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Mathematics > Numerical Analysis
Title: Structure-Preserving Oscillation-Eliminating Discontinuous Galerkin Schemes for Ideal MHD Equations: Locally Divergence-Free and Positivity-Preserving
(Submitted on 25 Apr 2024 (v1), last revised 2 May 2024 (this version, v2))
Abstract: Numerically simulating magnetohydrodynamics (MHD) poses notable challenges, including the suppression of spurious oscillations near discontinuities (e.g., shocks) and preservation of essential physical structures (e.g., the divergence-free constraint of magnetic field and the positivity of density and pressure). This paper develops structure-preserving oscillation-eliminating discontinuous Galerkin (OEDG) schemes for ideal MHD. The schemes leverage a locally divergence-free (LDF) oscillation-eliminating (OE) procedure to suppress spurious oscillations while retaining the LDF property of magnetic field and many desirable attributes of original DG schemes, such as conservation, local compactness, and optimal convergence rates. The OE procedure is based on the solution operator of a novel damping equation, a linear system of ordinary differential equations that are exactly solvable without any discretization. The OE procedure is performed after each Runge-Kutta stage and does not impact DG spatial discretization, facilitating its easy integration into existing DG codes as an independent module. Moreover, this paper presents a rigorous positivity-preserving (PP) analysis of the LDF OEDG schemes on Cartesian meshes, utilizing the optimal convex decomposition technique and the geometric quasi-linearization (GQL) approach. Efficient PP LDF OEDG schemes are derived by incorporating appropriate discretization of Godunov-Powell source terms into only the discrete equations of cell averages, under a condition achievable through a simple PP limiter. Several one- and two-dimensional MHD tests verify the accuracy, effectiveness, and robustness of the proposed structure-preserving OEDG schemes.
Submission history
From: Kailiang Wu [view email][v1] Thu, 25 Apr 2024 17:40:50 GMT (10566kb,D)
[v2] Thu, 2 May 2024 16:32:55 GMT (11603kb,D)
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