Abstract

We consider the dispersion of a pollutant along a 2D channel or river. We directly derive the 1D numerical model from the 2D physical dynamics for the evolution of the contaminant. The holistic discretisation of the 2D advection-diffusion equation is placed within the purview of centre manifold theory by diving the physical domain into 2D elements and introducing artifical insulating boundaries which are later removed. The use of centre manifold theory also provides a systematic treatment of the physical boundaries in the numerical discretisation. The resulting holistic discretisation is consistent with the 1D Taylor model for shear dispersion in a channel.

This new technique accurately models the subgrid scale processes and provides a direct link between the 1D numerical discretisation and the original 2D physical dynamics. This method is easily extended to nonlinear reaction-diffusion equations and more complex geometries.

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