Optimal time advancing dispersion relation preserving schemes

In this paper we examine the constrained optimization of explicit Runge-Kutta (RK) schemes coupled with central spatial discretization schemes to solve the one-dimensional convection equation The constraints are defined with respect to the correct error propagation equation which goes beyond the traditional von Neumann analysts developed in Sengupta et al [T K Sengupta, A. Dipankar, P Sagaut, Error dynamics beyond von Neumann analysis, J Comput Phys 226 (2007) 1211-1218]. The efficiency of these optimal schemes is demonstrated for the one-dimensional convection problem and also by solving the Navier-Stokes equations for a two-dimensional lid-driven cavity (LDC) problem For the LDC problem, results for Re = 1000 are compared with results using spectral methods in Botella and Peyret [O Botella, R Peyret, Benchmark spectral results on the lid-driven cavity flow, Comput Fluids 27 (1998) 421-433] to calibrate the method in solving the steady state problem We also report the results of the same flow at Re = 10,000 and compare them with some recent results to establish the correctness and accuracy of the scheme for solving unsteady flow problems Finally, we also compare our results for a wave-packet propagation problem with another method developed for computational aeroacoustics (C) 2010 Elsevier Inc All rights reserved