Comparison of some optimisation techniques for numerical schemes discretising equations with advection terms

Abstract

We have considered the measures of errors devised by Tam and Webb (1993), Bogey and Bailly (2002) and by Berland et al. (2007) to construct low dispersion, low dissipation and high order numerical schemes in computational aeroacoustics. We modify their measures to obtain three different techniques of optimisation. We investigate the strength and weak points of these three optimisation techniques together with our technique of minimised integrated exponential error for low dispersion and low dissipation (MIEELDLD) (Appadu and Dauhoo, 2009, 2011; Appadu, 2011) when controlling the grade and balance of dispersion and dissipation with reference to some numerical schemes applied to the 1D linear advection equation and the Korteweg-de-Vries Burgers equations. It is observed that the technique of MIEELDLD is more effective than the other three techniques to control the grade and balance of dissipation and dispersion in numerical schemes. Therefore, we use MIEELDLD to optimise parameters in the α-scheme and to obtain a modification to the beam-warming scheme which has improved shock capturing properties. Some numerical experiments are carried out to validate the results by showing that when the optimal parameters are used, better results in terms of shock-capturing properties are obtained.