On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows
The authors introduce a new reformulation of the first order hyperbolic model for unsteady turbulent shallow water flows introduced and studied in some previous works, e.g. [\textit{S. Gavrilyuk} et al., J. Comput. Phys. 366, 252--280 (2018; Zbl 1406.65068)]. The main key of the model reformulation proposed is the decomposition of the specific Reynolds stress tensor \({\textbf{P}}\) at the aid of a new object \({\textbf{Q}}\) such that \({\textbf{P}}={\textbf{Q}}{\textbf{Q}}^T\). This yields \(\mathrm{tr} {\textbf{P}}\geq 0\). The mathematical model is interesting since one important subset of evolution equations is nonconservative and the nonconservative products also act across genuinely nonlinear fields. A thermodynamically compatible \textit{viscous extension} of the model that is necessary to define a proper vanishing viscosity limit of the inviscid model and that is absolutely fundamental for the subsequent construction of a thermodynamically compatible numerical scheme is considered. Two families of numerical methods are introduced. The first scheme is a provably \textit{thermodynamically compatible} semi-discrete finite volume scheme that makes direct use of the \textit{Godunov form} of the equations and can therefore be called a \textit{discrete Godunov formalism}. The new method mimics the underlying continuous viscous system \textit{exactly} at the semi-discrete level and is thus consistent with the conservation of total energy, with the entropy inequality and with the vanishing viscosity limit of the model. The second scheme is a high order path-conservative ADER discontinuous Galerkin finite element method with a posteriori subcell finite volume limiter that can be applied to the inviscid as well as to the viscous form of the model. Both schemes use path integrals to define the jump terms at the element interfaces. Some different numerical methods are applied to the inviscid system and are compared with each other and with the scheme proposed in [the third author et al., J. Comput. Phys. 366, 252-280 (2018; Zbl 1406.65068)] on the example of three Riemann problems. An excelent agreement between all different methods was observed in all cases. For the high order ADER-DG schemes a numerical convergence study was carried out at the aid of a manufactured solution, since the analytic solution used in the above stated reference was too simple for a high order DG scheme. It is shown numerically that the convergence rates of ADER-DG schemes is up to sixth order in space and time.
thermodynamically compatible finite volume schemes
path-conservative ADER discontinuous Galerkin schemes
scientific article; zbMATH DE number 7384714
scientific article