Temporal discretization for improving kinetic-energy and entropy preservation properties in KEEP schemes

Abstract

A temporal discretization for improving kinetic-energy and entropy preservation on compressible flow simulations is proposed. The key of the proposed scheme is to introduce the auxiliary velocity, density, and internal energy to the temporal discretization and derive the quantities that satisfy the analytical relations of temporal-derivative terms in the kinetic-energy and entropy equations at the discrete level by utilizing the Maclaurin expansion and truncating high-order terms. The numerical fluxes in the kinetic-energy and entropy preserving (KEEP) schemes proposed by Kuya et al. (2018) are also evaluated by the auxiliary quantities to enhance kinetic-energy and entropy preservation in both spatial and temporal discretizations. The numerical experiments of the inviscid Taylor–Green vortex demonstrate that the proposed scheme significantly improves the kinetic energy and entropy preservation and achieves stable and non-dissipative compressible flow simulations at CFL ≈1.0, at which the temporal-discretization errors for the kinetic-energy and entropy preservation are not negligible when the widely used fourth-order four-step Runge–Kutta scheme is employed.