Comparison of the convergence to steady-state solution with weighted-type finite-difference schemes for the Euler equations

Abstract

Weighted-type finite-difference schemes are a class of widely used nonlinear schemes that can capture strong discontinuities accurately and efficiently. For the Euler equations without source terms, poor convergence of weighted-type schemes is a widely known difficulty in finding steady-state solutions with strong shock waves. The primary reason for this lies in the fact that classical weighted-type schemes produce spurious oscillations near strong discontinuities. Recently, a novel weighted-type scheme has been developed. The nonlinear weights of the new scheme are fourth-order accurate and do not reduce the accuracy at the high-order critical points, which is beneficial for steady-state convergence. In this paper, we compare the convergence performances of classical and new weighted-type schemes in detail. Several benchmark problems containing shock waves, contact discontinuities, and rarefaction waves were used to compare the convergence performance among different weighted-type schemes. The results show that the new weighted-type scheme basically eliminates slight post-shock oscillations, and the residual settles to machine zero. Compared to classical weighted-type schemes, the steady-state convergence performance of the new weighted-type scheme is significantly improved.