The application of Runge-Kutta schemes to singular initial value problems

Authors:
Frank de Hoog and Richard Weiss

Journal:
Math. Comp. **44** (1985), 93-103

MSC:
Primary 65L05

DOI:
https://doi.org/10.1090/S0025-5718-1985-0771033-0

MathSciNet review:
771033

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Abstract: A theory for explicit Runge-Kutta schemes applied to the initial value problem for a first-order 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.

**[1]**Louis Bauer, Edward L. Reiss, and Herbert B. Keller,*Axisymmetric buckling of hollow spheres and hemispheres*, Comm. Pure Appl. Math.**23**(1970), 529–568. MR**0278605**, https://doi.org/10.1002/cpa.3160230402**[2]**Frank R. de Hoog and Richard Weiss,*Difference methods for boundary value problems with a singularity of the first kind*, SIAM J. Numer. Anal.**13**(1976), no. 5, 775–813. MR**0440931**, https://doi.org/10.1137/0713063**[3]**F. de Hoog & R. Weiss, "The application of linear multistep methods to singular initial value problems,"*Math. Comp.*, v. 31, 1977, pp. 676-690.**[4]**Herbert B. Keller and Antoinette W. Wolfe,*On the nonunique equilibrium states and buckling mechanism of spherical shells*, J. Soc. Indust. Appl. Math.**13**(1965), 674–705. MR**0183174****[5]**H. Meissner & P. Tholfsen, "Cylindrically symmetric solutions of the Ginzburg-Landau equations,"*Phys. Rev.*, v. 169, 1968, pp. 413-416.**[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 Wisconsin-Madison, 1973.**[7]**P. Rentrop,*A Taylor series method for the numerical solution of two-point boundary value problems*, Numer. Math.**31**(1978/79), no. 4, 359–375. MR**516580**, https://doi.org/10.1007/BF01404566**[8]**Anthony Ralston,*A first course in numerical analysis*, McGraw-Hill Book Co., New York-Toronto-London, 1965. MR**0191070**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1985-0771033-0

Keywords:
Explicit Runge-Kutta schemes,
initial value problem,
singularity of the first kind,
convergence

Article copyright:
© Copyright 1985
American Mathematical Society