Hodiex: A Sixth Order Accurate Method for Solving Elliptical PDEs

Abstract

This paper describes a method for discretizing general linear two dimensional elliptical PDEs with variable coefficients, Lu=g, which achieves high orders of accuracy on an extended range of problems. The method can be viewed as an extension of the ELLPACK6 discretization module HODIE ("High Order Difference Approximation with Identity Expansion"), which achieves high orders of accuracy on a more limited class of problems. We thus call this method HODIEX. An advantage of HODIEX methods, including the one described here, is that they are based on a compact 9-point stencil which yields linear systems with a smaller bandwidth than if a larger stencil were used to achieve higher accuracy.