An optimal time-stepping algorithm for unsteady
advection-diffusion problems
Naresh M. Chadha and
Niall Madden,
Abstract
The transport of a pollutant in regions of strong tide-induced
currents is mainly governed by a time dependent advection-diffusion
problem. The nature of the problem may alternate from
being diffusion-dominated to advection-dominated throughout a tidal
cycle, see, e.g., [Wu, Fanconer and Lin, 2005]. Standard numerical methods
may not produce acceptable results in a case when a problem is
advection-dominated, whereas schemes designed for the
advection-dominated case are usually suboptimal if the processes are
mainly diffusive~[Roos, Stynes and Tobiska, 2008].
We address this problem by considering a two-weight finite difference
scheme for the one-dimensional advection-diffusion problem. We provide an
analysis that yields certain bounds on the weights involved in the
scheme to ensure that the scheme is von Neumann stable and
satisfies a discrete maximum principle.
The optimal value of the weights are obtained using the notion of an
equivalent differential equation.
Furthermore, we propose an algorithm for choosing the time step to optimize the
rate of convergence of the scheme.
The results of numerical experiments
that demonstrate the competitiveness of the new method are presented.
This material is based upon works supported by the Science Foundation
Ireland under Grant No. 08/RFP/CMS1205.
Submitted September 2011.
Published as in the Journal of Computational and Applied
Mathematics, 294 (2016), 57-77. doi:10.1016/j.cam.2015.07.029