Runge-Kutta methods for the solution of stiff two-point boundary value problems

Applied Numerical Mathematics

Volumn 22, Issue 1-3 SPEC. ISS., 1996, Pages 165-177

  Cash, J R ^a  

^a IMPERIAL COLLEGE LONDON   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

BOUNDARY CONDITIONS; BOUNDARY VALUE PROBLEMS; PERTURBATION TECHNIQUES; PROBLEM SOLVING;

RUNGE KUTTA METHODS; STIFF TWO POINT BOUNDARY VALUE PROBLEMS;

NUMERICAL METHODS;

EID: 0030283367     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/s0168-9274(96)00030-x     Document Type: Article

References (43)

1. U.M. Ascher, R.M.M. Matthcij and R.D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (Prentice-Hall, Englewood Cliffs, NJ, 1988).
2. U.M. Ascher, J. Christiansen and R.D. Russell, Collocation software for boundary value ODEs, ACM Trans. Math. Software 7 (1981) 209-222.
3. G. Bader and U.M. Ascher, A new basis implementation for a mixed order boundary value ODE solver, SIAM J. Sci. Statist. Comput. 8 (1987) 483-500.
4. K. Burrage, F. Chipman and P. Muir, Order results for mono-implicit Runge-Kutta methods, SIAM J. Numer. Anal. (1994) 876-891.
5. J.C. Butcher, The Numerical Analysis of Ordinary Differential Equations (Wiley, New York, 1987).
6. J.C. Butcher, J.R. Cash, G. Moore and R.D. Russell, Defect correction for two-point boundary value problems on nonequidistant meshes, Math. Comp. 64 (1995) 629-648.
7. J.R. Cash and M.H. Wright, A deferred correction method for nonlinear two-point boundary value problems. Implementation and numerical evaluation, SIAM J. Sci. Statist. Comput. 12 (1991) 971-989.
8. J.R. Cash and M.H. Wright, Algorithm TWPBVP in the ODE chapter of NETLIB.
9. J.R. Cash, A class of implicit Runge-Kutta methods for the numerical integration of stiff ordinary differential equations. J. Assoc. Comput. Mach. 22 (1975) 504-511.
10. J.R. Cash and A. Sighal, High order methods for the numerical solution of two-point boundary value problems. BIT 22 (1982) 184-199.
11. J.R. Cash and M.H. Wright, Implementation issues in solving nonlinear equations for two-point boundary value problems, Computing 45 (1990) 17-37.
12. J.R. Cash, On the numerical integration of nonlinear two-point boundary value problems using iterated deferred corrections, Part 1: A survey and comparison of some one-step formulae, Comput. Math. Appl. 12 (1986) 1029-1048.
13. J.R. Cash, On the numerical integration of nonlinear two-point boundary value problems using iterated deferred corrections, Part 2: The development and analysis of highly stable deferred correction formulae, SIAM J. Numer. Anal. 25 (1988) 862-882.
14. J.R. Cash and H.H.M. Silva, Iterated deferred correction for linear two-point boundary value problems, J. Braz. Math. Soc., to appear.
15. J.R. Cash, Z. Bashir-Ali and H.H.M. Silva, Lobatto deferred correction for stiff two-point boundary value problems, Comput. Math. Appl., to appear.
16. J.R. Cash, G. Moore and R.W. Wright, An automatic continuation strategy for the solution of singularly perturbed two-point boundary value problems, J. Comput. Phys. 122 (1995) 266-279.
17. J.R. Cash and H.H.M. Silva, On the numerical solution of a class of singular two-point BVPs, J. Comput. Appl. Math. 45 (1993) 91-102.
18. J.W. Daniel and A.J. Martin, Numerov's method with deferred corrections for two-point boundary value problems, SIAM J. Numer. Anal. 14 (1977) 1033-1050.
19. W.H. Enright and P.H. Muir, Efficient classes of Runge-Kutta methods for two-point boundary value problems, Computing 37 (1986) 315-334.
20. L. Fox, The Numerical Solution of Two-Point Boundary Value Problems in Ordinary Differential Equations (Oxford University Press, 1957).
21. S. Gupta, An adaptive boundary value Runge-Kutta solver for first order boundary value problems, SIAM J. Numer. Anal. 22 (1985) 114-126.
22. P. Hemker, A Numerical Study of Stiff Two-Point Boundary Value Problems (Mathematisch Centrum, Amsterdam, 1977).
23. H.B. Keller and V. Pereyra, Difference methods and deferred corrections for ordinary boundary value problems, SIAM J. Numer. Anal. 16 (1979) 241-259.
24. H.O. Kreiss, T.A. Manteuffel, B. Swartz, B. Wendroff and A.B. White, Supra-convergent schemes on irregular grids, Math. Comp. 47 (1986) 537-554.
25. H.O. Kreiss, N.K. Nichols and D.L. Brown, Numerical methods for stiff two-point boundary value problems, SIAM J. Numer. Anal. 23 (1986) 325-368.
26. M. Lentini and V. Pereyra, A variable order finite difference method for nonlinear multipoint boundary value problems, Math. Comp. 28 (1974) 981-1004.
27. M. Lentini and V. Pereyra, An adaptive finite difference solver for nonlinear two-point boundary value problems with mild boundary layers, SIAM J. Numer. Anal. 14 (1977) 91-111.
28. T.A. Manteuffel and A.B. White, The numerical solution of second-order boundary value problems on nonuniform meshes, Math. Comp. 47 (1986) 511-535.
29. P.H. Muir and W.H. Enright, Relationships among some classes of implicit Runge-Kutta methods and their stability functions, BIT 27 (1987) 403-423.
30. P.H. Muir and B. Owren, Order barriers and characterizations for continuous mono-implicit Runge-Kutta schemes, Math. Comp. 61 (1993) 675-699.
31. P.H. Muir and K. Remington, A parallel implementation of a Runge-Kutta code for systems of nonlinear boundary value ODEs, Congr. Numer. 99 (1994) 291-305.
32. R.E. O'Malley, Introduction to Singular Perturbations (Academic Press, New York, 1974).
33. R.D. Skeel, A theoretical framework for proving accuracy results for deferred corrections, SIAM J. Numer. Anal. 19 (1982) 171-196.
34. H.J. Stetter, The defect correction principle and discretization methods, Numer. Math. 29 (1978) 425-443.
35. W.M.G. Van Bokhoven, Efficient higher order implicit one-step methods for integration of stiff differential equations, BIT 20 (1980) 34-43.
36. D. Voss, A second order mono-implicit Runge-Kutta method for parallel computers, in: S.K. Dey and J. Zierbath, eds., Proceedings of Advances in Scientific Computing and Mathematical Modelling (1993) 71-75.
37. W. Wasow, Linear Turning Point Theory (Springer, New York, 1985).
38. K. Wright, Some relationships between implicit Runge-Kutta, collocation and Lanczos Tau methods and their stability properties, BIT 20 (1970) 217-227.
39. R.W. Wright, An automatic continuation strategy for the numerical solution of stiff two-point boundary value problems, Ph.D. Thesis, University of London (1995).
40. U.M. Ascher and R.M.M. Mattheij, General framework, stability, and error analysis for numerical stiff boundary value methods, Numer. Math. 54 (1988) 355-372.
41. E. Hairer, S.P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems (Springer, Berlin, 1991).
42. H.J. Stetter, Analysis of Discretization Methods for Ordinary Differential Equations (Springer, Berlin, 1973).
43. R. Weiss, The application of implicit Runge-Kutta and collocation methods to boundary-value problems, Math. Comp. 28 (1974).