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Originally introduced to describe a transition region in stars, the shallow water magnetohydrodynamics (SWMHD) model is now used throughout a number of solar physics and geophysical applications. In these applications, it is common to see phenomena that result from just a small perturbation of a steady-state solution. However, if using a standard method to try and capture these numerically, one may miss these small phenomena entirely unless the grid is refined significantly. This refinement may prove quite costly, and even completely unreasonable in large 3-dimensional simulations.

Well-Balanced (WB) schemes provide one alternative solution to this issue. Such methods preserve (non-trivial) steady-states of the system to order machine precision, in turn allowing one to capture small perturbations of these steady-states on coarse meshes. To this end, we propose a WB finite volume method for both the 1-D and 2-D SWMHD system. These methods also properly treat the divergence-free condition of the magnetic field on a discrete level. The WB and divergence-free properties of the proposed schemes are both provable, and several numerical experiments demonstrate a high resolution of obtained results and a lack of spurious oscillations. This figure presents the proposed WB method (top row) against the non-well-balanced (NWB) variation (bottom row) and their abilities (or lack thereof) to capture a small perturbation of a steady-state. It is clear that the WB method captures the proper structure of the solution even on a coarse mesh, while the NWB method has smearing of the solution due to numerical error -- even on a refined mesh.

Paper / ArXiv Preprint

Contact: Dr. Michael Redle