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

TL;DR

This paper introduces a high-order finite difference method on Cartesian grids for complex geometries, utilizing the ROD technique to accurately handle boundary conditions without explicit boundary expressions, achieving up to 6th-order accuracy.

Contribution

The paper presents the ROD method for boundary treatment in high-order finite difference schemes on arbitrary geometries, simplifying boundary representation and avoiding explicit boundary expressions.

Findings

Achieves at least 6th-order accuracy on smooth domains.

Handles Dirichlet, Neumann, and Robin boundary conditions universally.

Demonstrates effectiveness on 2D convection-diffusion problems.

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 condition. We introduce the Reconstruction Off-site Data (ROD) method, that transfers in polynomial functions the information located on the physical boundary. 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 the ghost cell values. Numerical evidences based on the simple 2D convection-diffusion operators are presented to prove the ability of the method to reach at least the 6th-order with arbitrary smooth domains.