Forward and backward finite differences for isotropic gradients on a square lattice derived from a rectangular lattice formulation

Abstract

In this research, forward and backward isotropic finite differences for the gradient operator are developed up to fourth order. Isotropic gradients are characterized by error terms which have small directional preference. Currently, only centred finite differences are available for isotropic discretization in the scientific literature; however, these finite differences are not suited for evaluation on domain boundaries or at multilevel lattice interfaces. We show that the order of accuracy with respect to the gradient direction with isotropic discretizations can be higher in some situations, and, at the same time, use neighbours that are closer to the evaluation point than with standard discretizations. The isotropic discretizations presented here were developed for a special rectangular lattice formulation, and general stencil weights are provided. When the rectangular lattice has a certain aspect ratio, forward and backward isotropic gradients can be obtained on a square lattice.

Keywords:

• numerical differentiation
• isotropic finite difference
• isotropic gradient
• forward stencil
• backward stencil
• square lattice

2010 AMS Subject Classifications::

• 65L99
• 65M99
• 65N99

Acknowledgements

The author is particularly grateful for the assistance given by Benoit Malouin in the typesetting of this article, and thanks the anonymous reviewers for their insightful and constructive comments.