Very high-order Cartesian-grid finite difference method on arbitrary geometries

Abstract

An arbitrary order finite difference method for curved boundary domains with Cartesian grid is proposed. The technique handles in a universal manner Dirichlet, Neumann or Robin conditions. We introduce the Reconstruction Off-site Data (ROD) method, that transfers in polynomial functions the information located on the physical boundary to the computational domain. Three major advantages are: (1) a simple description of the physical boundary with Robin condition using a collection of points; (2) no analytical expression (implicit or explicit) is required, particularly the ghost cell centroids projection are not needed; (3) we split up into two independent machineries the boundary treatment and the resolution of the interior problem, coupled by the ghost cell values. Numerical evidences based on the simple 2D convection-diffusion operators are presented to prove the capability of the method to reach at least the 6th-order with arbitrary smooth domains.