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Advanced seminar AG Mathematical Modeling
Monday, 05.05.2025 at 17.00 in B7 - Room 210
Lecture by Michael Radle (Aachen) on the topic "Well-Balanced Schemes for Shallow Water MHD".
Abstract: 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 on large 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. In this talk, I share our proposed 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 locally divergence-free properties of the proposed schemes are provable, and the proposed method has been successfully tested on several examples.