Exponential rosenbrock-type methods

SIAM Journal on Numerical Analysis

Volumn 47, Issue 1, 2008, Pages 786-803

  Hochbruck, Marlis ^a   Ostermann, Alexander ^b   Schweitzer, Julia ^a  

^a HEINRICH HEINE UNIVERSITY   (Germany)

^b UNIVERSITY OF INNSBRUCK   (Austria)

Author keywords

Convergence bounds; Embedded methods of high order; Exponential integrators; Exponential rosenbrock type methods; Stability bounds; Stiff order conditions; Variable stop size implementation

Indexed keywords

CONVERGENCE BOUNDS; EXPONENTIAL INTEGRATORS; EXPONENTIAL ROSENBROCK-TYPE METHODS; HIGH ORDER; ORDER CONDITIONS; TYPE METHODS;

BANACH SPACES; CAVITY RESONATORS; CONVERGENCE OF NUMERICAL METHODS; DIFFERENTIAL EQUATIONS; FUNCTION EVALUATION; LINEAR SYSTEMS; SYSTEM STABILITY; TOPOLOGY;

EXPONENTIAL FUNCTIONS;

EID: 70350671297     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/080717717     Document Type: Article

References (28)

1. P.N. BROWN, G.D. BYRNE, AND A.C. HINDMARSH, VODE: A variable-coefficient ODE solver, SIAM J. Sci. Statist. Comput., 10 (1989), pp. 1038-1051.
2. G.D. BYRNE, Pragmatic experiments with Krylov methods in the stiff ODE setting, in Computational Ordinary Differential Equations, J.R Cash and I. Gladwell, eds., Clarendon Press, Oxford, 1992, pp. 323-356.
3. M. CALIARI AND A. OSTERMANN, Implementation of exponential Rosenbrock-type integrators, Appl. Numer. Math., to appear.
4. M. CALIARI, M. VIANELLO, AND L. BEROAMASOHI, Interpolating discrete advection-diffusion propagators at Leja sequences, J. Comput. Appl. Math., 172 (2004), pp. 79-99.
5. M.P. CALVO AND C. PALENOIA, A class of explicit multistep exponential integrators for semilinear problems, Numer. Math., 102 (2006), pp. 367-381.
6. S.M. Cox AND P.C. MATTHEWS, Exponential time differencing for stiff systems, J. Comput. Phys., 176 (2002), pp. 430-455.
7. V.L. DRUSKIN AND L.A. KNIZHNERMAN, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic, Numer. Linear Algebra Appl., 2 (1995), pp. 205-217.
8. K.-J. ENGEL AND R. NAGEL, One-Parameter Semigroups for Linear Evolution Equations, Springer, Now York, 2000.
9. A. FRIEDLI, Verallgemeinerte Runge-Kutta Verfahren zur Lösung steif er Differentialgleichungssysteme, in Numerical Treatment of Differential Equations, Lect. Notes in Math. 631, R. Bulirsch, R.D. Grigorieff, and J. Schröder, ods., Springer, Berlin, 1978.
10. E. HAIRER, AND G. WANNER, Solving Ordinary Differential Equations II, Stiff and DifferentialAlgebraic Problems, 2nd rev. od., Springer, Now York, 1996.
11. M. HOCHBRUCK AND CH. LUBIOH, On Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal., 34 (1997), pp. 1911-1925.
12. M. HOCHBRUCK, CH. LUBICH, AND H. SELHOFER, Exponential integrators for large systems of differential equations, SIAM J. Sci. Comput., 19 (1998), pp. 1552-1574.
13. M. HOCHBRUCK AND J. NIEHOFF, Approximation of matrix operators applied to multiple vectors, Math. Comput. Simulation, 79 (2008), pp. 1270-1283.
14. M. HOCHBRUCK AND A. OSTERMANN, Explicit exponential Runge-Kutta methods for semilinear parabolic problems, SIAM J. Numer. Anal., 43 (2005), pp. 1069-1090.
15. M. HOCHBRUCK AND A. OSTERMANN, Exponential integrators of Rosenbrock-type, Oberwolfach Rep. 3 (2006), pp. 1107-1110.
16. A.-K. KASSAM AND L.N. TREFETHEN, Fourth-order time stepping for stiff PDEs, SIAM J. Sci. Comput., 26 (2005), pp. 1214-1233.
17. S. KROGSTAD, Generalized integrating factor methods for stiff PDEs, J. Comput. Phys., 203 (2005), pp. 72-88.
18. J.D. LAWSON, Generalized Runge-Kutta processes for stable systems with large Lipschitz constants, SIAM J. Numer. Anal., 4 (1967), pp. 372-380.
19. CH. LUBICH AND A. OSTERMANN, Linearly implicit time discretization of non-linear parabolic equations, IMA J. Numer. Anal., 15 (1995), pp. 555-583.
20. CH. LUBICH AND A. OSTERMANN, Runge-Kutta approximation of quasi-linear parabolic equations, Math. Comp., 64 (1995), pp. 601-627.
21. F. MAZZIA (Coordinator), Test Set for IVP Solvers, http://pitogora.dm. uniba.it/~testset.
22. A. OSTERMANN AND M. THALHAMMER, Convergence of Runge-Kutta methods for nonlinear parabolic equations, Appl. Numer. Math., 42 (2002), pp. 367-380.
23. A. OSTERMANN, M. THALHAMMER, AND W. WRIGHT, A class of explicit exponential general linear methods, BIT, 46 (2006), pp. 409-431.
24. A. PAZY, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, Now York, 1983.
25. B. SOMMEIJER, L. SHAMPINE, AND J. VERWER, RKC: An explicit solver for parabolic PDEs, J. Comput. Appl. Math., 88 (1998), pp. 315-326.
26. K. STREHMEL AND R. WEINER, B-convergence results for linearly implicit one step methods, BIT, 27 (1987), pp. 264-281.
27. M. TOKMAN, Efficient integration of large stiff systems of ODEs with exponential propagation iterative (EPI) methods, J. Comput. Phys., 213 (2006), pp. 748-776.
28. R. WEINER, B.A. SCHMITT, AND H. PODHAISKY, ROWMAP-a ROW-code with Krylov techniques for large stiff ODEs, Appl. Numer. Math., 25 (1997), pp. 303-319.