A Structure-Preserving and Thermodynamically Compatible Cell-Centered Lagrangian Finite Volume Scheme for Continuum Mechanics

Abstract

SIAM Journal on Numerical Analysis, Volume 64, Issue 3, Page 685-707, June 2026.
Abstract. In this work we present a novel structure-preserving scheme for the compatible discretization of the Godunov–Peshkov–Romenski (GPR) model of continuum mechanics written in Lagrangian form. The governing equations fall into the larger class of overdetermined hyperbolic and thermodynamically compatible systems of partial differential equations (PDE). This model admits an extra conservation law for the total energy (first principle of thermodynamics) and satisfies the entropy inequality (second principle of thermodynamics). Furthermore, in the absence of algebraic source terms, the distortion field of the continuum and the specific thermal impulse satisfy a curl-free condition, provided the initial data are curl-free. Last but not least, the determinant of the distortion field is related to the density of the medium, i.e., the system is also endowed with a nonlinear algebraic constraint. In the stiff relaxation limit, the system tends to the compressible Navier–Stokes equations, i.e., the GPR model is able to describe at the same time the dynamics of nonlinear solids as well as the one of fluids. The objective of this work is to construct and analyze a new semidiscrete thermodynamically compatible cell-centered Lagrangian finite volume scheme on moving unstructured meshes that satisfies the following structural properties of the governing PDE exactly at the discrete level: (i) compatibility with the first law of thermodynamics, i.e., discrete total energy conservation; (ii) compatibility with the second law of thermodynamics, i.e., discrete entropy inequality; (iii) exact discrete compatibility between the density and the determinant of the distortion field; (iv) exact preservation of the curl-free property of the distortion field and of the specific thermal impulse in the absence of algebraic source terms. We will show that it is indeed possible to achieve all the above properties simultaneously. Unlike in existing schemes, we choose to directly discretize the entropy inequality, hence obtaining total energy conservation as a consequence of an appropriate and thermodynamically compatible discretization of all the other equations. From this discretization, property (ii) above follows trivially by construction, while (i) leads to provable nonlinear stability, which is an important feature, in particular for the complex PDE system under consideration here. The thermodynamic compatibility and thus nonlinear stability in the sense of total energy conservation is achieved via a very simple and general approach recently introduced by Abgrall, Dumbser, and Maire. by using a scalar correction factor that is defined at the nodes of the grid. This perfectly fits into the formalism of nodal solvers which is typically adopted in cell-centered Lagrangian finite volume methods. The new scheme is run on some academic benchmark problems for computational fluid and solid mechanics to show that the properties also hold in the practical implementation of the scheme.