On Thermodynamically Compatible Finite Volume Methods and Path-Conservative ADER Discontinuous Galerkin Schemes for Turbulent Shallow Water Flows

Abstract In this paper we propose a new reformulation of the first order hyperbolic model for unsteady turbulent shallow water flows recently proposed in Gavrilyuk et al. (J Comput Phys 366:252–280, 2018). The novelty formulation forwarded here is use evolution variable that guarantees trace discrete Reynolds stress tensor to be always non-negative. mathematical particularly challenging because one important subset equations nonconservative and products also act across genuinely nonlinear fields. Therefore, consider thermodynamically compatible viscous extension necessary define proper vanishing viscosity limit inviscid absolutely fundamental subsequent construction numerical scheme. We then introduce two different, but related, families methods its solution. scheme provably semi-discrete finite volume makes direct Godunov form can therefore called formalism . method mimics underlying continuous system exactly at level thus consistent with conservation total energy, entropy inequality model. second general purpose high path-conservative ADER discontinuous Galerkin element posteriori subcell limiter applied as well Both schemes have common they make path integrals jump terms interfaces. different are compared each other (2018) on example three Riemann problems. Moreover, comparison fully resolved solution small parameter (vanishing limit). all cases an excellent agreement between achieved. furthermore show convergence rates ADER-DG up sixth space time present test problems where compare available experimental data.