The application of linear multistep methods to singular initial value problems

Authors:
Frank R. de Hoog and Weiss Richard

Journal:
Math. Comp. **31** (1977), 676-690

MSC:
Primary 65L05

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Abstract: A theory for linear multistep schemes applied to the initial value problem for a nonlinear first order system of differential equations with a singularity of the first kind is developed. Predictor-corrector schemes are also considered.

The specific examples given are systems derived from partial differential equations in the presence of symmetry.

**[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****[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****[3]**F. R. de HOOG & R. WEISS,*Collocation Methods for Singular Boundary Value Problems*, MRC Technical Summary Report #1547, 1975.**[4]**Peter Henrici,*Discrete variable methods in ordinary differential equations*, John Wiley & Sons, Inc., New York-London, 1962. MR**0135729****[5]**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****[6]**Yudell L. Luke,*The special functions and their approximations, Vol. I*, Mathematics in Science and Engineering, Vol. 53, Academic Press, New York-London, 1969. MR**0241700****[7]**H. MEISSNER & P. THOLFSEN, "Cylindrically symmetric solutions of the Ginzburg-Landau equations,"*Phys. Rev.*, v. 169, 1968, pp. 413-416.

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DOI:
http://dx.doi.org/10.1090/S0025-5718-77-99844-1

Keywords:
Initial value problem,
singularity of the first kind,
linear multistep method,
stability,
convergence,
predictor-corrector scheme

Article copyright:
© Copyright 1977
American Mathematical Society