A multirate time stepping strategy for stiff ordinary differential equations THANKSREF="*" ID="*"Received May 1, 2006. Accepted July 19, 2006. Communicated by Christian Lubich.
Tóm tắt
To solve ODE systems with different time scales which are localized over the components, multirate time stepping is examined. In this paper we introduce a self-adjusting multirate time stepping strategy, in which the step size for a particular component is determined by its own local temporal variation, instead of using a single step size for the whole system. We primarily consider implicit time stepping methods, suitable for stiff or mildly stiff ODEs. Numerical results with our multirate strategy are presented for several test problems. Comparisons with the corresponding single-rate schemes show that substantial gains in computational work and CPU times can be obtained.
Tài liệu tham khảo
A. Bartel, Generalised Multirate. Two ROW-type versions for circuit simulation, Unclass. Natlab Report No. 2000/84, Philips Electronics, 2000.
A. Bartel and M. Günther, A multirate W-method for electrical networks in state space formulation, J. Comput. Appl. Math., 147 (2002), pp. 411–425.
C. Engstler and C. Lubich, Multirate extrapolation methods for differential equations with different time scales, Computing, 58 (1997), pp. 173–185.
C. Engstler and C. Lubich, MUR8: A multirate extension of the eight-order Dormand–Prince method, Appl. Numer. Math., 25 (1997), pp. 185–192.
D. Estep, M. G. Larson, and R. D. Williams, Estimating the Error of Numerical Solutions of Systems of Reaction-Diffusion Equations, Mem. Am. Math. Soc., 146 (2000).
C. Gear and D. Wells, Multirate linear multistep methods, BIT, 24 (1984), pp. 484–502.
M. Günther, A. Kvaernø, and P. Rentrop, Multirate partitioned Runge–Kutta methods, BIT, 41 (2001), pp. 504–514.
E. Hairer and G. Wanner, Solving Ordinary Differential Equations II – Stiff and Differential-Algebraic Problems, 2nd edn., Springer Series in Comput. Math. 14, Springer, Berlin, 1996.
W. Hundsdorfer and J. G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer Series in Comput. Math. 33, Springer, Berlin, 2003.
J. Jansson and A. Logg, Algorithms for multi-adaptive time stepping, Chalmers Finite Element Center, Preprint 2004-13, 2004.
A. Kvaernø, Stability of multirate Runge–Kutta schemes, Int. J. Differ. Equ. Appl., 1A (2000), pp. 97–105.
A. Logg, Multi-adaptive Galerkin methods for ODEs I, SIAM J. Sci. Comput., 24 (2003), pp. 1879–1902.
A. Logg, Multi-adaptive Galerkin methods for ODEs II. Implementation and applications, SIAM J. Sci. Comput., 25 (2003), pp. 1119–1141.
L. F. Shampine, Numerical Solution of Ordinary Differential Equations, Chapman & Hall, New York, 1994.