An efficient IMEX-DG solver for the compressible Navier-Stokes equations for non-ideal gases

TL;DR

An e-cient, accurate and robust IMEX solver for the compressible Navier-Stokes equation describing non-ideal gases with general equations of state is proposed and allows to simulate low Mach regimes at a significantly reduced computational cost, while maintaining full second order accuracy also for higher Mach number regimes.

Abstract

We propose an efficient, accurate and robust IMEX solver for the compressible Navier-Stokes equation describing non-ideal gases with general equations of state. The method, which is based on an $h-$adaptive Discontinuos Galerkin spatial discretization and on an Additive Runge Kutta IMEX method for time discretization, is tailored for low Mach number applications and allows to simulate low Mach regimes at a significantly reduced computational cost, while maintaining full second order accuracy also for higher Mach number regimes. The method has been implemented in the framework of the $deal.II$ numerical library, whose adaptive mesh refinement capabilities are employed to enhance efficiency. Refinement indicators appropriate for real gas phenomena have been introduced. A number of numerical experiments on classical benchmarks for compressible flows and their extension to real gases demonstrate the properties of the proposed method.

Figures (32)

• Figure 1: Adaptive simulation of the inviscid isentropic vortex benchmark: a) computational mesh at $t = T_f$, b) contour plot of the density at $t = T_f$.
• Figure 2: Sod shock tube problem at $t = 0.2$, comparison with exact solution, a) density, b) velocity, c) pressure.
• Figure 3: Sod shock tube problem with van der Waals EOS at $t = 0.2$, comparison with exact solution, a) density, b) velocity, c) pressure.
• Figure 4: Sod shock tube problem with Peng-Robinson EOS at $t = 0.2$, comparison with reference solution, a) density, b) velocity, c) pressure.
• Figure 5: Computational results for the 2D lid-driven cavity, a) streamlines, b) comparison with the solutions in ghia:1982 and in tavelli:2017. Blue dots denote the results in ghia:1982, red crosses the results in tavelli:2017 and the black line our numerical results.