Software based on explicit RK formulas

Applied Numerical Mathematics

Volumn 22, Issue 1-3 SPEC. ISS., 1996, Pages 293-308

  Shampine, L F ^a   Gladwell, I ^a  

^a SOUTHERN METHODIST UNIVERSITY   (United States)

Author keywords

Continuous extension; Evaluation; Explicit RK formulas; Global error; History; Software; Step size control; Stiffness

Indexed keywords

COMPUTER SOFTWARE; DIFFERENTIAL EQUATIONS; ERROR ANALYSIS;

RUNGE KUTTA METHODS; STIFF DIFFERENTIAL EQUATIONS;

NUMERICAL METHODS;

EID: 0030283365     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/s0168-9274(96)00039-6     Document Type: Article

References (60)

1. P. Bogacki and L.F. Shampine, A 3(2) pair of Runge-Kutta formulas, Appl. Math. Lett. 2 (1989) 1-9.
2. P. Bogacki and L.F. Shampine, An efficient Runge-Kutta (4,5) pair, Report 89-20, Mathematics Department, SMU, Dallas (1989), Comput. Math. Appl., to appear.
3. P. Biddle, A technique for evaluating the performance of initial value solvers for non-stiff ordinary differential equations, M.Sc. Thesis, Department of Mathematics, University of Manchester, UK (1977).
4. R.W. Brankin and I. Gladwell, Using shape preserving local interpolants for plotting solutions of ordinary differential equations, IMA J. Numer. Anal. 9 (1989) 555-566.
5. R.W. Brankin, I. Gladwell, J.R. Dormand, P.J. Prince and W.L. Seward, ALGORITHM 670: A Runge-Kutta-Nyström code, ACM Trans. Math. Software 15 (1989) 31-40.
6. R.W. Brankin, I. Gladwell and L.F. Shampine, Starting BDF and Adams codes at optimal order, J. Comput. Appl. Math. 21 (1988) 357-368.
7. R.W. Brankin, I. Gladwell and L.F. Shampine, RKSUITE: a suite of Runge-Kutta codes for the initial value problem for ODEs, Softreport 91-1, Mathematics Department, SMU, Dallas (1991).
8. R.W. Brankin, I. Gladwell and L.F. Shampine, RKSUITE: A suite of explicit Runge-Kutta codes, in: R. Agarwal, ed., Contributions in Numerical Mathematics, World Scientific Series in Applicable Analysis 2 (World Scientific, Singapore, 1993) 85-98.
9. P.N. Brown, G.D. Byrne and A.C. Hindmarsh, VODE: a variable-coefficient ODE solver, SIAM J. Sci. Statist. Comput. 10 (1989) 1038-1051.
10. M.H. Carpenter, D. Gottlieb, S. Abarbanel and W.S. Don, The theoretical accuracy of Runge-Kutta discretizations for the initial boundary value problem: A study of the boundary error. SIAM J. Sci. Comput., to appear
11. J.R. Dormand and P.J. Prince, A family of embedded Runge-Kutta formulae. J. Comput. Appl. Math. 6 (1980) 19-26.
12. J.R. Dormand and P.J. Prince, Global error estimation with Runge-Kutta methods, IMA J. Numer. Anal. 4 (1984) 169-184.
13. J.R. Dormand and P.J. Prince, Global error estimation with Runge-Kutta methods II, IMA J. Numer. Anal. 5 (1985) 481-497.
14. W.H. Enright, Analysis of error control strategies for continuous Runge-Kutta methods, SIAM J. Numer. Anal. 26 (1989) 588-599.
15. E. Fehlberg, Klassische Runge-Kutta-Formeln vierter und niedrigerer Ordnung mit Schrittweiten-Kontrolle und ihre Anwendung auf Wärmeleitungsprobleme. Computing 6 (1970) 61-71.
16. C.W. Gear, Runge-Kutta starters for multistep methods, ACM Trans. Math. Software 6 (1980) 263-279.
17. I. Gladwell, Initial value routines in the NAG library, ACM Trans. Math. Software 5 (1979) 386-400.
18. I. Gladwell, J.A.I. Craigie and C.R. Crowther, Testing initial value routines as black boxes, Numerical Analysis Report 34, Department of Mathematics, University of Manchester (1979).
19. I. Gladwell, L.F. Shampine and R.W. Brankin, Automatic selection of the initial step size for an ODE solver, J. Comput. Appl. Math. 18 (1987) 175-192.
20. K. Gustafsson, Control theoretic techniques for stepsize selection in explicit Runge-Kutta methods, ACM Trans. Math. Software 17 (1991) 533-554.
21. [|21] K. Gustafsson, M. Lundh and G. Söderlind, A PI stepsize control for the numerical solution of ordinary differential equations. BIT 18 (1988) 270-287.
22. E. Hairer and D. Stoffer, Reversible long-time integration with variable step sizes, SIAM J. Sci. Comput., to appear.
23. E. Hairer, S.P. Nörsett and G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems (Springer, Berlin, 1987).
24. E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems (Springer, Berlin, 1991).
25. D.J. Higham, Robust defect control with Runge-Kutta schemes, SIAM J. Numer. Anal. 26 (1989) 1175-1183.
26. M.K. Horn, Fourth and fifth-order scaled Runge-Kutta algorithms for treating dense output, SIAM J. Numer. Anal. 20 (1983) 558-568.
27. F.Q. Hu, M.Y. Hussaini and J. Manthey, Low dissipation and dispersion Runge-Kutta schemes for computational acoustics, ICASE Report 94-102, NASA Langley Research Center (1994).
28. T.E. Hull, W.H. Enright, B.M. Fellen and A.E. Sedgwick, Comparing numerical methods for ordinary differential equations. SIAM J. Numer. Anal. 9 (1972) 603-637.
29. T.E. Hull, W.H. Enright and K.R. Jackson, User's guide for DVERK-a subroutine for solving non-stiff ODEs, Technical Report 100, Department of Computer Science, University of Toronto, Ontario (1976).
30. T.E. Hull, W.H. Enright and K.R. Jackson, Runge-Kutta research at Toronto, Appl. Numer. Math. 22 (1996) 225-236.
31. K.R. Jackson, W.H. Enright and T.E. Hull, A theoretical criterion for comparing Runge-Kutta formulas, SIAM J. Numer. Anal. 15 (1978) 618-641.
32. G.-S. Jiang and C.-W. Shu, Efficient implementation of weighted ENO schemes, ICASE Report 95-73, NASA Langley Research Center (1995).
33. R.H. Merson, An operational method for the study of integration processes, in: Proc. Sympos. Data Processing, Weapons Research Establishment, Salisbury, Australia (1957) 110-1-110-25.
34. M.L. Parker, Performance profile testing of ordinary differential equation routines, M.Sc. Thesis, Department of Mathematics, University of Manchester, UK (1979).
35. J.C. Polking, MATLAB Manual for Ordinary Differential Equations (Prentice-Hall, Englewood Cliffs, NJ, 1995).
36. P.J. Prince and J.R. Dormand, High order embedded Runge-Kutta formulae, J. Comput. Appl. Math. 7 (1981) 67-75.
37. L.F. Shampine, Local extrapolation in the solution of ordinary differential equations, Math. Comp. 27 (1973) 91-97.
38. L.F. Shampine, Stiffness and non-stiff differential equation solvers, in: L. Collatz et al., eds., Numerische Behandlung von Differentialgleichungen, International Series of Numerical Mathematics 27 (Birkhauser, Basel, 1975) 287-301.
39. L.F. Shampine, The step sizes used by one-step codes for ODEs, Appl. Numer. Math. 1 (1985) 95-106.
40. L.F. Shampine, Interpolation for Runge-Kutta methods, SIAM J. Numer. Anal. 22 (1985) 1014-1027.
41. L.F. Shampine, Conservation laws and the numerical solution of ODEs, Comput. Math. Appl. 12B (1986) 1287-1296.
42. L.F. Shampine, Diagnosing stiffness for Runge-Kutta methods, SIAM J. Sci. Statist. Comput. 12 (1991) 260-272.
43. L.F. Shampine, Numerical Solution of Ordinary Differential Equations (Chapman and Hall, New York, 1994).
44. L.F. Shampine, I. Gladwell and R. W. Brankin, Reliable solution of event location problems for ODEs, ACM Trans. Math. Software 17 (1991) 11-25.
45. L.F. Shampine and M.K. Gordon, Some numerical experiments with DIFSUB, SIGNUM Newsletter 7 (1972) 24-26.
46. L.F. Shampine and K.L. Hiebert, Detecting stiffness with the Fehlberg (4,5) formulas, Comput. Math. Appl. 3 (1977) 41-46.
47. L.F. Shampine and M.W. Reichelt, The MATLAB ODE suite, SIAM J. Sci. Comput., to appear.
48. L.F. Shampine and H.A. Watts, Global error estimation for ordinary differential equations and Algorithm 504, GERK: global error estimation for ordinary differential equations, ACM Trans. Math. Software 2 (1976) 172-186 and 200-203.
49. L.F. Shampine and H.A. Watts, The art of writing a Runge-Kutta code I, in: J.R. Rice, ed., Mathematical Software III (Academic Press, New York, 1977) 257-275; and The art of writing a Runge-Kutta code II, Appl. Math. Comput. 5 (1979) 93-121.
50. L.F. Shampine and H.A. Watts, The art of writing a Runge-Kutta code I, in: J.R. Rice, ed., Mathematical Software III (Academic Press, New York, 1977) 257-275; and The art of writing a Runge-Kutta code II, Appl. Math. Comput. 5 (1979) 93-121.
51. L.F. Shampine, H.A. Watts and S.M. Davenport, Solving nonstiff ordinary differential equations - the state of the art, SIAM Rev. 18 (1976) 376-411.
52. P.W. Sharp and E. Smart, Explicit Runge-Kutta pairs with one more derivative evaluation than the minimum, SIAM J. Sci. Comput. 14 (1993) 338-348.
53. P.W. Sharp and J.H. Verner, Explicit Runge-Kutta 4-5 pairs with interpolants, Math. Preprint 1995-03, Queen's University, Kingston, Ontario (1995).
54. C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes I, J. Comput. Phys 77 (1989) 439-471.
55. J.H. Verner, Explicit Runge-Kutta methods with estimates of the local truncation error, SIAM J. Numer. Anal. 15 (1978) 772-790.
56. J.H. Verner, Private communication (1992).
57. H.A. Watts, Step size control in ordinary differential equation solvers, Trans. Soc. Computer Simulation 1 (1984) 15-25.
58. H.A. Watts, RDEAM-an Adams ODE code with root solving capability, Report SAND85-1595, Sandia National Laboratories, Albuquerque, NM (1985).
59. H.A. Watts, Backward differentiation formulas revisited: improvements in DEBDF and a new root solving code RDEBD. Report SAND85-2676, Sandia National Laboratories, Albuquerque, NM (1986).
60. J.A. Zonneveld, Automatic Numerical Integration (Mathematisch Centrum, Amsterdam, 1964).