Decoupling schemes for predicting compressible fluid flows

In this paper we consider two implicit schemes for the compressible Euler equations, consisting of a scalar advection equation for the density, a vectorial advection equation for the velocity and a barotropic equation of state. The spatial discretization utilizes first order finite elements and the equations are integrated in time by means of the backward Euler scheme, however, the nonlinearity is handled in two different ways. The most straightforward possibility is to solve the nonlinear system by the Newton's method. We show that the resulting discrete fully-nonlinear scheme conserves mass and momentum. Further, we employ a two–level splitting scheme with respect to the physical processes and thus we decouple the mass and momentum conservation equations. The resulting scheme can be applied iteratively at each time level to approximate the solution of the fully nonlinear scheme. The capability of the proposed schemes are illustrated by numerical results for a two–dimensional model problem with an initial density perturbation. The results clearly demonstrate that two iterations at each time level are sufficient to produce non-oscillatory results, without employing any limiters or shock capturing. © 2018 Elsevier Ltd