Efficient shock-capturing numerical schemes using the approach of minimised integrated square difference error for hyperbolic conservation laws

Abstract

The relative phase error (RPE) is a measure of the amount of oscillations while the effective amplification factor measures the amount of damping in a numerical scheme. Recently [2], we have devised a new method baptised as the Minimised Integrated Square Difference Error, (MISDE) for obtaining efficient shock-capturing schemes. In this work, we show based on our concept that the 4-6 Low-Dissipation and Dispersion Runge-Kutta (4-6 LDDRK) [6] is very efficient to capture shocks. Using MISDE, we obtain a modified form of the Fromm's scheme in 1-D which is less dispersive. An optimal value of the parameter, ε at cfl = 0.9 in the LW - ε [10] scheme is also computed. It is shown that the Sαβ scheme [5] is more efficient than the 2-D MacCormack scheme to capture shocks. We solve the 2-D circular explosion problem [11] using the LWLF6 and MCLF6 [1] schemes at some cfl numbers and show that shock capturing ability is dependent on the cfl number. © Springer-Verlag Berlin Heidelberg 2007.