The application of RungeKutta schemes to singular initial value problems
Authors:
Frank de Hoog and Richard Weiss
Journal:
Math. Comp. 44 (1985), 93103
MSC:
Primary 65L05
MathSciNet review:
771033
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Abstract: A theory for explicit RungeKutta schemes applied to the initial value problem for a firstorder system of differential equations with a singularity of the first kind is developed. It is shown that, in general, the order of convergence is at most two but that the usual order up to a logarithmic term can be obtained for level three and four schemes applied to specific problems.
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 L. Bauer, E. L. Reiss & H. B. Keller, "Axisymmetric buckling of hollow spheres and hemispheres," Comm. Pure Appl. Math., v. 23, 1970, pp. 529568. MR 0278605 (43:4335)
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 F. de Hoog & R. Weiss, "Difference methods for boundary value problems with a singularity of the first kind," SIAM J. Numer. Anal., v. 13, 1976, pp. 775813. MR 0440931 (55:13799)
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 F. de Hoog & R. Weiss, "The application of linear multistep methods to singular initial value problems," Math. Comp., v. 31, 1977, pp. 676690.
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 H. B. Keller & A. W. Wolfe, "On the nonunique equilibrium states and buckling mechanism of spherical shells," J. Soc. Indust. Appl. Math., v. 13, 1965, pp. 674705. MR 0183174 (32:656)
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 H. Meissner & P. Tholfsen, "Cylindrically symmetric solutions of the GinzburgLandau equations," Phys. Rev., v. 169, 1968, pp. 413416.
 [6]
 S. V. Parter, M. L. Stein & P. R. Stein, On the Multiplicity of Solutions of a Differential Equation Arising in Chemical Reactor Theory, Computer Sciences Technical Report #194, University of WisconsinMadison, 1973.
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 P. Rentrop, "A Taylor series method for the numerical solution of two point boundary value problems," Numer. Math., v. 31, 1979, pp. 359375. MR 516580 (80c:65171)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198507710330
PII:
S 00255718(1985)07710330
Keywords:
Explicit RungeKutta schemes,
initial value problem,
singularity of the first kind,
convergence
Article copyright:
© Copyright 1985
American Mathematical Society
