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Volume 15, Issue 4
A Partial RKDG Method for Solving the 2D Ideal MHD Equations Written in Semi-Lagrangian Formulation on Moving Meshes with Exactly Divergence-Free Magnetic Field

Shijun Zou, Xijun Yu, Zihuan Dai, Fang Qing & Xiaolong Zhao

Adv. Appl. Math. Mech., 15 (2023), pp. 932-963.

Published online: 2023-04

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  • Abstract

A partial Runge-Kutta Discontinuous Galerkin (RKDG) method which preserves the exactly divergence-free property of the magnetic field is proposed in this paper to solve the two-dimensional ideal compressible magnetohydrodynamics (MHD) equations written in semi-Lagrangian formulation on moving quadrilateral meshes. In this method, the fluid part of the ideal MHD equations along with $z$-component of the magnetic induction equation is discretized by the RKDG method as our previous paper [47]. The numerical magnetic field in the remaining two directions (i.e., $x$ and $y$) are constructed by using the magnetic flux-freezing principle which is the integral form of the magnetic induction equation of the ideal MHD. Since the divergence of the magnetic field in 2D is independent of its $z$-direction component, an exactly divergence-free numerical magnetic field can be obtained by this treatment. We propose a new nodal solver to improve the calculation accuracy of velocities of the moving meshes. A limiter is presented for the numerical solution of the fluid part of the MHD equations and it can avoid calculating the complex eigen-system of the MHD equations. Some numerical examples are presented to demonstrate the accuracy, non-oscillatory property and preservation of the exactly divergence-free property of our method.

  • AMS Subject Headings

76M10, 76N15, 76W05

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COPYRIGHT: © Global Science Press

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@Article{AAMM-15-932, author = {Zou , ShijunYu , XijunDai , ZihuanQing , Fang and Zhao , Xiaolong}, title = {A Partial RKDG Method for Solving the 2D Ideal MHD Equations Written in Semi-Lagrangian Formulation on Moving Meshes with Exactly Divergence-Free Magnetic Field}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2023}, volume = {15}, number = {4}, pages = {932--963}, abstract = {

A partial Runge-Kutta Discontinuous Galerkin (RKDG) method which preserves the exactly divergence-free property of the magnetic field is proposed in this paper to solve the two-dimensional ideal compressible magnetohydrodynamics (MHD) equations written in semi-Lagrangian formulation on moving quadrilateral meshes. In this method, the fluid part of the ideal MHD equations along with $z$-component of the magnetic induction equation is discretized by the RKDG method as our previous paper [47]. The numerical magnetic field in the remaining two directions (i.e., $x$ and $y$) are constructed by using the magnetic flux-freezing principle which is the integral form of the magnetic induction equation of the ideal MHD. Since the divergence of the magnetic field in 2D is independent of its $z$-direction component, an exactly divergence-free numerical magnetic field can be obtained by this treatment. We propose a new nodal solver to improve the calculation accuracy of velocities of the moving meshes. A limiter is presented for the numerical solution of the fluid part of the MHD equations and it can avoid calculating the complex eigen-system of the MHD equations. Some numerical examples are presented to demonstrate the accuracy, non-oscillatory property and preservation of the exactly divergence-free property of our method.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2021-0366}, url = {http://global-sci.org/intro/article_detail/aamm/21597.html} }
TY - JOUR T1 - A Partial RKDG Method for Solving the 2D Ideal MHD Equations Written in Semi-Lagrangian Formulation on Moving Meshes with Exactly Divergence-Free Magnetic Field AU - Zou , Shijun AU - Yu , Xijun AU - Dai , Zihuan AU - Qing , Fang AU - Zhao , Xiaolong JO - Advances in Applied Mathematics and Mechanics VL - 4 SP - 932 EP - 963 PY - 2023 DA - 2023/04 SN - 15 DO - http://doi.org/10.4208/aamm.OA-2021-0366 UR - https://global-sci.org/intro/article_detail/aamm/21597.html KW - Ideal compressible MHD equations, semi-Lagrangian formulation, exactly divergence-free magnetic field, Runge-Kutta discontinuous Galerkin method. AB -

A partial Runge-Kutta Discontinuous Galerkin (RKDG) method which preserves the exactly divergence-free property of the magnetic field is proposed in this paper to solve the two-dimensional ideal compressible magnetohydrodynamics (MHD) equations written in semi-Lagrangian formulation on moving quadrilateral meshes. In this method, the fluid part of the ideal MHD equations along with $z$-component of the magnetic induction equation is discretized by the RKDG method as our previous paper [47]. The numerical magnetic field in the remaining two directions (i.e., $x$ and $y$) are constructed by using the magnetic flux-freezing principle which is the integral form of the magnetic induction equation of the ideal MHD. Since the divergence of the magnetic field in 2D is independent of its $z$-direction component, an exactly divergence-free numerical magnetic field can be obtained by this treatment. We propose a new nodal solver to improve the calculation accuracy of velocities of the moving meshes. A limiter is presented for the numerical solution of the fluid part of the MHD equations and it can avoid calculating the complex eigen-system of the MHD equations. Some numerical examples are presented to demonstrate the accuracy, non-oscillatory property and preservation of the exactly divergence-free property of our method.

Shijun Zou, Xijun Yu, Zihuan Dai, Fang Qing & Xiaolong Zhao. (2023). A Partial RKDG Method for Solving the 2D Ideal MHD Equations Written in Semi-Lagrangian Formulation on Moving Meshes with Exactly Divergence-Free Magnetic Field. Advances in Applied Mathematics and Mechanics. 15 (4). 932-963. doi:10.4208/aamm.OA-2021-0366
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