An implicit upwinding volume element method based on meshless radial basis function techniques for modelling transport phenomena

This work presents a control volume (CV) method for transient transport problems where the cell surface fluxes are reconstructed using local interpolation functions that besides interpolating the nodal values of the field variable, also satisfies the governing equation at some auxiliary points in the interpolation stencils. The interpolation function relies on a Hermitian radial basis function (HRBF) meshless collocation approach to find the solution of auxiliary local boundary/initial value problems, that are solved using the same time‐integration scheme adopted to update the global CV solution. By the use of interpolation functions that approximate the governing equation a form of analytical upwinding scheme is achieved, without the need of using predefined interpolation stencils according to the magnitude and direction of the local advective velocity. In this way, the interpolation formula retains the desired information about the advective velocity field allowing the use of centrally defined stencils even in the cases of advective dominant problems. This new CV approach is referred to as the CV‐HRBF method. Two time‐stepping formulations are considered: a full implicit approach and the weighted Crank–Nicholson one. The implicit upwinding scheme, intrinsic to the proposed CV‐HRBF, is tested by solving a travelling front problem at the Péclet number equal to 500, 1000 and infinity; with the latter corresponding to a shock front. Finally, the accuracy of the numerical method is validated against one‐ and three‐dimensional reactive transport problems characterized by smooth solutions. Copyright © 2009 John Wiley & Sons, Ltd.