A first order hyperbolic reformulation of the Navier-Stokes-Korteweg system based on the GPR model and an augmented Lagrangian approach

Abstract

In this paper we present a novel first order hyperbolic reformulation of the barotropic Navier-Stokes-Korteweg system. The new formulation is based on a combination of the first order hyperbolic Godunov-Peshkov-Romenski (GPR) model of continuum mechanics with an augmented Lagrangian approach that allows to rewrite nonlinear dispersive systems in first order hyperbolic form at the aid of new evolution variables. The governing equations for the new evolution variables introduced by the rewriting of the dispersive part are endowed with curl involutions that need to be taken care of. In this paper, we account for the curl involutions at the aid of a thermodynamically compatible generalized Lagrangian multiplier (GLM) approach, similar to the GLM divergence cleaning introduced by Munz et al. for the divergence constraint of the magnetic field in the Maxwell and MHD equations. A key feature of the new mathematical model presented in this paper is its ability to restore hyperbolicity even for non-convex equations of state, such as the van der Waals equation of state, thanks to the use of an augmented Lagrangian approach and the resulting inclusion of surface tension terms into the hyperbolic flux.The governing PDE system proposed in this paper is solved at the aid of a high order ADER discontinuous Galerkin finite element scheme with a posteriori subcell finite volume limiter in order to deal with shock waves, discontinuities and steep gradients in the numerical solution. We propose an exact solution of our new mathematical model, show numerical convergence rates of up to sixth order of accuracy and show numerical results for several standard benchmark problems, including travelling wave solutions, stationary bubbles and Ostwald ripening in one and two space dimensions.