Volume 32, Issue 2
A RKDG Method for 2D Lagrangian Ideal Magnetohydrodynamics Equations with Exactly Divergence-Free Magnetic Field

Commun. Comput. Phys., 32 (2022), pp. 547-582.

Published online: 2022-08

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• Abstract

In this paper, we present a Runge-Kutta Discontinuous Galerkin (RKDG) method for solving the two-dimensional ideal compressible magnetohydrodynamics (MHD) equations under the Lagrangian framework. The fluid part of the ideal MHD equations along with $z$-component of the magnetic induction equation are discretized using a DG method based on linear Taylor expansions. By using the magnetic flux-freezing principle which is the integral form of the magnetic induction equation of the ideal MHD, an exactly divergence-free numerical magnetic field can be obtained. The nodal velocities and the corresponding numerical fluxes are explicitly calculated by solving multidirectional approximate Riemann problems. Two kinds of limiter are proposed to inhibit the non-physical oscillation around the shock wave, and the second limiter can eliminate the phenomenon of mesh tangling in the simulations of the rotor problems. This Lagrangian RKDG method conserves mass, momentum, and total energy. Several numerical tests are presented to demonstrate the accuracy and robustness of the proposed scheme.

• Keywords

Lagrangian RKDG method, ideal compressible MHD equations, Taylor basis, exactly divergence-free magnetic field, limiter.

• AMS Subject Headings

76M10, 76N15, 76W05

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@Article{CiCP-32-547, author = {Shijun and Zou and and 24297 and and Shijun Zou and Xiaolong and Zhao and and 24298 and and Xiaolong Zhao and Xijun and Yu and and 24301 and and Xijun Yu and Zihuan and Dai and and 24302 and and Zihuan Dai}, title = {A RKDG Method for 2D Lagrangian Ideal Magnetohydrodynamics Equations with Exactly Divergence-Free Magnetic Field}, journal = {Communications in Computational Physics}, year = {2022}, volume = {32}, number = {2}, pages = {547--582}, abstract = {

In this paper, we present a Runge-Kutta Discontinuous Galerkin (RKDG) method for solving the two-dimensional ideal compressible magnetohydrodynamics (MHD) equations under the Lagrangian framework. The fluid part of the ideal MHD equations along with $z$-component of the magnetic induction equation are discretized using a DG method based on linear Taylor expansions. By using the magnetic flux-freezing principle which is the integral form of the magnetic induction equation of the ideal MHD, an exactly divergence-free numerical magnetic field can be obtained. The nodal velocities and the corresponding numerical fluxes are explicitly calculated by solving multidirectional approximate Riemann problems. Two kinds of limiter are proposed to inhibit the non-physical oscillation around the shock wave, and the second limiter can eliminate the phenomenon of mesh tangling in the simulations of the rotor problems. This Lagrangian RKDG method conserves mass, momentum, and total energy. Several numerical tests are presented to demonstrate the accuracy and robustness of the proposed scheme.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2021-0130}, url = {http://global-sci.org/intro/article_detail/cicp/20868.html} }
TY - JOUR T1 - A RKDG Method for 2D Lagrangian Ideal Magnetohydrodynamics Equations with Exactly Divergence-Free Magnetic Field AU - Zou , Shijun AU - Zhao , Xiaolong AU - Yu , Xijun AU - Dai , Zihuan JO - Communications in Computational Physics VL - 2 SP - 547 EP - 582 PY - 2022 DA - 2022/08 SN - 32 DO - http://doi.org/10.4208/cicp.OA-2021-0130 UR - https://global-sci.org/intro/article_detail/cicp/20868.html KW - Lagrangian RKDG method, ideal compressible MHD equations, Taylor basis, exactly divergence-free magnetic field, limiter. AB -

In this paper, we present a Runge-Kutta Discontinuous Galerkin (RKDG) method for solving the two-dimensional ideal compressible magnetohydrodynamics (MHD) equations under the Lagrangian framework. The fluid part of the ideal MHD equations along with $z$-component of the magnetic induction equation are discretized using a DG method based on linear Taylor expansions. By using the magnetic flux-freezing principle which is the integral form of the magnetic induction equation of the ideal MHD, an exactly divergence-free numerical magnetic field can be obtained. The nodal velocities and the corresponding numerical fluxes are explicitly calculated by solving multidirectional approximate Riemann problems. Two kinds of limiter are proposed to inhibit the non-physical oscillation around the shock wave, and the second limiter can eliminate the phenomenon of mesh tangling in the simulations of the rotor problems. This Lagrangian RKDG method conserves mass, momentum, and total energy. Several numerical tests are presented to demonstrate the accuracy and robustness of the proposed scheme.

Shijun Zou, Xiaolong Zhao, Xijun Yu & Zihuan Dai. (2022). A RKDG Method for 2D Lagrangian Ideal Magnetohydrodynamics Equations with Exactly Divergence-Free Magnetic Field. Communications in Computational Physics. 32 (2). 547-582. doi:10.4208/cicp.OA-2021-0130
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