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Efficient integration of large stiff systems of ODEs with exponential propagation iterative (EPI) methods. (English) Zbl 1089.65063
Summary: A new class of exponential propagation techniques which we call exponential propagation iterative (EPI) methods is introduced. It is demonstrated how for large stiff systems these schemes provide an efficient alternative to standard integrators for computing solutions over long time intervals. The EPI methods are constructed by reformulating the integral form of a solution to a nonlinear autonomous system of ordinary differential equations (ODEs) as an expansion in terms of products between special functions of matrices and vectors that can be efficiently approximated using Krylov subspace projections.
The methodology for constructing EPI schemes is presented and their performance is illustrated using numerical examples and comparisons with standard explicit and implicit integrators. The history of the exponential propagation type integrators and their connection with EPI schemes are also discussed.

MSC:
65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
Software:
Expokit
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[1] Knoll, D.A.; Keyes, D.E., Jacobian-free Newton-Krylov methods: a survey of approaches and applications, J. comput. phys., 193, 357-397, (2004) · Zbl 1036.65045
[2] Hairer, E.; Nørsett, S.P.; Wanner, G., Solving ordinary differential equations, (1987), Springer Berlin · Zbl 0638.65058
[3] S.M. Cox, Exponential time differencing (ETD) references. Available from: <www.maths.adelaide.edu.au/people/scox/etd.html>.
[4] J. Certaine, The solution of ordinary differential equations with large time constants, in: A. Ralston, H.S. Wilf (Eds.), Mathematical Methods for Digital Computers, 1960, pp. 128-132.
[5] Pope, D.A., An exponential method of numerical integration of ordinary differential equations, Commun. ACM, 6, 8, 491-493, (1963) · Zbl 0117.11204
[6] Lawson, J.D., Generalized Runge-Kutta processes for stable systems with large Lipschitz constants, SIAM J. numer. anal., 4, 372-380, (1967) · Zbl 0223.65030
[7] Nørsett, S.P., An A-stable modification of the Adams-bashforth methods, Lecture notes math., 109, 214-219, (1969)
[8] Moler, C.B.; Van Loan, C.F., Nineteen dubious ways to compute the exponential of a matrix, SIAM rev., 20, 4, 801-836, (1978) · Zbl 0395.65012
[9] Moler, C.B.; Van Loan, C.F., Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later, SIAM rev., 45, 1, 3-49, (2003) · Zbl 1030.65029
[10] Gallopoulos, E.; Saad, Y., Efficient solution of parabolic equations by Krylov approximation methods, SIAM J. sci. stat. comp., 13, 1236-1264, (1992) · Zbl 0757.65101
[11] Lawson, J.D.; Thomas, S.J.; Zahar, R.V.M., Weighted quadrature in Krylov methods, Utilitas Mathematica, 51, 165-182, (1997) · Zbl 0896.65050
[12] Beylkin, G.; Keiser, J.M.; Vozovoi, L., A new class of time discretization schemes for the solution of nonlinear pdes, J. comput. phys., 147, 362-387, (1998) · Zbl 0924.65089
[13] Cox, S.M.; Matthews, P.C., Exponential time differencing for stiff systems, J. comput. phys., 176, 430-455, (2002) · Zbl 1005.65069
[14] Kassam, A.K.; Trefethen, L.N., Fourth-order time stepping for stiff pdes, SIAM J. sci. comp., (2004)
[15] Krogstad, S., Generalized integrating factor methods for stiff pdes, J. comput. phys., 203, 1, 72-88, (2005) · Zbl 1063.65097
[16] Hochbruck, M.; Ostermann, A., Exponential Runge-Kutta methods for parabolic problems, Appl. numer. math., 53, 323-339, (2005) · Zbl 1070.65099
[17] Saad, Y., Iterative methods for sparse linear systems, (1996), PWS Publishing Company · Zbl 1002.65042
[18] Gear, C.W.; Saad, Y., Iterative solution of linear equations in ODE codes, SIAM J. sci. stat. comput., 4, 4, 583-601, (1983) · Zbl 0541.65051
[19] Nauts, A.; Wyatt, R.E., New approach to many-state quantum dynamics: the recursive-residue-generation method, Phys. rev. lett., 51, 5, 2238-2241, (1983)
[20] Park, T.J.; Light, J.C., Unitary quantum time evolution by iterative Lanczos reduction, J. chem. phys., 85, 5870-5876, (1986)
[21] Van der Vorst, H.A., An iterative solution method for solving f(a)x=b using Krylov subspace information obtained for the symmetric positive definite matrix a, J. comput. appl. math., 18, 249-263, (1987) · Zbl 0621.65022
[22] Friesner, R.A.; Tuckerman, L.S.; Dornblaser, B.C.; Russo, T.V., A method for exponential propagation of large systems of stiff nonlinear differential equations, J. sci. comput., 4, 327-354, (1989)
[23] M. Tokman, Magnetohydrodynamic modeling of solar coronal arcades using exponential propagation methods, Ph.D. Thesis, Caltech, 2000.
[24] Hochbruck, M.; Lubich, Ch., On Krylov subspace approximations to the matrix exponential operator, SIAM J. numer. anal., 34, 1911-1925, (1997) · Zbl 0888.65032
[25] Hochbruck, M.; Lubich, Ch.; Selhofer, H., Exponential integrators for large systems of differential equations, SIAM J. sci. comput., 19, 1552-1574, (1998) · Zbl 0912.65058
[26] Tokman, M.; Bellan, P.M., Three-dimensional model of the structure and evolution of coronal mass ejections, Astrophys. J., 567, 2, 1202-1210, (2002)
[27] Arnoldi, W.E., The principle of minimized iteration in the solution of the matrix eigenvalue problem, Quart. appl. math., 9, 17-29, (1951) · Zbl 0042.12801
[28] Knizhnerman, L.A., Calculation of functions of unsymmetric matrices using Arnoldi method, Comput. math. math. phys., 31, 1, 1-9, (1991) · Zbl 0774.65021
[29] Saad, Y., Analysis of some Krylov subspace approximations to the matrix exponential operator, SIAM J. numer. anal., 29, 209-228, (1992) · Zbl 0749.65030
[30] Sidje, R.B., Expokit: a software package for computing matrix exponentials, ACM T. math. software, 24, 1, 130-156, (1998) · Zbl 0917.65063
[31] R.H. Merson, An operational method for the study of integration processes, in: Proceedings of the Symposium on Data Processing, Weapons Research Establishment, Salisbury, Australia, 1957, pp. 110-1-110-25.
[32] Butcher, J.C., Coefficients for the study of Runge-Kutta integration processes, J. austr. math. soc., 3, 185-201, (1963) · Zbl 0223.65031
[33] Lefever, R.; Nicolis, G., Chemical instabilities and sustained oscillations, J. theor. biol., 30, 267-284, (1971) · Zbl 1170.92344
[34] FitzHugh, R., Mathematical models of excitation and propagation in nerve, (), 1-85
[35] Nagumo, J.; Arimoto, S.; Yoshizawa, S., An active pulse transmission line simulating nerve axon, Proc. IRE, 50, 2061-2070, (1962)
[36] Zeeman, E.C., Differential equations for the heartbeat and nerve impulse, (), 8-67 · Zbl 0289.92004
[37] Dembo, R.S.; Eisenstat, S.C.; Steihaug, T., Inexact Newton methods, SIAM J. numer. anal., 19, 2, 400-408, (1982) · Zbl 0478.65030
[38] Eisenstat, S.C.; Walker, H.F., Choosing the forcing terms in an inexact Newton method, SIAM J. sci. comput., 17, 1, 16-32, (1996) · Zbl 0845.65021
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