On the order of accuracy for finite difference approximations of partial differential equations using stencil composition

TL;DR

This paper explores how stencil composition can be used to create finite difference schemes for PDEs, analyzing their accuracy, stability, and applying them to complex models like the Cahn-Hilliard equation.

Contribution

It introduces a systematic study of stencil composition for PDE discretization, including properties, accuracy order, stability, and practical applications to benchmark problems.

Findings

Stencil composition can produce higher-order accurate finite difference schemes.

The composed stencils maintain stability properties comparable to traditional methods.

Numerical experiments confirm the theoretical order of accuracy and applicability to complex PDEs.

Abstract

Stencil composition uses the idea of function composition, wherein two stencils with arbitrary orders of derivative are composed to obtain a stencil with a derivative order equal to sum of the orders of the composing stencils. In this paper, we show how stencil composition can be applied to form finite difference stencils in order to numerically solve partial differential equations (PDEs). We present various properties of stencil composition and investigate the relationship between the order of accuracy of the composed stencil and that of the composing stencils. We also present comparisons between the stability restrictions of composed higher-order PDEs to their compact versions and numerical experiments wherein we verify the order of accuracy by convergence tests. To demonstrate an application to PDEs, a boundary value problem involving the two-dimensional biharmonic equation is numerically solved using stencil composition and the order of accuracy is verified by performing a convergence test. The method is then applied to the Cahn-Hilliard phase-field model. In addition to sample results in 2D and 3D for this benchmark problem, the scalability, spectral properties, and sparsity is explored.