High-order conservative finite difference GLM-MHD schemes for cell-centered MHD

TL;DR

This paper introduces high-order conservative finite difference schemes for ideal MHD equations, utilizing advanced WENO and MP reconstructions, with a cell-centered divergence control approach that enhances robustness and efficiency.

Contribution

It develops and compares third- and fifth-order accurate schemes for MHD, employing a cell-centered divergence cleaning method that simplifies implementation and improves computational efficiency.

Findings

High-order schemes achieve accurate results in smooth regions.

The divergence-free condition is effectively maintained without elliptic cleaning.

Numerical tests confirm robustness for smooth and discontinuous flows.

Abstract

We present and compare third- as well as fifth-order accurate finite difference schemes for the numerical solution of the compressible ideal MHD equations in multiple spatial dimensions. The selected methods lean on four different reconstruction techniques based on recently improved versions of the weighted essentially non-oscillatory (WENO) schemes, monotonicity preserving (MP) schemes as well as slope-limited polynomial reconstruction. The proposed numerical methods are highly accurate in smooth regions of the flow, avoid loss of accuracy in proximity of smooth extrema and provide sharp non-oscillatory transitions at discontinuities. We suggest a numerical formulation based on a cell-centered approach where all of the primary flow variables are discretized at the zone center. The divergence-free condition is enforced by augmenting the MHD equations with a generalized Lagrange multiplier yielding a mixed hyperbolic/parabolic correction, as in Dedner et al. (J. Comput. Phys. 175 (2002) 645-673). The resulting family of schemes is robust, cost-effective and straightforward to implement. Compared to previous existing approaches, it completely avoids the CPU intensive workload associated with an elliptic divergence cleaning step and the additional complexities required by staggered mesh algorithms. Extensive numerical testing demonstrate the robustness and reliability of the proposed framework for computations involving both smooth and discontinuous features.