On the Accuracy of a Finite-Difference Method for Parabolic Partial Differential Equations with Discontinuous Boundary Conditions

Abstract

Although the numerical solution of parabolic partial differential equations (PDEs) is widely documented, the effect of discontinuous boundary conditions on numerical accuracy is not. This article employs the Keller box finite-difference method to study the effect of such discontinuities when solving the linear one-demensional transient heat equation. We demonstrate that this formally second-order-accurate scheme can lose accuracy, but that an analytical understanding of the behavior of the solution helps in providing an accuracy-restoring formulation. Benchmark computations are presented that will provide guidance in the numerical solution of nonlinear parabolic PDEs for which there are no closed-form analytical solutions.

Acknowledgments

The authors acknowledge the support of the Mathematics Applications Consortium for Science and Industry (MACSI, www.macsi.ul.ie), funded by the Science Foundation Ireland Mathematics Initiative Grant 06/MI/005.