An adaptive RBF finite collocation approach to track transport processes across moving fronts

We develop a radial basis function finite collocation method (RBF-FC) that utilises an adaptive quadtree dataset to cluster nodes around critical features in the domain, for the solution of transport processes occurring in moving front problems.An adaptive quadtree dataset is used to generate an improved distribution of solution centres in the domain, around which local Hermitian collocation systems are formed. In these local systems, the governing PDE and boundary operators of the problem are enforced by collocation. Globally, the systems are linked via reconstruction of the solution variable in terms of the solution value at neighbouring nodes producing a sparse global matrix with a solution cost that scales linearly with the number of nodes in the domain. By generating an adaptive set of nodal points with the quadtree dataset, we can reduce the global solution error in the RBF-FC method whilst maintaining low solution times.The proposed method is validated on a steady-state boundary layer capture problem and a transient advection problem (infinite Peclet number) in a 2D domain. We then couple the method with an interface tracking technique to solve the transient heat transfer in a Hele-Shaw cell viscous fingering problem. We demonstrate the effectiveness of the proposed method for the solution of convection dominated transport in problems across moving fronts.