TY - GEN

T1 - High order finite difference schemes for the heat equation whose convergence rates are higher than their truncation errors

AU - Ditkowski, A.

N1 - Publisher Copyright:
© Springer International Publishing Switzerland 2015.

PY - 2015

Y1 - 2015

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84951913668&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-19800-2_13

DO - 10.1007/978-3-319-19800-2_13

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AN - SCOPUS:84951913668

SN - 9783319197999

T3 - Lecture Notes in Computational Science and Engineering

SP - 167

EP - 178

BT - Spectral and High Order Methods for Partial Differential Equations, ICOSAHOM 2014, Selected papers from the ICOSAHOM

A2 - Kirby, Robert M.

A2 - Berzins, Martin

A2 - Hesthaven, Jan S.

PB - Springer Verlag

Y2 - 23 June 2014 through 27 June 2014

ER -