Numerical solution of Navier–Stokes–Korteweg systems by Local Discontinuous Galerkin methods in multiple space dimensions

Abstract

Compressible liquid–vapor flow with phase transitions can be described by systems of Navier–Stokes–Korteweg type. They extend the Navier–Stokes equations by nonlinear higher-grade terms which take the form of either differential or nonlocal integral operators. A numerical approximation method on the basis of the Local Discontinuous Galerkin method in multiple space dimensions is suggested for isothermal flows. It relies on a specific discretization of a non-conservative formulation. To enhance the performance of the overall scheme two techniques are used: (i) local spatial adaptivity based on gradient indicators for the density and (ii) parallelism based on domain decomposition.The paper concludes with numerical experiments in two and three space dimensions. They show the reliability and efficiency of the proposed approach as well as they demonstrate the applicability of the models for several important phase transition phenomena.