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

Author:
Richard Weiss

Journal:
Math. Comp. **28** (1974), 449-464

MSC:
Primary 65L10

MathSciNet review:
0341881

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[1]**Owe Axelsson,*A class of 𝐴-stable methods*, Nordisk Tidskr. Informationsbehandling (BIT)**9**(1969), 185–199. MR**0255059****[2]**J. C. Butcher,*Implicit Runge-Kutta processes*, Math. Comp.**18**(1964), 50–64. MR**0159424**, 10.1090/S0025-5718-1964-0159424-9**[3]**Carl de Boor and Blâir Swartz,*Collocation at Gaussian points*, SIAM J. Numer. Anal.**10**(1973), 582–606. MR**0373328****[4]**F. de Hoog, "Implicit Runge-Kutta methods for Volterra integro-differential equations,"*Nordisk Tidskr. Informationsbehandling*(*BIT*). (To appear.)**[5]**F. de Hoog and R. Weiss,*Implicit Runge-Kutta methods for second kind Volterra integral equations*, Numer. Math.**23**(1974/75), 199–213. MR**0373349****[6]**Bernie L. Hulme,*Discrete Galerkin and related one-step methods for ordinary differential equations*, Math. Comp.**26**(1972), 881–891. MR**0315899**, 10.1090/S0025-5718-1972-0315899-8**[7]**Herbert B. Keller,*Accurate difference methods for linear ordinary differential systems subject to linear constraints*, SIAM J. Numer. Anal.**6**(1969), 8–30. MR**0253562****[8]**Herbert B. Keller,*Accurate difference methods for nonlinear two-point boundary value problems*, SIAM J. Numer. Anal.**11**(1974), 305–320. MR**0351098****[9]**M. R. Osborne,*Minimising truncation error in finite difference approximations to ordinary differential equations*, Math. Comp.**21**(1967), 133–145. MR**0223107**, 10.1090/S0025-5718-1967-0223107-X**[10]**R. D. Russell and L. F. Shampine,*A collocation method for boundary value problems*, Numer. Math.**19**(1972), 1–28. MR**0305607****[11]**R. Weiss,*Numerical Procedures for Volterra Integral Equations*, Thesis, Computer Centre, The Australian National University, Canberra, 1972.**[12]**K. Wright,*Some relationships between implicit Runge-Kutta, collocation Lanczos 𝜏 methods, and their stability properties*, Nordisk Tidskr. Informationsbehandling (BIT)**10**(1970), 217–227. MR**0266439**

Retrieve articles in *Mathematics of Computation*
with MSC:
65L10

Retrieve articles in all journals with MSC: 65L10

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1974-0341881-2

Keywords:
Implicit Runge-Kutta method,
collocation method,
boundary-value problem

Article copyright:
© Copyright 1974
American Mathematical Society