A Kernel Based High Order "Explicit" Unconditionally Stable Constrained Transport Method for Ideal Magnetohydrodynamics

TL;DR

This paper introduces a kernel-based, unconditionally stable constrained transport method for ideal MHD that effectively maintains divergence-free magnetic fields, improves robustness in shock problems, and eliminates the need for diffusion limiters.

Contribution

It develops a new kernel-based approach for vector potential in 2D and 3D that is A-stable and removes diffusion limiters, enhancing stability and robustness in MHD simulations.

Findings

Successfully tested on 2D and 3D shock and blast problems

Eliminates diffusion limiters in constrained transport method

Demonstrates robustness in strong shock scenarios

Abstract

The ideal Magnetohydrodynamics (MHD) equations are challenging because one needs to maintain the divergence free condition, $\nabla \cdot \Bv = 0$. Many numerical methods have been developed to enforce this condition. In this work, we further our work on mesh aligned constrained transport by developing a new kernel based approach for the vector potential in 2D and 3D. The approach for solving the vector potential is based on the method of lines transpose and is A-stable, eliminating the need for diffusion limiters needed in our previous work in 3D. The work presented here is an improvement over the previous method in the context of problems with strong shocks due to the fact that we could eliminate the diffusion limiter that was needed in our previous version of constrained transport. The method is robust and has been tested on the 2D and 3D cloud shock, blast wave and field loop problems.