A PI stepsize control for the numerical solution of ordinary differential equations. (English) Zbl 0645.65039

Using ideas of control theory the authors derive a new step size control algorithm for the numerical integration of ordinary differential equations. The essential formula (Formula (11) on page 277) is astonishing simple and easy to implement. The proposed “proportional integral (PI)” algorithm is more robust than earlier ones and avoids oscillations of the step sizes when they are limited by numerical stability. It usually decreases the number of rejected steps. Several numerical experiments illustrate these properties.
Reviewer: E.Hairer


65L05 Numerical methods for initial value problems involving ordinary differential equations
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
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[1] H. Elmqvist, K. J. Åström and T. Schönthal,SIMNON, User’s Guide for MS-DOS Computers, Department of Automatic Control, Lund Institute of Technology, Lund, Sweden (1986).
[2] W. H. Enright, T. E. Hull and B. Lindberg,Comparing numerical methods for stiff systems of ODE’s, BIT 15 (1975), 28–33. · Zbl 0301.65040
[3] G. F. Franklin, J. D. Powell and A. Emami-Naeini,Feedback Control Systems, Addison-Wesley (1986), pp. 99–103. · Zbl 0615.93001
[4] C. W. Gear,Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall (1971). · Zbl 1145.65316
[5] E. Hairer, S. P. Nørsett and G. Wanner,Solving Ordinary Differential Equations, I, Springer (1987). · Zbl 0638.65058
[6] G. Hall,Equilibrium states of Runge-Kutta schemes: Part I, ACM Transactions on Mathematical Software, 11, 3 (1985), 289–301. · Zbl 0601.65056
[7] G. Hall,Equilibrium states of Runge-Kutta schemes: Part II, ACM Transactions on Mathematical Software, 12, 3 (1986), 183–192. · Zbl 0633.65062
[8] G. Hall and D. J. Higham,Analysis of stepsize selection for Runge-Kutta codes, NA report No. 137, University of Manchester (1987). · Zbl 0661.65077
[9] D. J. Higham and G. Hall,Embedded Runge-Kutta formulae with stable equilibrium states, NA report No. 140, University of Manchester (1987). · Zbl 0704.65054
[10] B. Lindberg,Characterization of optimal stepsize sequences for methods for stiff differential equations, SINUM, 14 (1977), 859–887. · Zbl 0373.65029
[11] L. F. Shampine,Stiffness and nonstiff differential equation solvers, in L. Collatz (Ed.):Numerische Behandlung von Differentialgleichungen, Information Series of Numerical Mathematics 27, Birkhäuser Verlag, Basel (1975), pp. 287–301. · Zbl 0303.65065
[12] K. J. Åström and B. Wittenmark,Computer Controlled Systems, Prentice-Hall, Englewood Cliffs, N.J. (1984), pp. 369–373.
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