A Provable Positivity-Preserving Local Discontinuous Galerkin Method for the Viscous and Resistive MHD Equations

doi:10.1007/s42967-022-00247-5

Communications on Applied Mathematics and Computation ›› 2024, Vol. 6 ›› Issue (1): 279-310.doi: 10.1007/s42967-022-00247-5

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A Provable Positivity-Preserving Local Discontinuous Galerkin Method for the Viscous and Resistive MHD Equations

Mengjiao Jiao, Yan Jiang, Mengping Zhang

School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui, China

收稿日期:2022-08-01 修回日期:2022-11-30 发布日期:2024-04-16
通讯作者: Yan Jiang,E-mail:jiangy@ustc.edu.cn;Mengjiao Jiao,E-mail:jmj123@mail.ustc.edu.cn;Mengping Zhang,E-mail:mpzhang@ustc.edu.cn E-mail:jiangy@ustc.edu.cn;jmj123@mail.ustc.edu.cn;mpzhang@ustc.edu.cn
基金资助:
Research supported by the NSFC Grant 11901555,12271499 and the Cyrus Tang Foundation. Research supported by the NSFC Grant 11871448 and 12126604.

A Provable Positivity-Preserving Local Discontinuous Galerkin Method for the Viscous and Resistive MHD Equations

Mengjiao Jiao, Yan Jiang, Mengping Zhang

School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui, China

Received:2022-08-01 Revised:2022-11-30 Published:2024-04-16
Contact: Yan Jiang,E-mail:jiangy@ustc.edu.cn;Mengjiao Jiao,E-mail:jmj123@mail.ustc.edu.cn;Mengping Zhang,E-mail:mpzhang@ustc.edu.cn E-mail:jiangy@ustc.edu.cn;jmj123@mail.ustc.edu.cn;mpzhang@ustc.edu.cn
Supported by:
Research supported by the NSFC Grant 11901555,12271499 and the Cyrus Tang Foundation. Research supported by the NSFC Grant 11871448 and 12126604.

摘要/Abstract

摘要： In this paper, we construct a high-order discontinuous Galerkin (DG) method which can preserve the positivity of the density and the pressure for the viscous and resistive magnetohydrodynamics (VRMHD). To control the divergence error in the magnetic field, both the local divergence-free basis and the Godunov source term would be employed for the multi-dimensional VRMHD. Rigorous theoretical analyses are presented for one-dimensional and multi-dimensional DG schemes, respectively, showing that the scheme can maintain the positivity-preserving (PP) property under some CFL conditions when combined with the strong-stability-preserving time discretization. Then, general frameworks are established to construct the PP limiter for arbitrary order of accuracy DG schemes. Numerical tests demonstrate the effectiveness of the proposed schemes.

关键词: Viscous and resistive MHD equations, Positivity-preserving, Discontinuous Galerkin (DG) method, High order accuracy

Abstract: In this paper, we construct a high-order discontinuous Galerkin (DG) method which can preserve the positivity of the density and the pressure for the viscous and resistive magnetohydrodynamics (VRMHD). To control the divergence error in the magnetic field, both the local divergence-free basis and the Godunov source term would be employed for the multi-dimensional VRMHD. Rigorous theoretical analyses are presented for one-dimensional and multi-dimensional DG schemes, respectively, showing that the scheme can maintain the positivity-preserving (PP) property under some CFL conditions when combined with the strong-stability-preserving time discretization. Then, general frameworks are established to construct the PP limiter for arbitrary order of accuracy DG schemes. Numerical tests demonstrate the effectiveness of the proposed schemes.

Key words: Viscous and resistive MHD equations, Positivity-preserving, Discontinuous Galerkin (DG) method, High order accuracy

Mengjiao Jiao, Yan Jiang, Mengping Zhang. A Provable Positivity-Preserving Local Discontinuous Galerkin Method for the Viscous and Resistive MHD Equations[J]. Communications on Applied Mathematics and Computation, 2024, 6(1): 279-310.

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A Provable Positivity-Preserving Local Discontinuous Galerkin Method for the Viscous and Resistive MHD Equations

A Provable Positivity-Preserving Local Discontinuous Galerkin Method for the Viscous and Resistive MHD Equations

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