Numerical experiments using high-resolution schemes for unsteady, inviscid, compressible flows

Abstract

The performance of seven high-resolution schemes is investigated in various unsteady, inviscid, compressible flows. We employ the Roe, HLL (Harten, Lax and van Leer), and HLLC (Toro et al.) Riemann solvers, two variants of the van Leer and Steger–Warming flux vector splitting (FVS) schemes, Rusanov's scheme, and a hybrid total variation diminishing (TVD) scheme that combines a high-order Riemann solver with a flux vector splitting scheme. The above schemes have been implemented in conjunction with an implicit-unfactored method which is based on Newton-type sub-iterations and Gauss–Seidel relaxation. The performance of the schemes has been assessed in six unsteady flow problems: two one-dimensional shock tube problems, shock-wave reflection from a wedge, shock-wave diffraction around a cylinder, blast-wave propagation in an enclosure, and interaction of a shock wave with a gas bubble. More dissipative schemes do not necessarily provide faster convergence per time step and also suppress instabilities that occur in certain unsteady flow problems. The efficiency of the solution depends strongly on the advective (high-resolution) scheme. The results reveal that the Roe, HLLC and hybrid TVD schemes provide similar and overall the best results. For the unsteady problems considered here, the computations show that an explicit implementation based on a TVD, fourth-order Runge–Kutta method results in longer computing times than the implicit-unfactored method.