A class of explicit exponential general linear methods

BIT Numerical Mathematics

Volumn 46, Issue 2, 2006, Pages 409-431

  Ostermann, A ^a   Thalhammer, M ^a   Wright, W M ^b  

^a UNIVERSITY OF INNSBRUCK   (Austria)

^b LA TROBE UNIVERSITY   (Australia)

Author keywords

Abstract evolution equations; Convergence; Explicit schemes; Exponential integrators; General linear methods; High order methods; Semilinear parabolic problems

Indexed keywords

EID: 33745319450     PISSN: 00063835     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10543-006-0054-3     Document Type: Article

References (20)

1. N. Y. Bakaev, On variable stepsize Rungs-Kutta approximations of a Cauchy problem for the evolution equation, BIT, 38 (1998) pp. 462-485.
2. H. Berland, B. Skaflestad, and W. M. Wright, Expint - A Matlab package for exponential integrators, Tech. report 4/05, Department of Mathematics, NTNU, 2005.
3. G. Beylkin, J. M. Keiser, and L. Vozovoi, A new class of time discretization schemes for the solution of nonlinear PDEs, J. Comput. Phys., 147 (1998), pp. 362-387.
4. H. Brunner and P. J. van der Houwen, The Numerical Solution of Volterra Equations, CWI Monographs 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. P. M. Cox and P. C. Matthews, Exponential time differencing for stiff systems, J. Comput. Phys., 176 (2002), pp. 430-455.
7. A. Friedli, Verallgemeinerte Runge-Kutta Verfahren zur Lösung steifer Differential-gleichungssysteme, in Numerical Treatment of Differential Equations, R. Bulirsch, R. Grigorieff, and J. Schröder, (eds.), Lecture Notes in Mathematics, vol. 631, pp. 35-50, Springer, Berlin, 1978.
8. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer, Berlin, 1981.
9. M. Hochbruck and M. E. Hochstenbach, Subspace extraction for matrix functions, preprint, Department of Mathematics, Case Western Reserve University, 2005.
10. M. Hochbruck and A. Ostermann, Exponential Runge-Kutta methods for parabolic problems, Appl. Numer. Math., 53 (2005), pp. 323-339.
11. M. Hochbruck and A. Ostermann, Explicit exponential Runge-Kutta methods for semilinear parabolic problems, SIAM J. Numer. Anal., 43 (2005), pp. 1069-1090.
12. A. K. Kassam and L. N. Trefethen, Fourth-order time stepping for stiff PDEs, SIAM J. Sci. Comput., 26 (2005), pp. 1214-1233.
13. S. Krogstad, Generalized integrating factor methods for stiff PDEs, J. Comput. Phys., 203 (2005), pp. 72-88.
14. A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, 1995.
15. 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, NTNU, 2005.
16. S. P. Nørsett, An A-stable modification of the Adams-Bashforth methods, in Conference on the Numerical Solution of Differential Equations, J. Morris, ed., Lecture Notes in Mathematics, vol. 109, pp. 214-219, Springer, Berlin, 1969.
17. 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.
18. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
19. K. Strehmel and R. Weiner, B-convergence results for linearly-implicit one step methods, BIT, 27 (1987), pp. 264-281.
20. J. G. Verwer, On generalized linear multistep methods with zero-parasitic roots and an adaptive principal root, Numer. Math., 27 (1977), pp. 143-155.