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 \(\|\tau \|\). 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 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.
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Notes
- 1.
This scheme was presented for demonstrating the phenomenon that the error, due to high frequency modes, is lower than the truncation error. As this is not a practical scheme, full analysis of the error is not presented, only a demonstration of the dynamics of high frequency error modes is presented. Full analysis is given for the scheme presented in the next section.
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Acknowledgements
The author would like to thank Jennifer K. Ryan, Chi-Wang Shu and Sigal Gottlieb for the fruitful discussions and their help. The author would also like to thank the anonymous reviewers for their useful remarks.
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Ditkowski, A. (2015). High Order Finite Difference Schemes for the Heat Equation Whose Convergence Rates are Higher Than Their Truncation Errors. In: Kirby, R., Berzins, M., Hesthaven, J. (eds) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014. Lecture Notes in Computational Science and Engineering, vol 106. Springer, Cham. https://doi.org/10.1007/978-3-319-19800-2_13
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DOI: https://doi.org/10.1007/978-3-319-19800-2_13
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