A new combined stable and dispersion relation preserving compact scheme for non-periodic problems

Abstract

A new compact scheme is presented for computing wave propagation problems and Navier–Stokes equation. A combined compact difference scheme is developed for non-periodic problems (called NCCD henceforth) that simultaneously evaluates first and second derivatives, improving an existing combined compact difference (CCD) scheme. Following the methodologies in Sengupta et al. [T.K. Sengupta, S.K. Sircar, A. Dipankar, High accuracy schemes for DNS and acoustics, J. Sci. Comput. 26 (2) (2006) 151–193], stability and dispersion relation preservation (DRP) property analysis is performed here for general CCD schemes for the first time, emphasizing their utility in uni- and bi-directional wave propagation problems – that is relevant to acoustic wave propagation problems. We highlight: (a) specific points in parameter space those give rise to least phase and dispersion errors for non-periodic wave problems; (b) the solution error of CCD/NCCD schemes in solving Stommel Ocean model (an elliptic p.d.e.) and (c) the effectiveness of the NCCD scheme in solving Navier–Stokes equation for the benchmark lid-driven cavity problem at high Reynolds numbers, showing that the present method is capable of providing very accurate solution using far fewer points as compared to existing solutions in the literature.