An optimal time-stepping algorithm for unsteady advection–diffusion problems

Abstract

We consider the numerical solution of a linear time dependent advection–diffusion problem by an implicit two-weight, three-point finite difference scheme. We extend the scheme proposed by Chadha and Madden (2011), to incorporate an optimal time step selection algorithm for the method. The optimal values of the weights involved in the scheme have been obtained by using the notion of an equivalent differential equation (Warming and Hyett, 1974), and, by enforcing certain constraints due to von Neumann stability and a discrete maximum principle, we obtain sharp bounds for optimal time-stepping in terms of mesh width and problem data. Furthermore, we formulate the time step selection as a root-finding problem, and use this to show the optimal time step exists and is unique. The resulting method, based on optimal values of weights and optimal time-stepping, is of fifth-order in space, and third-order in time. The results of extensive numerical experiments are presented for several classic test problems for various values of the advection and diffusion parameters. These show that our proposed method is superior (quantitatively and qualitatively) to other schemes that can be represented within the framework of two-weighted schemes. We also present comparisons with other high-order methods, and an application of the method to a problem with non-smooth data.