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A three-time-level explicit difference scheme of the second order of accuracy for parabolic equations

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Abstract

The difference schemes of Richardson [1] and of Crank-Nicolson [2] are schemes providing second-order approximation. Richardson's three-time-level difference scheme is explicit but unstable and the Crank-Nicolson two-time-level difference scheme is stable but implicit. Explicit numerical methods are preferable for parallel computations. In this paper, an explicit three-time-level difference scheme of the second order of accuracy is constructed for parabolic equations by combining Richardson's scheme with that of Crank-Nicolson. Restrictions on the time step required for the stability of the proposed difference scheme are similar to those that are necessary for the stability of the two-time-level explicit difference scheme, but the former are slightly less onerous.

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References

  1. L. F. Richardson, “The approximate solution by finite differences of physical problems involving differential equations with an application to the stresses in a masonry dam,”Roy. Soc. Philos. Trans.,210A, 307–357 (1910).

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  2. J. Crank and P. Nicolson, “A practical method for numerical integration of solutions of partial differential equations of heat conduction type. I,”Proc. Cambridge Philos. Soc.,43, 50–67 (1947).

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  3. Yuan' Chzhao-din,Certain Difference Schemes for the Solution of the First Boundary Value Problem for Linear Partial Differential liquations, Kandidat thesis in the physico-mathematical sciences [in Russian], Moscow State University, Moscow (1958).

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  4. A. S. Shvedov,Construction of a Three-Time-Level Explicit Difference Scheme of the Second Order of Accuracy for Parabolic Equations on the Basis of the Richardson and the Crank-Nicolson Difference Scheme [in Russian], Preprint No. 104, Keldysh Inst. Appl. Math., Moscow (1995).

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Authors and Affiliations

  1. M. V. Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, USSR

    A. S. Shvedov

Authors
  1. A. S. Shvedov
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Translated fromMatematicheskie Zametki, Vol. 60, No. 5, pp. 751–759, November, 1996.

This research was supported by the Russian Foundation for Basic Research under grant No. 95-01-00489 and by the International Science Foundation under grants No. N8Q300 and No. JBR100.

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Shvedov, A.S. A three-time-level explicit difference scheme of the second order of accuracy for parabolic equations. Math Notes 60, 562–568 (1996). https://doi.org/10.1007/BF02309170

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  • DOI: https://doi.org/10.1007/BF02309170

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