High Order Semi-implicit WENO Schemes for All Mach Full Euler System of Gas Dynamics

TL;DR

This paper introduces a high order semi-implicit WENO scheme for the full Euler system of gas dynamics that efficiently handles all Mach number flows by combining explicit treatment of material waves with implicit treatment of acoustic waves, ensuring accuracy and stability.

Contribution

It develops a novel high order semi-implicit scheme using IMEX-RK and WENO methods that is asymptotic preserving and effective for low Mach number flows, addressing limitations of traditional explicit schemes.

Findings

Scheme achieves high order accuracy in space and time.

Effectively captures discontinuities in compressible flows.

Proven to be asymptotic preserving and divergence-free.

Abstract

In this paper, we propose a new high order semi-implicit scheme for the all Mach full Euler equations of gas dynamics. Material waves are treated explicitly, while acoustic waves are treated implicitly, thus avoiding severe CFL restrictions for low Mach flows. High order accuracy in time is obtained by semi-implicit temporal integrator based on the IMEX Runge-Kutta (IMEX-RK) framework. High order in space is achieved by finite difference WENO schemes with characteristic-wise reconstructions adapted to the semi-implicit IMEX-RK time discretization. Type A IMEX schemes are constructed to handle not well-prepared initial conditions. Besides, these schemes are proven to be asymptotic preserving and asymptotically accurate as the Mach number vanishes for well-prepared initial conditions. Divergence-free property of the time-discrete schemes is proved. The proposed scheme can also well capture discontinuous solutions in the compressible regime, especially for two dimensional Riemann problems. Numerical tests in one and two space dimensions will illustrate the effectiveness of the proposed schemes.