LDF projection are non-intrusive, in the sense that they are fully decoupled from the Runge-Kutta stage update."
"The damping equation of the OE procedure can be solved exactly, making the LDF-OEDG method remain stable under normal CFL conditions."

Key Insights Distilled From

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

by Wei Zeng,Qia... at arxiv.org 04-23-2024

https://arxiv.org/pdf/2404.14274.pdf

Deeper Inquiries

How can the LDF-OEDG method be extended to handle more complex MHD phenomena, such as resistive effects or relativistic flows

The LDF-OEDG method can be extended to handle more complex MHD phenomena by incorporating additional terms in the governing equations to account for resistive effects or relativistic flows. For resistive effects, terms representing resistivity can be added to the ideal MHD equations to model the dissipation of magnetic fields. This can be achieved by including terms related to the resistive diffusion of the magnetic field in the flux equations. The damping coefficients in the OE procedure can be adjusted to account for the additional resistive effects and ensure stability and accuracy in the simulations.
In the case of relativistic flows, the ideal MHD equations need to be modified to include relativistic effects such as Lorentz contraction, time dilation, and relativistic energy-momentum conservation. The LDF-OEDG method can be adapted to solve these modified equations by incorporating the relativistic terms into the flux calculations and adjusting the numerical scheme to handle the increased complexity of the equations. The damping coefficients in the OE procedure may need to be modified to account for the relativistic effects and ensure accurate simulations of relativistic MHD phenomena.

What are the potential applications of the LDF-OEDG method beyond ideal MHD, for example in other areas of computational physics or engineering

The LDF-OEDG method has potential applications beyond ideal MHD in various areas of computational physics and engineering. One potential application is in astrophysics, where the method can be used to simulate complex phenomena such as accretion disks, magnetized stellar winds, and relativistic jets. The ability of the LDF-OEDG method to handle magnetic divergence-free constraints and suppress oscillations near discontinuities makes it well-suited for modeling astrophysical processes involving strong magnetic fields and shocks.
In engineering, the LDF-OEDG method can be applied to problems in plasma physics, fusion research, and space propulsion. The method's ability to accurately capture shocks and resolve complex flow structures makes it valuable for studying plasma instabilities, magnetic confinement in fusion reactors, and the interaction of plasma with electromagnetic fields in propulsion systems. Additionally, the LDF-OEDG method can be used in geophysics to model the behavior of Earth's magnetic field and its interaction with the solar wind.

The authors focus on rectangular meshes in this work. How would the LDF-OEDG method perform on unstructured meshes, and what modifications would be required

The performance of the LDF-OEDG method on unstructured meshes would depend on the specific implementation and the adaptability of the method to handle irregular mesh geometries. To apply the LDF-OEDG method to unstructured meshes, modifications would be required in the spatial discretization and the basis functions used for the numerical solution representation.
One approach to extend the LDF-OEDG method to unstructured meshes would be to use general polyhedral elements and basis functions that can accommodate the irregular geometry of the mesh. The LDF projection onto divergence-free spaces would need to be adapted to work with the new basis functions and element types. Additionally, the damping coefficients in the OE procedure may need to be adjusted to account for the different mesh configurations and element shapes.
Overall, the LDF-OEDG method can be applied to unstructured meshes with appropriate modifications to the numerical scheme and basis functions, allowing for the simulation of complex MHD phenomena on a wider range of mesh geometries.