Abstract
The design of a code which uses different stepsizes for different components of a system of ordinary differential equations is discussed. Methods are suggested which achieve moderate efficiency for problems having some components with a much slower rate of variation than others. Techniques for estimating errors in the different components are analyzed and applied to automatic stepsize and order control. Difficulties, absent from non-multirate methods, arise in the automatic selection of stepsizes, leading to a suggested organization of the code that is counter-intuitive. An experimental code and some initial experiments are described.
Similar content being viewed by others
References
J. F. Andrus,Numerical solution of systems of ordinary differential equations separated into subsystems, SIAM J. Num. Anal. 16(4), 1979, 605–611.
C. W. Gear,Automatic multirate methods for ordinary differential equations, Proc. IFIP 1980, 717–722. North-Holland Publishing Company.
P. Henrici,Discrete Variable Methods in Ordinary Differential Equations, John Wiley & Sons, Inc.: New York, 1962.
A. Orailoglu,Software design issues in the implementation of hierarchical, display editors, Rept. No. UIUCDCS-4-83-1139, Dept. Computer Sci., Univ. Illinois, 1983.
O. A. Paulusinski and J. V. Wait,Simulation methods for combined linear and nonlinear systems, Simulation 30(3), 1978, 85–94.
L. Petzold,Automatic selection of methods for solving stiff and nonstiff systems of ordinary differential equations, SIAM J. Scientific and Statistical Computing 4(1), 1983, 136–148.
L. F. Shampine,Type-insensitive ODE codes based on implicit A (α)-stable formulas, Math. Comp. 39 (159), 1982, 109–124.
D. R. Wells,Multirate linear multistep methods for the solution of systems of ordinary differential equations, Rept. No. UIUCDCS-R-82-1093, Dept. Computer Sci., Illinois, 1982.
Author information
Authors and Affiliations
Additional information
Dedicated to Professor Germund Dahlquist on the occasion of his 60th birthday
Supported in part by the Department of Energy under grant DOE DEAC0276ERO2383.
Work done while attending the University of Illinois.
Rights and permissions
About this article
Cite this article
Gear, C.W., Wells, D.R. Multirate linear multistep methods. BIT 24, 484–502 (1984). https://doi.org/10.1007/BF01934907
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01934907