The application of implicit Runge-Kutta and collection methods to boundary-value problems

Weiss, Richard

doi:10.1090/S0025-5718-1974-0341881-2

The application of implicit Runge-Kutta and collection methods to boundary-value problems
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Math. Comp. 28 (1974), 449-464 Request permission

Abstract:

The solution of a nonlinear system of first order differential equations with nonlinear boundary conditions by implicit Runge-Kutta methods based on interpolatory quadrature formulae is examined. An equivalence between implicit Runge-Kutta and collocation schemes is established. It is shown that the difference equations obtained have a unique solution in a neighbourhood of an isolated solution of the continuous problem, that this solution can be computed by Newton iteration and that it converges to the isolated solution. The order of convergence is equal to the degree of precision of the related quadrature formula plus one. The efficient implementation of the methods is discussed and numerical examples are given.

References

Owe Axelsson, A class of $A$-stable methods, Nordisk Tidskr. Informationsbehandling (BIT) 9 (1969), 185–199. MR 255059, DOI 10.1007/bf01946812
J. C. Butcher, Implicit Runge-Kutta processes, Math. Comp. 18 (1964), 50–64. MR 159424, DOI 10.1090/S0025-5718-1964-0159424-9
Carl de Boor and Blâir Swartz, Collocation at Gaussian points, SIAM J. Numer. Anal. 10 (1973), 582–606. MR 373328, DOI 10.1137/0710052

Nordisk Tidskr. Informationsbehandling

BIT

F. de Hoog and R. Weiss, Implicit Runge-Kutta methods for second kind Volterra integral equations, Numer. Math. 23 (1974/75), 199–213. MR 373349, DOI 10.1007/BF01400303
Bernie L. Hulme, Discrete Galerkin and related one-step methods for ordinary differential equations, Math. Comp. 26 (1972), 881–891. MR 315899, DOI 10.1090/S0025-5718-1972-0315899-8
Herbert B. Keller, Accurate difference methods for linear ordinary differential systems subject to linear constraints, SIAM J. Numer. Anal. 6 (1969), 8–30. MR 253562, DOI 10.1137/0706002
Herbert B. Keller, Accurate difference methods for nonlinear two-point boundary value problems, SIAM J. Numer. Anal. 11 (1974), 305–320. MR 351098, DOI 10.1137/0711028
M. R. Osborne, Minimising truncation error in finite difference approximations to ordinary differential equations, Math. Comp. 21 (1967), 133–145. MR 223107, DOI 10.1090/S0025-5718-1967-0223107-X
R. D. Russell and L. F. Shampine, A collocation method for boundary value problems, Numer. Math. 19 (1972), 1–28. MR 305607, DOI 10.1007/BF01395926

Numerical Procedures for Volterra Integral Equations

K. Wright, Some relationships between implicit Runge-Kutta, collocation Lanczos $\tau$ methods, and their stability properties, Nordisk Tidskr. Informationsbehandling (BIT) 10 (1970), 217–227. MR 266439, DOI 10.1007/bf01936868