On thermodynamically compatible finite volume schemes for continuum mechanics

S. Busto, M. Dumbser, I. Peshkov, E. Romenski
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引用次数: 17

Abstract

In this paper we present a new family of semi-discrete and fully-discrete finite volume schemes for overdetermined, hyperbolic and thermodynamically compatible PDE systems. In the following we will denote these methods as HTC schemes. In particular, we consider the Euler equations of compressible gasdynamics, as well as the more complex Godunov-Peshkov-Romenski (GPR) model of continuum mechanics, which, at the aid of suitable relaxation source terms, is able to describe nonlinear elasto-plastic solids at large deformations as well as viscous fluids as two special cases of a more general first order hyperbolic model of continuum mechanics. The main novelty of the schemes presented in this paper lies in the fact that we solve the \textit{entropy inequality} as a primary evolution equation rather than the usual total energy conservation law. Instead, total energy conservation is achieved as a mere consequence of a thermodynamically compatible discretization of all the other equations. For this, we first construct a discrete framework for the compressible Euler equations that mimics the continuous framework of Godunov's seminal paper \textit{An interesting class of quasilinear systems} of 1961 \textit{exactly} at the discrete level. All other terms in the governing equations of the more general GPR model, including non-conservative products, are judiciously discretized in order to achieve discrete thermodynamic compatibility, with the exact conservation of total energy density as a direct consequence of all the other equations. As a result, the HTC schemes proposed in this paper are provably marginally stable in the energy norm and satisfy a discrete entropy inequality by construction. We show some computational results obtained with HTC schemes in one and two space dimensions, considering both the fluid limit as well as the solid limit of the governing partial differential equations.
连续介质力学的热相容有限体积格式
在本文中,我们提出了一类新的半离散和全离散有限体积格式,适用于超定、双曲和热力学相容的PDE系统。下面我们将把这些方法称为HTC方案。特别地,我们考虑了可压缩气体动力学的欧拉方程,以及更复杂的连续介质力学的Godunov-Peshkov-Romenski (GPR)模型,该模型在合适的松弛源项的帮助下,能够描述大变形下的非线性弹塑性固体以及粘性流体,作为更一般的连续介质力学一阶双曲模型的两种特殊情况。本文提出的方案的主要新颖之处在于我们将\textit{熵不等式}求解为初级演化方程,而不是通常的总能量守恒定律。相反,总能量守恒的实现仅仅是所有其他方程的热力学相容离散化的结果。为此,我们首先为可压缩欧拉方程构造了一个离散框架,该框架模仿了Godunov在1961年的开创性论文\textit{《一类有趣的拟线性系统》}中的连续框架,\textit{完全}在离散水平上。更一般的GPR模型的控制方程中的所有其他项,包括非保守乘积,都被明智地离散化,以实现离散的热力学兼容性,总能量密度的精确守恒是所有其他方程的直接结果。结果表明,本文提出的HTC方案在能量范数上是边际稳定的,并且通过构造满足一个离散熵不等式。在考虑控制偏微分方程的流体极限和固体极限的情况下,我们给出了用HTC格式在一维和二维空间中得到的一些计算结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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