An analysis of the order of Runge-Kutta methods that use an iterative scheme to compute their internal stage values

BIT Numerical Mathematics

Volumn 36, Issue 4, 1996, Pages 713-765

  Jackson, K R ^a   Kvaerno A ^b   Norsett S P ^b  

^a UNIVERSITY OF TORONTO   (Canada)

^b NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Norway)

Author keywords

Initial value problem; Internal stage values; Iterative scheme; Newton's method; Order; Ordinary differential equation; Runge Kutta method

Indexed keywords

EID: 0000087072     PISSN: 00063835     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF01733789     Document Type: Article

References (27)

1. N. G. R. Broderick, C. M. de Sterke, and K. R. Jackson, Coupled mode equations with free earner effects: A numerical solution, J. Optical and Quantum Electronics, 26 (1994), pp. S219-S234.
2. K. Burrage, The error behaviour of a general class of predictor-corrector methods, Appl. Numer. Math., 8 (1991), pp. 201-216.
3. K. Burrage, Efficient block predictor-corrector methods with a small number of corrections., J. Comp. Appl. Math., 45 (1993), pp. 139-150.
4. K. Burrage, Parallel methods for initial value problems, Appl. Numer. Math., 11 (1993), pp. 5-25.
5. K. Burrage, The search for the Holy Grail, or: Predictor-corrector methods for solving ODEIVPs, Appl. Numer. Math., 11 (1993), pp. 125-141.
6. K. Burrage, Parallel and Sequential Methods for Ordinary Differential Equations, Oxford University Press, Oxford, 1995.
7. K. Burrage, J. C. Butcher, and F. H. Chipman, An implementation of singly-implicit Runge-Kutta methods, BIT, 20 (1980), pp. 326-340.
8. J. C. Butcher, The Numerical Analysis of Ordinary Differential Equations, John Wiley & Sons, New York, 1987.
9. C. M. de Sterke, K. R. Jackson, and B. D. Robert, Nonlinear coupled mode equations on a finite interval: A numerical procedure, J. of the Optical Society of America B, 8 (1991), pp. 403-412.
10. J. Dorsselaer and M. N. Spijker, The error committed by stopping Newton iterations in the numerical solution of stiff initial value problems, IMA J. Numer. Anal., 14 (1994), pp. 183-209.
11. R. F. Enenkel, The implementation of parallel Runge-Kutta methods. Master's thesis, Computer Science Dept., University of Toronto, Toronto, Canada, 1988.
12. E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems, Springer-Verlag, Berlin, 1987.
13. E. Hairer and G. Wanner, On the Butcher group and general multi-value methods, Computing, 13 (1974), pp. 1-15.
14. P. Houwen and B. P. Sommeijer, Parallel iteration of high-order Runge-Kutta methods with stepsize control, J. Comp. Appl. Math., 29 (1990), pp. 111-127.
15. P. Houwen and B. P. Sommeijer, Parallel step-by-step methods, Appl. Numer. Math., 11 (1993), pp. 69-81.
16. P. Houwen, B. P. Sommeijer, and W. Couzy, Embedded diagonally implicit Runge-Kutta algorithms on parallel computers, Math. Comp., 58 (1992), pp. 135-159.
17. K. R. Jackson, A survey of parallel numerical methods for initial value problems for ordinary differential equations, IEEE Trans. on Magnetics, 27 (1991), pp. 3792-3797.
18. K. R. Jackson, A. Kværnø, and S. P. Nørsett, The use of Butcher series in the analysis of Newton-like iterations in Runge-Kutta formulas, Appl. Numer. Math., 15 (1994), pp. 341-356.
19. K. R. Jackson and S. P. Nørsett, The potential for parallelism in Runge-Kutta methods. Part 1: RK formulas in standard form, SIAM J. Numer. Anal., 32 (1995), pp. 49-82.
20. J. Kalvenes, Experimentation with parallel ODE-solvers, Tech. Rep. NM-R8912, Mathematical Sciences Div., Norwegian Institute of Technology, Trondheim, Norway, 1986.
21. A. Kværnø, The order of Runge-Kutta methods applied to semi-explicit DAEs of index 1, using Newton-type iterations to compute the internal stage values, Tech. Rep. 2/1992, Mathematical Sciences Div., Norwegian Institute of Technology, Trondheim, Norway, 1992.
22. S. P. Nørsett and H. H. Simonsen, Aspects of parallel Runge-Kutta methods, in Numerical Methods for Ordinary Differential Equations, A. Bellen, C. W. Gear and E. Russo, eds., Berlin, 1989, Springer Verlag, pp. 103-107. Lecture Notes in Mathematics #1386, Proceedings of the l'Aquila Symposium 1987.
23. S. P. Nørsett and H. H. Simonsen, Aspects of parallel Runge-Kutta methods, in Numerical Methods for Ordinary Differential Equations, A. Bellen, C. W. Gear and E. Russo, eds., Berlin, 1989, Springer Verlag, pp. 103-107. Lecture Notes in Mathematics #1386, Proceedings of the l'Aquila Symposium 1987.
24. S. P. Nørsett and P. G. Thomsen, User's guide for SIMPLE, a stiff system solver. Working notes.
25. S. P. Nørsett and A. Wolfbrandt, Order conditions for Rosenbrock type methods, Numer. Math., 32 (1979), pp. 1-15.
26. L. F. Shampine, Numerical Solution of Ordinary Differential Equations, Chapman & Hall, New York, 1994.
27. B. P. Sommeijer, Parallel-iterated Runge-Kutta methods for stiff ordinary differential equations., J. Comp. Appl. Math., 45 (1993), pp. 151-168.