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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 $S^{\beta}_{\alpha}$ 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. [ABSTRACT FROM AUTHOR]

Liska and Wendroff[4] blended the Lax-Wendroff scheme (LWS) and Lax-Friedrichs scheme (LFS) in order to reduce oscillations and damping from dispersive and diffusive schemes. In this work, we combine the MacCormack Scheme (MCS) and LFS to obtain a new composite scheme which we have formulated as MacCormack/Lax-Friedrichs (MCLF) scheme. Based on the numerical results for conservation laws in 1-D and in 2-D, we observe that the MCLF scheme is more efficient than LWLF to capture shocks. We show analytically that MCLF scheme is less diffusive than LWLF scheme in 1-D and 2-D. Also, we have used another composite scheme in 2-D [3] consisting of Corrected Lax-Friedrichs scheme (CFS) and of Lax-Friedrichs scheme, developed by Liska and Wendroff to solve a 2-D contact discontinuity problem given in [2]. Lastly, we present some preliminary work on a composite scheme made up of MCS and MCLF scheme. [ABSTRACT FROM AUTHOR]