High Order Finite Difference Schemes for the Heat Equation Whose Convergence Rates are Higher Than Their Truncation Errors

TL;DR

This paper introduces high-order finite difference schemes for the heat equation that achieve convergence rates exceeding their truncation errors, enabling more efficient and accurate numerical solutions.

Contribution

It demonstrates the construction of stable schemes with errors smaller than their truncation errors, allowing for improved efficiency and potential post-processing enhancements.

Findings

Stable schemes with errors smaller than truncation errors are possible.

Enhanced accuracy through post-processing techniques.

More flexible scheme design for the heat equation.

Abstract

Typically when a semi-discrete approximation to a partial differential equation (PDE) is constructed a discretization of the spatial operator with a truncation error $\tau$ is derived. This discrete operator should be semi-bounded for the scheme to be stable. Under these conditions, the Lax--Ricchtmyer equivalence theorem assures that the scheme converges and that the error will be, at most, of the order of $\| \tau \|$. In most cases, the error is in indeed of the order of $\| \tau \|$. We demonstrate that for the Heat equation stable schemes can be constructed, whose truncation errors are $\tau$, however, the actual errors are much smaller. This gives more degrees of freedom in the design of schemes which can make them more efficient (more accurate or compact) than standard schemes. In some cases, the accuracy of the schemes can be further enhanced using post-processing procedures.