High-order accurate finite difference discretisations on fully unstructured dual quadrilateral meshes

TL;DR

This paper introduces a new high-order finite difference method for accurate numerical differentiation on unstructured quadrilateral meshes, applicable to various PDEs in multiple dimensions.

Contribution

It generalizes finite difference schemes to arbitrary unstructured meshes and higher dimensions, enhancing accuracy and flexibility in numerical PDE solutions.

Findings

Achieves high-order accuracy on unstructured dual quadrilateral meshes

Successfully applied to linear, nonlinear, second, and fourth order PDEs

Demonstrates effectiveness with subdivision surface refinement

Abstract

We present a novel approach for high-order accurate numerical differentiation on unstructured meshes of quadrilateral elements. To differentiate a given function, an auxiliary function with greater smoothness properties is defined which when differentiated provides the derivatives of the original function. The method generalises traditional finite difference methods to meshes of arbitrary topology in any number of dimensions for any order of derivative and accuracy. We demonstrate the accuracy of the numerical scheme using dual quadrilateral meshes and a refinement method based on subdivision surfaces. The scheme is applied to the solution of a range of partial differential equations, including both linear and nonlinear, second and fourth order equations, and a time-dependent first order equation.