In this paper we propose a novel thermodynamically compatible finite volume scheme for the numerical solution of the equations of magnetohydrodynamics (MHD) in one and two space dimensions. As shown by Godunov in 1972, the MHD system can be written as overdetermined symmetric hyperbolic and thermodynamically compatible (SHTC) system. More precisely, the MHD equations are symmetric hyperbolic in the sense of Friedrichs and satisfy the first and second principles of thermodynamics. In a more recent work on SHTC systems, \cite{Rom1998}, the entropy density is a primary evolution variable, and total energy conservation can be shown to be a \textit{consequence} that is obtained after a judicious linear combination of all other evolution equations. The objective of this paper is to mimic the SHTC framework also on the discrete level by directly discretizing the \textit{entropy inequality}, instead of the total energy conservation law, while total energy conservation is obtained via an appropriate linear combination as a \textit{consequence} of the thermodynamically compatible discretization of all other evolution equations. As such, the proposed finite volume scheme satisfies a discrete cell entropy inequality \textit{by construction} and can be proven to be nonlinearly stable in the energy norm due to the discrete energy conservation. In multiple space dimensions the divergence-free condition of the magnetic field is taken into account via a new thermodynamically compatible generalized Lagrangian multiplier (GLM) divergence cleaning approach. The fundamental properties of the scheme proposed in this paper are mathematically rigorously proven. The new method is applied to some standard MHD benchmark problems in one and two space dimensions, obtaining good results in all cases.
A fully discretization, unconditionally energy stable finite element method solving the thermodynamically consistent diffuse interface model for incompressible two-phase MHD flows with large density ratios
On thermodynamically compatible finite volume schemes for continuum mechanics
本文对超定双曲型和双曲型相容偏微分方程组提出了一族新的半离散和全离散有限体积格式。在下文中，我们将这些方法称为HTC方案。特别是，我们认为欧拉方程的可压缩气体动力学，以及更复杂的Godunov-Peshkov-Romenski（GPR）模型的连续介质力学，其中，在适当的松弛源项的帮助下，能够描述非线性弹塑性固体在大变形以及粘性流体作为两个特殊情况下的一个更一般的一阶双曲模型的连续介质力学。本文提出的方案的主要新颖之处在于，我们解决的\textit{entropy inequality}作为一个主要的发展方程，而不是通常的总能量守恒律。相反，实现总能量守恒仅仅是所有其他方程的一个相容离散化的结果。为此，我们首先构建了一个离散的可压缩欧拉方程的框架，模仿连续框架Goddom的开创性论文\textit{An interesting class of quasilinear systems}的1961年\textit{exactly}在离散水平。在更一般的GPR模型，包括非保守产品的控制方程中的所有其他条款，明智地离散，以实现离散的热力学兼容性，与作为所有其他方程的直接后果的总能量密度的精确守恒。结果表明，本文提出的HTC格式在能量范数下是可证明的边缘稳定的，并且通过构造满足离散熵不等式。我们显示了一些计算结果与HTC计划在一个和两个空间维度，考虑流体限制以及固体限制的控制偏微分方程。
A thermodynamically compatible semi-implicit finite volume scheme for the compressible Euler equations
University of Trento（特伦托大学）；Southern University of Science and Technology（南方科技大学）；La Sapienza Università di Roma（罗马第一大学）；Université de Strasbourg（斯特拉斯堡大学）；Inria
A Flux-Differencing Formula for Split-Form Summation By Parts Discretizations of Non-Conservative Systems: Applications to Subcell Limiting for magneto-hydrodynamics
Fully Discrete Multi-Entropy Stability of High-Order Schemes for Compressible MHD: A Weak-to-Strong Framework
Southern University of Science and Technology（南方科技大学）；Shenzhen International Center for Mathematics, Southern University of Science and Technology（南方科技大学深圳数学中心）
