Conservative, pressure-equilibrium-preserving discontinuous Galerkin method for compressible, multicomponent flows

TL;DR

This paper develops a novel discontinuous Galerkin method that preserves velocity and pressure equilibria in compressible multicomponent flows, ensuring better physical fidelity in simulations.

Contribution

It introduces a pressure-equilibrium-preserving DG scheme with energy conservation, addressing limitations of standard methods for multicomponent flow simulations.

Findings

The scheme maintains velocity and pressure equilibria in test flows.

It demonstrates improved accuracy in multi-dimensional interfacial flow simulations.

Extensions to the method enhance its applicability to complex flow scenarios.

Abstract

This paper concerns preservation of velocity and pressure equilibria in smooth, compressible, multicomponent flows in the inviscid limit. First, we derive the velocity-equilibrium and pressure-equilibrium conditions of a standard discontinuous Galerkin method that discretizes the conservative form of the compressible, multicomponent Euler equations. We show that under certain constraints on the numerical flux, the scheme is velocity-equilibrium-preserving. However, standard discontinuous Galerkin schemes are not pressure-equilibrium-preserving. Therefore, we introduce a discontinuous Galerkin method that discretizes the pressure-evolution equation in place of the total-energy conservation equation. Semidiscrete conservation of total energy, which would otherwise be lost, is restored via the correction terms of [Abgrall, J. Comput. Phys., 372, 2018, pp. 640-666] and [Abgrall et al., J. Comput. Phys., 453, 2022, 110955]. Since the addition of the correction terms prevents exact preservation of pressure and velocity equilibria, we propose modifications that then lead to a velocity-equilibrium-preserving, pressure-equilibrium-preserving, and (semidiscretely) energy-conservative discontinuous Galerkin scheme, although there are certain tradeoffs. Additional extensions are also introduced. We apply the developed scheme to smooth, interfacial flows involving mixtures of thermally perfect gases initially in pressure and velocity equilibria to demonstrate its performance in one, two, and three spatial dimensions.