High resolution rotated flux scheme for two-dimensional magnetohydrodynamics equations

doi:10.3785/j.issn.1008-9497.2024.01.005

Journal of Zhejiang University (Science Edition)

2024, Vol. 51

Issue (1): 29-40 DOI: 10.3785/j.issn.1008-9497.2024.01.005

Mathematics and Computer Science

High resolution rotated flux scheme for two-dimensional magnetohydrodynamics equations

Supei ZHENG,Mengqing ZHAI(

),Qi LI,Mangmang JIAN

School of Science，Chang'an University，Xi'an 710064，China

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Abstract

The rotated flux method could be used to effectively eliminate the shock instability of the approximate Riemann solver and suppress the generation of non-physical phenomenon if the equations to be solved meet the rotational invariance. For the 2D ideal magnetohydrodynamics (MHD) and shallow water magnetohydrodynamics (SWMHD) equations, the rotation-like matrix of flux function was constructed, and the corresponding rotational invariance theorem was given with proof, which was then used to deal with the governing equations applying quasi-1D method to derive the semi-discrete rotated flux scheme. Combining entropy stable flux and anti-diffusive flux by a flux limiter, a new flux that can adaptively adjust the dissipation term was obtained. Numerical experiments show that the new scheme can accurately capture the structure of solution, has high resolution, strong robustness and can be easily extended to higher dimensions.

Key words： ideal magnetohydrodynamics equations shallow water magnetohydrodynamics equations rotational invariance high resolution entropy stable flux

Received: 27 September 2022 Published: 10 January 2024

CLC:

O 241.82

Corresponding Authors: Mengqing ZHAI E-mail: zhaimengqing2016@163.com

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Cite this article:

Supei ZHENG,Mengqing ZHAI,Qi LI,Mangmang JIAN. High resolution rotated flux scheme for two-dimensional magnetohydrodynamics equations. Journal of Zhejiang University (Science Edition), 2024, 51(1): 29-40.

URL:

https://www.zjujournals.com/sci/EN/Y2024/V51/I1/29

二维磁流体方程的高分辨率旋转通量格式

若待解方程满足旋转不变性，则可通过旋转通量法有效消除近似Riemann求解器的激波不稳定现象，抑制非物理现象的产生。针对二维理想磁流体（MHD）方程和浅水波磁流体（SWMHD）方程，构造了通量函数的类旋转矩阵，给出了方程的旋转不变性证明；根据该性质对控制方程做类一维处理，推导了方程的半离散旋转通量格式；利用通量限制器，将熵稳定通量和反扩散通量进行加权组合，得到能够自适应调整耗散量的高分辨率旋转通量格式。数值实验表明，此格式能精确捕捉解的结构，分辨率高、鲁棒性强，且易向高维推广。

关键词： 理想磁流体方程, 浅水波磁流体方程, 旋转不变性, 高分辨率熵稳定通量

变量	区域
变量	I	II	III	IV
$ρ$	0.930 8	1.030 4	1.000 0	1.888 7
$ρ u$	1.455 7	1.577 4	1.750 0	0.233 4
$ρ v$	-0.463 3	-1.045 5	-1.000 0	-1.742 2
$ρ w$	0.057 5	-0.101 6	0.000 0	0.073 3
$ρ e$	5.083 8	5.781 3	6.000 0	12.999 0
$B 1$	0.350 1	0.350 1	0.564 2	0.564 2
$B 2$	0.983 0	0.507 8	0.507 8	0.983 0
$B 3$	0.350 0	0.157 6	0.253 9	0.491 5

Table 1 The initial condition for 2D Riemann problem

Fig.1 Numerical results of ES scheme for 2D Riemann problem

Fig.2 Numerical results of ESLR scheme for 2D Riemann problem

Fig.3 Change of total entropy with time for 2D Riemann problem

Fig.4 Numerical results of ES scheme for First Rotor problem

Fig.5 Numerical results of ESLR scheme for First Rotor problem

Fig.6 Change of total entropy with time for First Rotor problem

Fig.7 Numerical results of ES scheme for blast wave problem

Fig.8 Numerical results of ESLR scheme for blast wave problem

Fig.9 Change of total entropy with time for blast wave problem

Fig.10 Numerical results of Orszag-Tang-like turbulence problemNote Left is ES scheme， right is ESLR scheme，

Fig.11 Change of total entropy with time for Orszag-Tang-like turbulence problem

Fig.12 Change of total entropy with time for two separated conducting fluids problem

Fig.13 Numerical results of two separated conducting fluids problem

Fig.14 Numerical results of Rotor-like problem

Fig.15 Change of total entropy with time for Rotor-like problem


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