New analysis of the du Fort-Frankel methods

Neta Corem; Adi Ditkowski

doi:https://doi.org/10.1007/s10915-012-9627-2

New analysis of the du Fort-Frankel methods

Neta Corem, Adi Ditkowski

School of Mathematical Sciences

Tel Aviv University

Research output: Contribution to journal › Article › peer-review

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 τ= O(k ²}+h ²+k ²h 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.

Original language	English
Pages (from-to)	35-54
Number of pages	20
Journal	Journal of Scientific Computing
Volume	53
Issue number	1
DOIs	https://doi.org/10.1007/s10915-012-9627-2
State	Published - Oct 2012

Keywords

Du Fort-Frankel
Finite difference
Finite difference stability
Generalized Du Fort-Frankel

All Science Journal Classification (ASJC) codes

Software
General Engineering
Computational Mathematics
Theoretical Computer Science
Applied Mathematics
Numerical Analysis
Computational Theory and Mathematics

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Cite this

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title = "New analysis of the du Fort-Frankel methods",

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 τ= O(k 2}+h 2+k 2h 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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AB - 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 τ= O(k 2}+h 2+k 2h 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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KW - Finite difference

KW - Finite difference stability

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JO - Journal of Scientific Computing

JF - Journal of Scientific Computing

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ER -

New analysis of the du Fort-Frankel methods

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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