Håvard Berland
Abstract
Recently, a great deal of attention has been focused on the construction of exponential integrators for semilinear problems. In this article we describe a MATLAB1 package which aims to facilitate the quick deployment and testing of exponential integrators, of Runge--Kutta, multistep, and general linear type. A large number of integrators are included in this package along with several well-known examples. The so-called φ functions and their evaluation is crucial for accuracy, stability, and efficiency of exponential integrators, and the approach taken here is through a modification of the scaling and squaring technique, the most common approach used for computing the matrix exponential.
- Berland, H., Owren, B., and Skaflestad, B. 2005. B-series and order conditions for exponential integrators. SIAM J. Numer. Anal. 43, 4, 1715--1727. Google Scholar
Digital Library
- Beylkin, G., Keiser, J. M., and Vozovoi, L. 1998. A new class of time discretization schemes for the solution of nonlinear PDEs. J. Comp. Phys. 147, 362--387. Google ScholarDigital Library
- Burrage, K. and Butcher, J. C. 1980. Nonlinear stability for a general class of differential equation methods. BIT 20, 185--203.Google ScholarCross Ref
- Butcher, J. C. 2003. Numerical methods for ordinary differential equations. John Wiley & Sons Ltd., Chichester, U.K. Google ScholarDigital Library
- Calvo, M. and Palencia, C. 2006. A class of multistep exponential integrators for semilinear problems. Numer. Math. 102, 367--381.Google ScholarDigital Library
- Celledoni, E., Marthinsen, A., and Owren, B. 2003. Commutator-free Lie group methods. Fut. Gen. Comput. Syst. 19, 3, 341--352. Google ScholarDigital Library
- Certaine, J. 1960. The solution of ordinary differential equations with large time constants. In Mathematical Methods for Digital Computers. Wiley, New York, NY, 128--132.Google Scholar
- Cox, S. M. and Matthews, P. C. 2002. Exponential time differencing for stiff systems. J. Comput. Phys. 176, 2, 430--455. Google ScholarDigital Library
- Ehle, B. L. and Lawson, J. D. 1975. Generalized Runge--Kutta processes for stiff initial-value problems. J. Inst. Math. Appl. 16, 11--21.Google ScholarCross Ref
- Friedli, A. 1978. Verallgemeinerte Runge--Kutta Verfahren zur Lösung steifer Differentialgleichungssysteme. In Numerical Treatment of Differential Equations. Lecture Notes in Mathematics, vol. 631. Springer, Berlin, Germany, 35--50.Google Scholar
- Friesner, R. A., Tuckerman, L. S., Dornblaser, B. C., and Russo, T. V. 1989. A method for the exponential propagation of large stiff nonlinear differential equations. J. Sci. Comput. 4, 4, 327--354. Google ScholarDigital Library
- Gallopoulos, E. and Saad, Y. 1992. Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13, 1236--1264. Google ScholarDigital Library
- Higham, N. J. 2002. Accuracy and stability of numerical algorithms, 2nd ed. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA. Google ScholarDigital Library
- Hochbruck, M. and Lubich, C. 1997. On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34, 5, 1911--1925. Google ScholarDigital Library
- Hochbruck, M., Lubich, C., and Selhofer, H. 1998. Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput. 19, 5, 1552--1574. Google ScholarDigital Library
- Hochbruck, M. and Ostermann, A. 2005. Explicit exponential Runge--Kutta methods for semilinear parabolic problems. SIAM J. Numer. Anal. 43, 3, 1069--1090. Google ScholarDigital Library
- Kassam, A.-K. and Trefethen, L. N. 2005. Fourth-order time-stepping for stiff PDEs. SIAM J. Sci. Comput. 26, 4, 1214--1233 (electronic). Google ScholarDigital Library
- Krogstad, S. 2005. Generalized integrating factor methods for stiff PDEs. J. Comp. Phys. 203, 1, 72--88. Google ScholarDigital Library
- Lawson, J. D. 1967. Generalized Runge--Kutta processes for stable systems with large Lipschitz constants. SIAM J. Numer. Anal. 4, 372--380.Google ScholarCross Ref
- Lu, Y. Y. 2003. Computing a matrix function for exponential integrators. J. Comput. Appl. Math 161, 1, 203--216. Google ScholarDigital Library
- Minchev, B. and Wright, W. M. 2005. A review of exponential integrators for semilinear problems. Tech. rep. 2/05. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway. Available online at http://www.math.ntnu.no/preprint/.Google Scholar
- Moler, C. B. and van Loan, C. F. 2003. Nineteen dubious ways to compute the exponential of a matrix, twenty five years later. SIAM Review 45, 1, 3--49.Google ScholarDigital Library
- Munthe-Kaas, H. 1999. High order Runge--Kutta methods on manifolds. In Proceedings of the NSF/CBMS Regional Conference on Numerical Analysis of Hamiltonian Differential Equations (Golden, CO, 1997). Appl. Numer. Math. 29,1, 115--127. Google ScholarCross Ref
- N&phis;rsett, S. P. 1969. An A-stable modification of the Adams--Bashforth methods. In Conference on Numerical Solution of Differential Equations (Dundee, Scotland, 1969). Springer, Berlin, Germany, 214--219.Google Scholar
- Ostermann, A., Thalhammer, M., and Wright, W. M. 2006. A class of explicit exponential general linear methods. BIT 46, 409--431.Google ScholarDigital Library
- Saad, Y. 1981. Krylov subspace methods for solving large unsymmetric linear systems. Math. Comp. 37, 105--126.Google ScholarCross Ref
- Skaflestad, B. and Wright, W. M. 2006. The scaling and modified squaring method for functions related to the exponential. appl. Numer. Math. Submitted. Google ScholarDigital Library
- Strehmel, K. and Weiner, R. 1982. Behandlung steifer Anfangswertprobleme gewöhnlicher Differentialgleichungen mit adaptiven Runge--Kutta-Methoden. Computing 29, 2, 153--165.Google ScholarCross Ref
- Strehmel, K. and Weiner, R. 1987. B-convergence results for linearly implicit one step methods. BIT 27, 264--281. Google ScholarDigital Library
- van der Houwen, P. J. 1977. Construction of Integration Formulas for Initial Value Problems. North-Holland Series in Applied Mathematics and Mechanics, Vol. 19. North-Holland, Amsterdam, The Netherlands.Google Scholar
Index Terms
- EXPINT---A MATLAB package for exponential integrators
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Reviews
John Charles Butcher
Exponential integrators have a long history, involving many variants and many rediscoveries. The basic idea is very simple. For a differential equation system = Ly + N ( y,t ), in which stiff behavior is expressed principally through the linear term Ly , a classical method is generalized to include terms involving exp( hL ), and related operators, so that it reverts to the classical approximation for the problem = N ( y,t ) when L =0 and gives the exact solution to = Ly when N =0. Furthermore, for a problem in which both L and N are present, the method is required to maintain high order and good stability. In the implementation of these methods, the evaluations of φ i ( hL ), where φ0( z )=exp( z ), φ1( z )=(exp( z )-1)/ z , and so on, present the major computational challenges. Thus, the package EXPINT features carefully researched and written routines for these evaluations, and incorporates them into a variety of Runge-Kutta, linear multistep, and general linear schemes. Much is still to be learned about the capabilities of exponential integrators and the problems that they can solve effectively. This package provides a convenient laboratory for learning about the current state of these methods and for taking knowledge about them further. Online Computing Reviews Service
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