High order finite difference schemes for the heat equation whose convergence rates are higher than their truncation errors

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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 τ is derived. This discrete operator should be semi-bounded for the scheme to be stable. Under these conditions the Lax–Richtmyer equivalence theorem assures that the scheme converges and that the error will be, at most, of the order of ║τ║. In most cases the error is in indeed of the order of ║τ║. We demonstrate that for the Heat equation stable schemes can be constructed, whose truncation errors are τ, 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.

Original languageEnglish
Title of host publicationSpectral and High Order Methods for Partial Differential Equations, ICOSAHOM 2014, Selected papers from the ICOSAHOM
EditorsRobert M. Kirby, Martin Berzins, Jan S. Hesthaven
PublisherSpringer Verlag
Pages167-178
Number of pages12
ISBN (Print)9783319197999
DOIs
StatePublished - 2015
Event10th International Conference on Spectral and High-Order Methods, ICOSAHOM 2014 - Salt Lake City, United States
Duration: 23 Jun 201427 Jun 2014

Publication series

NameLecture Notes in Computational Science and Engineering
Volume106
ISSN (Print)1439-7358

Conference

Conference10th International Conference on Spectral and High-Order Methods, ICOSAHOM 2014
Country/TerritoryUnited States
CitySalt Lake City
Period23/06/1427/06/14

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