On the convergence of a finite volume method for the Navier–Stokes–Fourier system

Abstract

Abstract The goal of the paper is to study the convergence of finite volume approximations of the Navier–Stokes–Fourier system describing the motion of compressible, viscous and heat-conducting fluids. The numerical flux uses upwinding with an additional numerical diffusion of order $\mathcal O(h^{ \varepsilon +1})$, $0<\varepsilon <1$. The approximate solutions are piecewise constant functions with respect to the underlying polygonal mesh. We show that the numerical solutions converge strongly to the classical solution as long as the latter exists. On the other hand, any uniformly bounded sequence of numerical solutions converges unconditionally to the classical solution of the Navier–Stokes–Fourier system without assuming a priori its existence. A similar unconditional convergence result is obtained for a sequence of numerical solutions with uniformly bounded densities and temperatures if the bulk viscosity vanishes.