An order-N complexity meshless algorithm for transport-type PDEs, based on local Hermitian interpolation

This work describes a new numerical method utilising radial basis function interpolants. Based on local Hermitian interpolation of function values and boundary operators, and using an explicit time advancement formulation, the method is of order-N complexity. Computational cost to advance the solution in time is minimal, and is largely dependent on local system support size. The explicit time advancement formulation allows a novel solution technique for many nonlinear partial differential equations.The performance of the method is examined for a variety of linear convection–diffusion–reaction problems, featuring both steady and unsteady solutions. The method is also demonstrated with a nonlinear Richards’ equation model, solving an unsaturated flow in porous media problem. The technique is named the local Hermitian interpolation (LHI) method.