Abstract
In 1953 Du Fort and Frankel (Math. Tables Other Aids Comput., 7(43):135–152, 1953) proposed to solve the heat equation u t =u xx using an explicit scheme, which they claim to be unconditionally stable, with a truncation error is of order of \(\tau= O({{k}}^{2}+{{h}}^{2}+\frac{{{k}}^{2}}{{{h}}^{2}})\). Therefore, it is not consistent when k=O(h).
In the analysis presented below we show that the Du Fort–Frankel schemes are not unconditionally stable. However, when properly defined, the truncation error vanishes as h,k→0.
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Notes
|T| is the operator’s discrete norm, i.e. \(|T|= \|T\|_{h} = \sup_{{\|u\|_{h}=1}} \|T u\|_{h} \), where for u=(u 1,…,u N ) and equidistance grid, \(\|u\|_{h}=\sqrt{\sum_{j=1}^{N}{|u|^{2}}{h}}\). In case of nonuniform grid definition is dependent on the variable space distance.
i.e. Q is similar to a matrix that has only 2×2 blocks on its main diagonal.
References
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Acknowledgements
The authors would like to thank Prof. Saul Abarbanel, Sigal Gottlieb, Bertil Gustafsson and Chi-Wang Shu, for useful discussions and their help. We would also like to thanks the anonymous referees for their constructive remarks.
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Corem, N., Ditkowski, A. New Analysis of the Du Fort–Frankel Methods. J Sci Comput 53, 35–54 (2012). https://doi.org/10.1007/s10915-012-9627-2
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DOI: https://doi.org/10.1007/s10915-012-9627-2