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

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