Divergence-Free Reconstruction of Magnetic Fields and WENO Schemes for Magnetohydrodynamics

TL;DR

This paper develops higher-order divergence-free WENO schemes for magnetohydrodynamics, improving accuracy beyond second order while maintaining the divergence-free condition crucial for stable and reliable simulations.

Contribution

It introduces divergence-free reconstruction methods that achieve better than second order accuracy and designs corresponding WENO schemes for MHD.

Findings

Schemes achieve higher than second order accuracy on smooth problems.

The divergence-free property is maintained at higher order.

Numerical tests confirm the schemes' robustness and accuracy.

Abstract

Balsara (2001, J. Comput. Phys., 174, 614) showed the importance of divergence-free reconstruction in adaptive mesh refinement problems for magnetohydrodynamics (MHD) and the importance of the same for designing robust second order schemes for MHD was shown in Balsara (2004, ApJS, 151, 149). Second order accurate divergence-free schemes for MHD have shown themselves to be very useful in several areas of science and engineering. However, certain computational MHD problems would be much benefited if the schemes had third and higher orders of accuracy. In this paper we show that the reconstruction of divergence-free vector fields can be carried out with better than second order accuracy. As a result, we design divergence-free weighted essentially non-oscillatory (WENO) schemes for MHD that have order of accuracy better than second. A multistage Runge-Kutta time integration is used to ensure that the temporal accuracy matches the spatial accuracy. Accuracy analysis is carried out and it is shown that the schemes meet their design accuracy for smooth problems. Stringent tests are also presented showing that the schemes perform well on those tests.