Holistic discretisation of shear dispersion in a two-dimensional channel

Consider the spread of a contaminant along a 2D channel or river. We directly derive the 1D discrete numerical model from the 2D advective and diffusive 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 dividing the physical domain into rectangular 2D elements through introducing artificial insulating boundaries which are later removed. The resulting holistic discretisation is consistent with the 1D Taylor model for shear dispersion in the 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. Centre manifold theory also systematically incorporates the physical inlet and outlet conditions into the 1D discretisation. This method is straightforwardly extended to nonlinear reaction-diffusion equations and more complex geometries