HIGH ORDER SEMI-IMPLICIT WENO SCHEMES FOR ALL-MACH FULL EULER SYSTEM OF GAS DYNAMICS

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 a 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 non-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. The 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.

Automated summary

In this paper, a new semi-implicit scheme is proposed to solve the high order problems in gas dynamics equations. The scheme combines the semi-implicit temporal integrator and finite difference WENO schemes for high order accuracy in time and space. Special IMEX schemes are constructed to handle non-well-prepared initial conditions, and the asymptotic preserving and accuracy properties are proven.