Entropy Stable Discontinuous Galerkin Schemes on Moving Meshes for Hyperbolic Conservation Laws

This work is focused on the entropy analysis of a semi-discrete nodal discontinuous Galerkin spectral element method (DGSEM) on moving meshes for hyperbolic conservation laws. The DGSEM is constructed with a local tensor-product Lagrange-polynomial basis computed from Legendre-Gauss-Lobatto points. Furthermore, the collocation of interpolation and quadrature nodes is used in the spatial discretization. This approach leads to discrete derivative approximations in space that are summation-by-parts (SBP) operators. On a static mesh, the SBP property and suitable two-point flux functions, which satisfy the entropy condition from Tadmor, allow to mimic results from the continuous entropy analysis, if it is ensured that properties such as positivity preservation (of the water height, density or pressure) are satisfied on the discrete level. In this paper, Tadmor's condition is extended to the moving mesh framework. We show that the volume terms in the semi-discrete moving mesh DGSEM do not contribute to the discrete entropy evolution when a two-point flux function that satisfies the moving mesh entropy condition is applied in the split form DG framework. The discrete entropy behavior then depends solely on the interface contributions and on the domain boundary contribution. The interface contributions are directly controlled by proper choice of the numerical element interface fluxes. If an entropy conserving two-point flux is chosen, the interface contributions vanish. To increase the robustness of the discretization we use so-called entropy stable two-point fluxes at the interfaces that are guaranteed entropy dissipative and thus give a bound on the interface contributions in the discrete entropy balance. The remaining boundary condition contributions depend on the type of the considered boundary condition. E.g. for periodic boundary conditions that are of entropy conserving type, our methodology with the entropy conserving interface fluxes is fully entropy conservative and with the entropy stable interface fluxes is guaranteed entropy stable. The presented proof does not require any exactness of quadrature in the spatial integrals of the variational forms. As it is the case for static meshes, these results rely on the assumption that additional properties like positivity preservation are satisfied on the discrete level. Besides the entropy stability, the time discretization of the moving mesh DGSEM will be investigated and it will be proven that the moving mesh DGSEM satisfies the free stream preservation property for an arbitrary s-stage Runge-Kutta method, when periodic boundary conditions are used. The theoretical properties of the moving mesh DGSEM will be validated by numerical experiments for the compressible Euler equations with periodic boundary conditions

A simple and general framework for the construction of thermodynamically compatible schemes for computational fluid and solid mechanics

Financiado para publicación en acceso aberto: Universidade de Vigo/CISUGWe introduce a simple and general framework for the construction of thermodynamically compatible schemes for the numerical solution of overdetermined hyperbolic PDE systems that satisfy an extra conservation law. As a particular example in this paper, we consider the general Godunov-Peshkov-Romenski (GPR) model of continuum mechanics that describes the dynamics of nonlinear solids and viscous fluids in one single unified mathematical formalism. A main peculiarity of the new algorithms presented in this manuscript is that the entropy inequality is solved as a primary evolution equation instead of the usual total energy conservation law, unlike in most traditional schemes for hyperbolic PDE. Instead, total energy conservation is obtained as a mere consequence of the proposed thermodynamically compatible discretization. The approach is based on the general framework introduced in Abgrall (2018) [1]. In order to show the universality of the concept proposed in this paper, we apply our new formalism to the construction of three different numerical methods. First, we construct a thermodynamically compatible finite volume (FV) scheme on collocated Cartesian grids, where discrete thermodynamic compatibility is achieved via an edge/face-based correction that makes the numerical flux thermodynamically compatible. Second, we design a first type of high order accurate and thermodynamically compatible discontinuous Galerkin (DG) schemes that employs the same edge/face-based numerical fluxes that were already used inside the finite volume schemes. And third, we introduce a second type of thermodynamically compatible DG schemes, in which thermodynamic compatibility is achieved via an element-wise correction, instead of the edge/face-based corrections that were used within the compatible numerical fluxes of the former two methods. All methods proposed in this paper can be proven to be nonlinearly stable in the energy norm and they all satisfy a discrete entropy inequality by construction. We present numerical results obtained with the new thermodynamically compatible schemes in one and two space dimensions for a large set of benchmark problems, including inviscid and viscous fluids as well as solids. An interesting finding made in this paper is that, in numerical experiments, one can observe that for smooth isentropic flows the particular formulation of the new schemes in terms of entropy density, instead of total energy density, as primary state variable leads to approximately twice the convergence rate of high order DG schemes for the entropy density.Agencia Estatal de Investigación | Ref. PID2021-122625OB-I0

A simple and general framework for the construction of thermodynamically compatible schemes for computational fluid and solid mechanics

We introduce a simple and general framework for the construction of thermodynamically compatible schemes for the numerical solution of overdetermined hyperbolic PDE systems that satisfy an extra conservation law. As a particular example in this paper, we consider the general Godunov-Peshkov-Romenski (GPR) model of continuum mechanics that describes the dynamics of nonlinear solids and viscous fluids in one single unified mathematical formalism. A main peculiarity of the new algorithms presented in this manuscript is that the entropy inequality is solved as a primary evolution equation instead of the usual total energy conservation law, unlike in most traditional schemes for hyperbolic PDE. Instead, total energy conservation is obtained as a mere consequence of the proposed thermodynamically compatible discretization. The approach is based on the general framework introduced in Abgrall (2018) [1]. In order to show the universality of the concept proposed in this paper, we apply our new formalism to the construction of three different numerical methods. First, we construct a thermodynamically compatible finite volume (FV) scheme on collocated Cartesian grids, where discrete thermodynamic compatibility is achieved via an edge/face-based correction that makes the numerical flux thermodynamically compatible. Second, we design a first type of high order accurate and thermodynamically compatible discontinuous Galerkin (DG) schemes that employs the same edge/face-based numerical fluxes that were already used inside the finite volume schemes. And third, we introduce a second type of thermodynamically compatible DG schemes, in which thermodynamic compatibility is achieved via an element-wise correction, instead of the edge/face-based corrections that were used within the compatible numerical fluxes of the former two methods. All methods proposed in this paper can be proven to be nonlinearly stable in the energy norm and they all satisfy a discrete entropy inequality by construction. We present numerical results obtained with the new thermodynamically compatible schemes in one and two space dimensions for a large set of benchmark problems, including inviscid and viscous fluids as well as solids. An interesting finding made in this paper is that, in numerical experiments, one can observe that for smooth isentropic flows the particular formulation of the new schemes in terms of entropy density, instead of total energy density, as primary state variable leads to approximately twice the convergence rate of high order DG schemes for the entropy density