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

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. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

An arbitrary order finite difference method for curved boundary domains with Cartesian grid is proposed, which can handle Dirichlet, Neumann or Robin conditions in a universal manner. The method introduces the Reconstruction Off-site Data (ROD) method to transfer information from the physical boundary to the computational domain through polynomial functions. Three major advantages include simple description of the physical boundary with Robin condition, no requirement for analytical expressions, and splitting the boundary treatment and resolution of the interior problem into two independent parts. Numerical evidence shows the method's capability to reach at least 6th-order accuracy with arbitrary smooth domains.