Runge-Kutta software with defect control for boundary value odes

SIAM Journal on Scientific Computing

Volumn 17, Issue 2, 1996, Pages 479-497

  Enright, W H ^a   Muir, P H ^b  

^a UNIVERSITY OF TORONTO   (Canada)

^b SAINT MARY S UNIVERSITY   (Canada)

Author keywords

Boundary value ordinary differential equations; Defect control; Interpolants; Numerical software; Runge Kutta methods

Indexed keywords

EID: 2142782818     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1064827593251496     Document Type: Article

References (37)

1. U. ASCHER, J. CHRISTIANSEN, AND R. D. RUSSELL, Collocation software for boundary value ODEs, Trans. Math. Software, 7 (1981), pp. 209-222.
2. U. ASCHER, R. M. M. MATTHEU, AND R. RUSSELL, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Prentice Hall, Englewood Cliffs, New Jersey, 1988.
3. U. ASCHER AND S. JACOBS, On collocation implementation for singularly perturbed two-point problems, SIAM J. Sci. Stat. Comp., 10 (1989), pp. 533-549.
4. G. BADER AND U. ASCHER, A new basis implementation for a mixed order boundary value ode solver, SIAM J. Sci. Stat. Comp., 8 (1987), pp. 483-500.
5. G. K. BATCHELOR, A note on a class of solutions of the Navier-Stokes equations representing rotationally symmetric flow, Quart. J. Mech. Appl. Math., 4 (1951), pp. 29-41.
6. W. M. G. VAN BOKHOVEN, Efficient higher order implicit one-step methods for integration of stiff differential equations, BIT, 20 (1980), pp. 34-43.
7. D. L. BROWN AND J. LORENZ, A high-order method for stiff boundary value problems with turning points, SIAM J. Sci. Stat. Comp., 8 (1987), pp. 790-808.
8. K. BURRAGE, F. CHIPMAN, AND P. H. MUIR, Order results for mono-implicit Runge-Kutta methods, SIAM J. Numer. Anal., 31 (1994), pp. 876-891.
9. J. C. BUTCHER, The Numerical Analysis of Ordinary Differential Equations, John Wiley & Sons, Toronto, 1987.
10. 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, Comp. Math. Appl., 12a (1986), pp. 1029-1048.
11. _, 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), pp. 862-882.
12. J. R. CASH AND H. H. M. SILVA, On the numerical solution of a class of singular two-point boundary value problems, J. Comput. Appl.Math., 45 (1993), pp. 91-102.
13. J. R. CASH AND A. SINGHAL, Mono-implicit Runge-Kutta formulae for the numerical integration of stiff differential systems, IMA J. Numer. Anal., 2 (1982a), pp. 211-227.
14. _, High order methods for the numerical solution of two-point boundary value problems, BIT, 22 (1982b), pp. 184-199.
15. J. R. CASH AND M. H. WRIGHT, Implementation issues in solving nonlinear equations for two-point boundary value problems, Computing, 45 (1990), pp. 17-37.
16. _, A deferred correction method for nonlinear two-point boundary value problems: implementation and numerical evaluation, SIAM J. Sci. Stat. Comp., 12 (1991), pp. 971-989.
17. W. H. ENRIGHT, A new error-control for initial value solvers, Appl. Math. Comp., 31 (1989a), pp. 288-301.
18. _, Analysis of error control strategies for continuous Runge-Kutta methods, SIAM J. Numer. Anal., 26 (1989b), pp. 588-599.
19. W. H. ENRIGHT, K. R. JACKSON, S. P. NØRSETT, AND P. G. THOMSEN, Interpolants for Runge-Kutta formulas, ACM Trans. Math. Software, 12 (1986), pp. 193-218.
20. W. H. ENRIGHT AND P. H. MUIR, Efficient classes of Runge-Kutta methods for two-point boundary value problems, Computing, 37 (1986), pp. 315-334.
21. _, A Runge-Kutta type boundary value ODE solver with defect control, Tech. Rep. No. 267/93, Dept. Comp. Sci., Univ. of Toronto, Toronto, Ontario, Canada, 1993.
22. I. GLADWELL, L. F. SHAMPINE, L. S. BACA, AND R. W. BRANKIN, Practical aspects of interpolation in Runge-Kutta codes, SIAM J. Sci. Stat. Comp., 8 (1987), pp. 322-341.
23. S. GUPTA, An adaptive boundary value Runge-Kutta solver for first order boundary value problems, SIAM J. Numer. Anal., 22 (1985), pp. 114-126.
24. P. M. HANSON AND W. H. ENRIGHT, Controlling the defect in existing variable-order Adams codes for initial-value problems, ACM Trans. Math. Software, 9 (1983), pp. 71-97.
25. D. J. HIGHAM, Defect estimation in Adams PECE codes, SIAM J. Sci. Stat. Comput., 10 (1989a), pp. 964-976.
26. _, Robust defect control with Runge-Kutta schemes, SIAM J. Numer. Anal., 26 (1989b), pp. 1175-1183.
27. _, Runge-Kutta defect control using Hermite-Birkhoff interpolation, SIAM J. Sci. Stat. Comput., 12 (1991), pp. 991-999.
28. 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), pp. 325-368.
29. 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), pp. 91-111,
30. P. H. MUIR AND B. OWREN, Order barriers and characterizations of continuous mono-implicit Runge-Kutta schemes, Math. Comp., 61 (1993), pp. 675-699.
31. B. OWREN AND M. ZENNARO, Order barriers for continuous explicit Runge-Kutta methods, Math. Comp., 56 (1991), pp. 645-661.
32. _, Derivation of optimal continuous explicit Runge-Kutta methods, SIAM J. Sci. Stat. Comp., 13 (1992), pp. 1488-1501.
33. V. PEREYRA AND R. D. RUSSELL, Difficulties of comparing complex mathematical software: General comments and the BVODE case, Acta Cient. Venezolana, 33 (1982), pp. 15-22.
34. A. PROTHERO AND A. ROBINSON, On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations, Math. Comp., 28 (1974), pp. 145-162.
35. P. W. SHARP AND E. SMART, Explicit Runge-Kutta pairs with one more derivative evaluation than the minimum, SIAM J. Sci. Comp., 14 (1993), pp. 338-348.
36. J. VERNER, Differentiable interpolants for high-order Runge-Kutta methods, SIAM J. Numer. Anal., 30 (1993), pp. 1446-1466.
37. R. WEISS, The application of implicit Runge-Kutta and collocation methods to boundary value problems, Math. Comp., 28 (1974), pp. 449-464.