# On Thermodynamically Compatible Finite Volume Schemes for Continuum Mechanics

## Abstract

*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.

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## References

*A general framework to construct schemes satisfying additional conservation relations: Application to entropy conservative and entropy dissipative schemes*, J. Comput. Phys., 372 (2018), pp. 640--666.

*A high-order nonconservative approach for hyperbolic equations in fluid dynamics*, Comput. Fluids, 169 (2018), pp. 10--22.

*The internal consistency, stability, and accuracy of the discrete, compatible formulation of Lagrangian hydrodynamics*, J. Comput. Phys., 218 (2006), pp. 572--593.

*Stosswelle und detonation*, Physik, 8 (1923), 321.

*A second-order projection method for the incompressible Navier-Stokes equations*, J. Comput. Phys., 85 (1989), pp. 257--283.

*A staggered semi-implicit hybrid FV/FE projection method for weakly compressible flows*, J. Comput. Phys., 421 (2020), 109743.

*A structure-preserving staggered semi-implicit finite volume scheme for continuum mechanics*, J. Comput. Phys., 424 (2021), 109866.

*Line Integral Methods for Conservative Problems*, Chapman et Hall/CRC, Boca Raton, FL, 2016.

*Line integral solution of differential problems*, Axioms, 7 (2018), 36, https://doi.org/10.3390/axioms7020036.

*Asymptotic preserving and positive schemes for radiation hydrodynamics*, J. Comput. Phys., 215 (2006), pp. 717--740.

*High order ADER schemes for continuum mechanics*, Front. Phys., 8 (2020), 32.

*On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows*, J. Sci. Comput., 88 (2021), 28.

*A projection hybrid high order finite volume/finite element method for incompressible turbulent flows*, J. Comput. Phys., 353 (2018), pp. 169--192.

*A semi-implicit hybrid finite volume/finite element scheme for all Mach number flows on staggered unstructured meshes*, Appl. Math. Comput., 402 (2021), 126117.

*Efficient high order accurate staggered semi-implicit discontinuous Galerkin methods for natural convection problems*, Comput. & Fluids, 198 (2020), 104399.

*Uniformly accurate schemes for hyperbolic systems with relaxation*, SIAM J. Numer. Anal., 34 (1997), pp. 246--281.

*The force/work differencing of exceptional points in the discrete, compatible formulation of Lagrangian hydrodynamics*, J. Computat. Phys., 216 (2006), pp. 1--18.

*High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products: Applications to shallow-water systems*, Math. Comput., 75 (2006), pp. 1103--1134.

*Convergence of second-order, entropy stable methods for multi-dimensional conservation laws*, ESAIM Math. Model. Numer. Anal., 54 (2020), pp. 1415--1428.

*Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws*, J. Comput. Phys., 345 (2017), pp. 427--461.

*Ideal GLM-MHD: About the entropy consistent nine-wave magnetic field divergence diminishing ideal magnetohydrodynamics equations*, J. Comput. Phys., 364 (2018), pp. 420--467.

*High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: Viscous heat--conducting fluids and elastic solids*, J. Comput. Phys., 314 (2016), pp. 824--862.

*High order ADER schemes for a unified first order hyperbolic formulation of Newtonian continuum mechanics coupled with electro-dynamics*, J. Comput. Phys., 348 (2017), pp. 298--342.

*A simple extension of the Osher Riemann solver to non-conservative hyperbolic systems*, J. Sci. Comput., 48 (2011), pp. 70--88.

*Accurate numerical discretizations of non-conservative hyperbolic systems*, ESAIM Math. Model. Numer. Anal., 46 (2012), pp. 187--206.

*Symmetric positive linear differential equations*, Comm. Pure Appl. Math., 11 (1958), pp. 333--418.

*Systems of conservation equations with a convex extension*, Proc. Natl. Acad. Sci. USA, 68 (1971), pp. 1686--1688.

*A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations*, Appl. Math. Comput., 272 (2016), pp. 291--308.

*High-Re solutions for incompressible flow using Navier-Stokes equations and multigrid method*, J. Comput. Phys., 48 (1982), pp. 387--411.

*An interesting class of quasilinear systems*, Dokl. Akad. Nauk SSSR, 139 (1961), pp. 521--523.

*Symmetric form of the magnetohydrodynamic equation*, Numer. Methods Mech. Contin. Medium, 3 (1972), pp. 26--34.

*Nonstationary equations of the nonlinear theory of elasticity in Euler coordinates.*, J. Appl. Mech. Tech. Phys., 13 (1972), pp. 868--885.

*Thermodynamics, conservation laws, and symmetric forms of differential equations in mechanics of continuous media*, in Computational Fluid Dynamics Review 95, John Wiley & Sons, New York, 1995, pp. 19--31.

*Elements of Continuum Mechanics and Conservation Laws*, Kluwer Academic/Plenum Publishers, Norwell, MA, 2003.

*Thermodynamic formalization of the fluid dynamics equations for a charged dielectric in an electromagnetic field*, Comput. Math. Math. Phys., 52 (2012), pp. 787--799.

*Total variation diminishing Runge-Kutta schemes*, Math. Comput., 67 (1998), pp. 73--85.

*A provably entropy stable subcell shock capturing approach for high order split form DG for the compressible Euler equations*, J. Comput. Phys., 426 (2021), 109935.

*Weighted essentially non-oscillatory schemes on triangular meshes*, J. Comput. Phys., 150 (1999), pp. 97--127.

*Uniformly accurate diffusive relaxation scheme for multiscale transport equations*, SIAM J. Numer. Anal., 38 (2001), pp. 913--936.

*Entropy stable high order discontinuous Galerkin methods for ideal compressible MHD on structured meshes*, J. Comput. Phys., 354 (2018), pp. 163--178.

*Numerical schemes for hyperbolic systems of conservation laws with stiff diffusive relaxation*, SIAM J. Numer. Anal., 37 (2000), pp. 1246--1270.

*Upwind difference schemes for hyperbolic conservation laws*, Math. Comput., 38 (1982), pp. 339--374.

*Numerical methods for nonconservative hyperbolic systems: A theoretical framework*, SIAM J. Numer. Anal., 44 (2006), pp. 300--321.

*Implicit-explicit Runge-Kutta schemes for stiff systems of differential equations*, Adv. Theory Comput. Math., 3 (2000), pp. 269--288.

*Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation*, J. Sci. Comput., 25 (2005), pp. 129--155.

*Simulation of non-Newtonian viscoplastic flows with a unified first order hyperbolic model and a structure-preserving semi-implicit scheme*, Comput. & Fluids, (2021), 104963.

*Continuum mechanics and thermodynamics in the Hamilton and the Godunov-type formulations*, Contin. Mech. Thermodyn., 30 (2018), pp. 1343--1378.

*A hyperbolic model for viscous Newtonian flows*, Contin. Mech. Thermodyn., 28 (2016), pp. 85--104.

*Continuum mechanics with torsion*, Contin. Mech. Thermodyn., 31 (2019), pp. 1517--1541.

*Fully discrete explicit locally entropy-stable schemes for the compressible Euler and Navier--Stokes equations*, Comput. Math. Appl., 80 (2020), pp. 1343--1359.

*Hyperbolic systems of thermodynamically compatible conservation laws in continuum mechanics*, Math. Comput. Model., 28 (1998), pp. 115--130.

*Conservative models and numerical methods for compressible two-phase flow*, J. Sci. Comput., 42 (2010), pp. 68--95.

*A new continuum model for general relativistic viscous heat-conducting media*, Philos. Trans. Roy. Soc. A, 378 (2020), 20190175.

*Efficient implementation of essentially non-oscillatory shock capturing schemes*, J. Comput. Phys., 77 (1988), pp. 439--471.

*A fully discrete, kinetic energy consistent finite--volume scheme for compressible flows*, J. Comput. Phys., 228 (2009), pp. 1347--1364.

*The numerical viscosity of entropy stable schemes for systems of conservation laws I*, Math. Comput., 49 (1987), pp. 91--103.

*A pressure-based semi-implicit space-time discontinuous Galerkin method on staggered unstructured meshes for the solution of the compressible Navier-Stokes equations at all Mach numbers*, J. Comput. Phys., 341 (2017), pp. 341--376.

*Space-time adaptive ADER discontinuous Galerkin schemes for nonlinear hyperelasticity with material failure*, J. Comput. Phys., 422 (2020), 109758.

*Riemann Solvers and Numerical Methods for Fluid Dynamics*, Springer-Verlag, Berlin, 2009.

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**Submitted**: 4 May 2021

**Accepted**: 19 January 2022

**Published online**: 28 June 2022

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