An entropy stable essentially oscillation-free discontinuous Galerkin method for solving ideal magnetohydrodynamic equations

• Published in: Journal of computational physics Vol. 530; p. 113911
• Main Authors: Liu, Yong, Lu, Jianfang, Shu, Chi-Wang
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.06.2025
• Subjects: Discontinuous Galerkin; Entropy stability; Ideal magnetohydrodynamic equations; Oscillation-free; Discontinuous Galerkin; Ideal magnetohydrodynamic equations; Entropy stability; Oscillation-free; 65M60; 65M12; 76W05; 35L65
• ISSN: 0021-9991
• DOI: 10.1016/j.jcp.2025.113911

Recently, we developed an innovative entropy-stable oscillation-free discontinuous Galerkin (OFDG) scheme, referred to as ESOFDG, for hyperbolic conservation laws [36]. Through the incorporation of a strategically designed damping term, this scheme can effectively suppress the numerical oscillations without compromising high-order accuracy. Building on this foundation, in this paper we extend the ESOFDG framework to the ideal compressible magnetohydrodynamic (MHD) equations. Unlike the conventional hyperbolic conservation laws, the MHD system is usually subject to an additional divergence-free constraint, and the wave structure is not immediately apparent from the MHD model of conservative form. To address this, we employ a modified MHD model that includes a non-conservative source term, originally introduced by Godunov [25], to establish appropriate entropy pairs. Additionally, we have carefully designed a damping term specifically tailored for the MHD equations in the ESOFDG framework. The resulting scheme not only maintains high-order accuracy and ensures entropy stability at the semi-discrete level, but also satisfies the properties of affine invariance and evolution invariance. Several numerical experiments are shown to confirm the robustness and efficiency of the proposed scheme for MHD equations.