Application of point implicit Runge–Kutta methods to inviscid and laminar flow problems using AUSM and AUSM⁺ upwinding

Abstract

Explicit Runge–Kutta methods preconditioned by a pointwise matrix valued preconditioner can significantly improve the convergence rate to approximate steady state solutions of laminar flows. This has been shown for central discretisation schemes and Roe upwinding. Since the first-order approximation to the inviscid flux assuming constant weighting of the dissipative terms is given by the absolute value of the Roe matrix, the construction of the preconditioner is rather simple compared to other upwind techniques. However, in this article we show that similar improvements in the convergence rates can also be obtained for the AUSM⁺ scheme. Following the ideas for the central and Roe schemes, the preconditioner is obtained by a first-order approximation to the derivative of the convective flux. Viscous terms are included into the preconditioner considering a thin shear layer approximation. A complete derivation of the derivative terms is shown. In numerical examples, we demonstrate the improved convergence rates when compared with a standard explicit Runge–Kutta method accelerated with local time stepping.

Keywords:

• AUSM and AUSM⁺ upwinding
• point implicit preconditioner
• inviscid flow
• laminar flow
• Runge–Kutta methods