A fourth-order commutator-free exponential integrator for nonautonomous differential equations

SIAM Journal on Numerical Analysis

Volumn 44, Issue 2, 2006, Pages 851-864

  Thalhammer, Mechthild ^a  

^a UNIVERSITY OF INNSBRUCK   (Austria)

Author keywords

Commutator free methods; Convergence; Exponential integrators; Nonautonomous differential equations; Parabolic evolution equations; Stability

Indexed keywords

AUTONOMOUS AGENTS; BANACH SPACES; CONVERGENCE OF NUMERICAL METHODS; INTEGRAL EQUATIONS; LINEAR SYSTEMS;

COMMUTATOR FREE METHODS; EXPONENTIAL INTEGRATORS; NONAUTONOMOUS DIFFERENTIAL EQUATIONS; PARABOLIC EVOLUTION EQUATIONS;

BOUNDARY VALUE PROBLEMS;

EID: 34247278698     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/05063042     Document Type: Article

References (30)

1. CH. BESSE, B. BIDEGARAY, AND S. DESCOMBES, Order estimates in time of splitting methods for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 40 (2002), pp. 26-40.
2. S. BLANES, F. CASAS, J.A. OTEO, AND J. ROS, Magnus and Fer expansions for matrix differential equations: The convergence problem, J. Phys. A, 31 (1998), pp. 259-268.
3. S. BLANES AND P.C. MOAN, Splitting methods for the time-dependent Schrödinger equation, Phys. Lett. A, 265 (2000), pp. 35-42.
4. H. BRUNNNER AND P.J. VAN DER HOUWEN, The Numerical Solution of Volterra Equations, CWI Monogr. 3, North-Holland, Amsterdam, 1986.
5. M.P. CALVO AND C. PALENCIA, A class of explicit multistep exponential integrators for semilinear problems, Numer. Math., 102 (2006), pp. 367-381.
6. E. CELLEDONI, Eulerian and Semi-Lagrangian Schemes Based on Commutator Free Exponential Integrators, CRM Proc. Lecture Notes, ISSN 1065-8580, 39 (2005).
7. B. CELLEDONI, A. MARTINSEN, AND B. OWREN, Commutator-free Lie group methods, Future Generation Computer Systems (FGCS), 19 (2003), pp. 341-352.
8. S.M. COX AND P.C. MATTHEWS, Exponential time differencing for stiff systems, J. Comput. Phys., 176 (2002), pp. 430-455.
9. P.E. CROUCH AND R. GROSSMAN, Numerical integration of ordinary differential equations on manifolds, J. Nonlinear Sci., 3 (1993), pp. 1-33.
10. J. VAN DEN ESHOF AND M. HOCHBRUCK, Preconditioning Lanczos approximations to the matrix exponential, SIAM J. Sci. Comput., 27 (2006), pp. 1438-1457.
11. C. GONZALEZ, A. OSTERMANN, AND M. THALHAMMER, A second-order Magnus integrator for non-autonomous parabolic problems, J. Comput. Appl. Math., 189 (2006), pp. 142-156.
12. C. GONZÁLEZ AND M. THALHAMMER, A second-order Magnus type integrator for quasilinear parabolic problems, Math. Comp., in press.
13. D. HENRY, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer, Berlin, 1981.
14. M. HOCHBRUCK AND CH. LUBICH, On Magnus integrators for time-dependent Schrödinger equations, SIAM J. Numer. Anal., 41 (2003), pp. 945-963.
15. M. HOCHBRUCK AND CH. LUBICH, On Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal., 34 (1997), pp. 1911-1925.
16. M. HOCHBRUCK AND A. OSTERMANN, Explicit exponential Runge-Kutta methods for semilinear parabolic problems, SIAM J. Numer. Anal., 43 (2005), pp. 1069-1090.
17. A. ISERLES, On the global error of discretization methods for highly-oscillatory ordinary differential equations, BIT, 42 (2002), pp. 561-599.
18. T. JAHNKE AND CH. LUBICH, Error bounds for exponential operator splittings, BIT, 40 (2000), pp. 735-744.
19. A.K. KASSAM AND L.N. TREFETHEN, Fourth-order time stepping for stiff PDEs, SIAM J. Sci. Comput., 26 (2005), pp. 1214-1233.
20. S. KROGSTAD, Generalized integrating factor methods for stiff PDEs, J. Comput. Phys., 203 (2005), pp. 72-88.
21. Y.Y. LU, A fourth order Magnus scheme for Helmholtz equation, J. Comput. Appl. Math., 173 (2005), pp. 247-258.
22. A. LUNARDI, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, Switzerland, 1995.
23. W. MAGNUS, On the exponential solution of a differential equation for a linear operator, Comm. Pure Appl. Math., 7 (1954), pp. 649-673.
24. B.V. MINCHEV AND W.M. WRIGHT, A Review of Exponential Integrators for First Order Semi-linear Problems, Tech. report 2/05, Department of Mathematics, Norwegian University of Science and Technolgy (NTNU), April 2005.
25. C. MOLER AND CH. VAN LOAN, Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later, SIAM Rev., 45 (2003), pp. 3-49.
26. A. OSTERMANN AND M. THALHAMMER, Non-smooth data error estimates for linearly implicit Runge-Kutta methods, IMA J. Numer. Anal., 20 (2000), pp. 167-184.
27. A. PAZY, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
28. Q. SHENG, Global error estimates for exponential splitting, IMA J. Numer. Anal., 14 (1993), pp. 27-56.
29. H. TANABE, Equations of Evolution, Pitman, London, 1979.
30. M. THALHAMMER, A second-order Magnus type integrator for non-autonomous semilinear parabolic problems, Universität Innsbruck, 2005, preprint.