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On the Crank-Nicolson procedure for solving parabolic partial differential equations

Published online by Cambridge University Press:  24 October 2008

M. L. Juncosa  and
David Young
M. L. Juncosa
Affiliation:
RAND Corporation Santa Monica, California
David Young
Affiliation:
University of Maryland CollegePark, Maryland
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Abstract

Proof of convergence of the Crank-Nicolson procedure, an ‘implicit’ numerical method for solving parabolic partial differential equations, is given for the case of the classical ‘problem of limits’ for one-dimensional diffusion with zero boundary conditions. Orders of convergence are also given for different classes of initial functions. Results do not support the validity of so-called h2-extrapolation in some cases.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 53 , Issue 2 , April 1957 , pp. 448 - 461
Copyright
Copyright © Cambridge Philosophical Society 1957

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References

REFERENCES

(1)Crank, J. and Nicolson, P.A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type. Proc. Camb. Phil. Soc. 43 (1947), 50–67.CrossRefGoogle Scholar
(2)Hardy, G. H.Divergent series (Oxford, 1949).Google Scholar
(3)Hartman, P. and Wintner, A.On the solutions of the equation of heat conduction. Amer. J. Math. 72 (1950), 367–95.CrossRefGoogle Scholar
(4)Jackson, D.The theory of approximation (Amer. Math. Soc. Colloq. Publ., vol II).Google Scholar
(5)Juncosa, M. L. and Young, D. M.On the convergence of a solution of a difference equation to a solution of the equation of diffusion. Proc. Amer. Math. Soc. 5 (1954), 168–74.CrossRefGoogle Scholar
(6)Juncosa, M. L. and Young, D. M.On the order of convergence of solutions of a difference equation to a solution of the diffusion equation. J. Soc. Indust. Appl. Math. 1 (1953), 111–35.Google Scholar
(7)Juncosa, M. L. and Young, D. M.A uniform approximation to Fourier coefficients. Proc. Amer. Math. Soc. 4 (1953), 373–74.CrossRefGoogle Scholar
(8)Richardson, L. F.The approximate arithmetical solution by finite differences of physical problems involving differential equations. Phil. Trans. A, 210 (1910), 307–57.Google Scholar
(9)Steffensen, J. F.Interpolation, 2nd ed. (New York, 1950).Google Scholar
(10)Walsh, J. L. and Sewell, W. E.Note on degree of approximation to an integral by Riemann sums. Amer. Math. Mon. 44 (1937), 155–60.CrossRefGoogle Scholar
(11)Walsh, J. L. and Young, D. M.On the degree of convergence of solutions of difference equations to the solution of the Dirichlet problem. J. Math. Phys. 33 (1954), 80–93.Google Scholar
(12)Wasow, W.On the truncation error in solution of Laplace's equation by finite differences. J. Res. Nat. Bur. Stand. 48 (1952), 345–8.CrossRefGoogle Scholar