Methods and Algorithms for Scientific Computing

On Thermodynamically Compatible Finite Volume Schemes for Continuum Mechanics

Abstract

In this paper we present a new family of semidiscrete 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, with 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 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 [Dokl. Akad. Nauk SSSR, 139(1961), pp. 521--523] exactly at the discrete level. All other terms in the governing equations of the more general GPR model, including nonconservative 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 PDEs.

Keywords

  1. thermodynamically compatible finite volume schemes
  2. semidiscrete and fully discrete Godunov formalism
  3. hyperbolic thermodynamically compatible PDE systems
  4. overdetermined hyperbolic PDE systems
  5. entropy inequality
  6. unified model for solid mechanics and fluid mechanics

MSC codes

  1. 35L40
  2. 65M08

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Information & Authors

Information

Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: A1723 - A1751
ISSN (online): 1095-7197

History

Submitted: 4 May 2021
Accepted: 19 January 2022
Published online: 28 June 2022

Keywords

  1. thermodynamically compatible finite volume schemes
  2. semidiscrete and fully discrete Godunov formalism
  3. hyperbolic thermodynamically compatible PDE systems
  4. overdetermined hyperbolic PDE systems
  5. entropy inequality
  6. unified model for solid mechanics and fluid mechanics

MSC codes

  1. 35L40
  2. 65M08

Authors

Affiliations

Funding Information

akademgorodok : 075-15-2019-1613
Leibniz Rechenzentrum : pr63qo
Istituto Nazionale di Alta Matematica "Francesco Severi" https://doi.org/10.13039/100009112
Ministero dell'Istruzione, dell'Università e della Ricerca https://doi.org/10.13039/501100003407
Università degli Studi di Trento https://doi.org/10.13039/501100004004

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