Finite volume discretization of heat equation and compressible Navier-Stokes equations with weak Dirichlet boundary condition on triangular grids

A vertex-based finite volume method for Laplace operator on triangular grids is proposed in which Dirichlet boundary conditions are implemented weakly. The scheme satisfies a summation-by-parts (SBP) property including boundary conditions which can be used to prove energy stability of the scheme for the heat equation. A Nitsche-type penalty term is proposed which gives improved accuracy. The scheme exhibits second order convergence in numerical experiments. For the compressible Navier-Stokes equations we construct a finite volume scheme in which Dirichlet boundary conditions on the velocity and temperature are applied in a weak manner. Using the centered kinetic energy preserving flux, the scheme is shown to be consistent with the global kinetic energy equation. The SBP discretization of viscous and heat conduction terms together with penalty terms are combined with upwind fluxes in a Godunov-MUSCL scheme. Numerical results on some standard test cases for compressible flows are given to demonstrate the performance of the scheme.