Evaluating matrix functions for exponential integrators via carathéodory-fejér approximation and contour integrals

Electronic Transactions on Numerical Analysis

Volumn 29, Issue , 2007, Pages 1-18

  Schmelzer, Thomas ^a   Trefethen, Lloyd N ^a  

^a UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

Exponential integrators; Hankel contour; Matrix exponential; Numerical quadrature; Rational uniform approximation; Stiff semilinear parabolic PDEs

Indexed keywords

ERROR ANALYSIS; FUNCTION EVALUATION; POLES;

EXPONENTIAL INTEGRATORS; HANKEL CONTOUR; MATRIX EXPONENTIALS; NUMERICAL QUADRATURE; SEMILINEAR PARABOLIC PDES; UNIFORM APPROXIMATION;

EXPONENTIAL FUNCTIONS;

EID: 58149333483     PISSN: None     EISSN: 10689613     Source Type: Journal    
DOI: None     Document Type: Article

References (46)

1. S. M. ALLEN AND J. W. CAHN, A microscope theory for antipliase boundary motion and its application to antiphase domain coarsening, ActaMetall., 27 (1979), pp. 1085-1095.
2. A. I. APTEKAREV, Sharp constants for rational approximations of analytic functions, Mat. Sb., 193 (2002). pp. 1-72.
3. H. BERLAND, B. SKAFLESTAD, AND W. WRIGHT, EXPINT-A MATLAB package for exponential integrators, ACM Trans. Math. Softw., 33 (2007).
4. 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.
5. M. P. CALVO AND C. PALENCIA, A class of explicit multistep exponential integrators for semilinear problems, Nuiner. Math., 102 (2006), pp. 367-381.
6. A. J. CARPENTER, A. RUTTAN, AND R. S. VARGA, Exteruled numerical computations on the "1/9" conjecture in rational approximation theory, in Rational approximation and interpolation, P. Graves-Morris, E. Saff, andR. Varga, eds.. Lecture Notes in Mathematics, 1105, Springer, Berlin, 1984, pp. 383-411.
7. -x in [0, +∞) and applications to heat-conduction problems, J. Approx. Theory, 2 (1969), pp. 50-65.
8. S. M. COX AND P. C. MATTHEWS, Exponential time differencing for stiff systems, J. Comput. Phys., 176 (2002), pp. 430-455.
9. T. A. DAVIS, Direct methods for sparse linear systems, SIAM, Philadelphia, PA, 2006.
10. P. DEUFLHARD AND F. BORNEMANN, Scientific computing with ordinary differential equations, Springer-Verlag, New York, 2002.
11. S. W. ELLACOTT, On the Faber transform and efficient numerical rational approximation, SIAM J. Numer. Anal., 20 (1983), pp. 989-1000.
12. R. FISHER, The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), pp. 355-369.
13. G. H. GOLUB AND C. F. VAN LOAN, Matrix computations, 3rd ed., Johns Hopkins L'niversity Press, Baltimore, 1996.
14. A. A. GONCHAR AND E. A. R.AKHMANOV, Equilibrium distributions and degree of rational approximation of analytic functions, Math. USSR Sbornik, 62 (1989), pp. 305-348.
15. E. HAIRER AND G. WANNER, Solving ordinary differential equations. II. Stiff and differential-algebraic problems, Springer-Verlag, Berlin, 1991.
16. G. H. HALPHEN, Traité des fonctions elliptiques et de leurs applications. Premiere partie. Théorie des fonctions elliptiques et de leurs développement en series, Gauthier-Villars, Paris, 1886.
17. N. J. HIGHAM, Accuracy and stability of numerical algorithms, 2nd ed., SIAM, Philadelphia, 2002.
18. -, The scaling and squaring method for the matrix exponential revisited, SIAM J. Matrix Anal. Appl., 26 (2005), pp. 1179-1193.
19. M. HOCHBRUCK AND C. LUBICH, On Krylov subspace approximations to the matrix exponential operator, SIAM J. Nuiner. Anal., 34 (1997), pp. 1911-1925.
20. M. HOCHBRUCK, C. LUBICH, AND H. SELHOFER, Exponential integrators for large systems of differential equations, SIAM J. Sci. Comput., 19 (1998), pp. 1552-1574.
21. M. HOCHBRUCK AND A. OSTERMANN, Explicit exponential Runge-Kutta methods for semilinear parabolic problems, SIAM J. Numer. Anal., 43 (2005), pp. 1069-1090.
22. -, Exponential Runge-Kutta methods for parabolic problems, Appl. Numer. Math., 53 (2005), pp. 323-339.
23. A.-K. KASSAM, Solving reaction-diffusion equations ten times faster, Tech. Report NA 03/16, Oxford University Computing Laboratory, 2003.
24. A.-K. KASSAM AND L. N. TREFETHEN, Fourth-order time-stepping for stiff PDEs, SIAM J. Sci. Comput., 26 (2005), pp. 1214-1233.
25. S. KROGSTAD, Generalized integrating factor methods for stiff PDEs, J. Comput. Phys., 203 (2005), pp. 72-88.
26. P. LIVERMORE, An implementation of the exponential time differencing scheme to the magnetohydrodynamic equations in a spherical shell, J. Comput. Phys., 220 (2007), pp. 824-838.
27. Y. Y. Lu, Computing a matrix function for exponential integrators, J. Comput. Appl. Math., 161 (2003), pp. 203-216.
28. A. P. MAGNUS, CFGT determination of Varga's constant '1/9', 1985. handout of the Padé approximation meeting organized by C. Brezinski at Marseille-Luminy, October 14-18, 1985.
29. -, Asymptotics and super asymptotics of best rational approximation error norms for the exponential function (the '1/9' problem) by the Carathéodory-F ejér method, in Nonlinear Methods and Rational Approximation II, A. Cuyt et a]., ed., Kluwer, Dordrecht, 1994, pp. 173-185.
30. G. MEINARDUS, Approximation of functions: Theory and numerical methods, Springer, Berlin, 1967.
31. B. V. MINCHEV, Computing analytic matrix functions for a class of exponential integrators, Reports in Informatics 278, University of Bergen, Bergen, Norway, 2004.
32. -, Exponential integrators for semilinear problems, PhD thesis, University of Bergen, 2004.
33. B. V. MINCHEV AND W. M. WRIGHT, A review of exponential integrators for first order semi-linear problems. Tech. Report 2/05, Norwegian University of Science and Technology, Trondheim, Norway, 2005.
34. C. MOLER AND C. VAN LOAN, Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later, SIAM Rev., 45 (2003), pp. 3-49.
35. M. MORI, Discovery of the double exponential transformation and its developments, Publ. RIMS, Kyoto University, 41 (2005), pp. 897-935.
36. S. P. N0RSETT,AN A-stable modification of the Adams-Bashforth methods, in Conf. on Numerical Solution of Differential Equations (Dundee, Scotland, 1969), Springer, Berlin, 1969, pp. 214-219.
37. A. OSTERMANN, M. THALHAMMER, AND W. WRIGHT, A class of explicit exponential general linear methods, BIT, 46 (2006), pp. 409-431.
38. Y. SACACD, Analysis of some Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal., 29 (1992), pp. 209-228.
39. T. SCHMELZER AND L.N. TREFETHEN, Computing the gamma function using contour integrals and rational approximations, SIAM J. Numer. Anal., 45 (2007), pp. 558-571.
40. L. F. SHAMPINE AND M. W. REICHELT, The MATLAB ODE suite, SIAM J. Sci. Comput., 18 (1997), pp. 1-22.
41. T.-W. TEE, An adaptive rational spectral method for differential equations with rapidly varying solutions, PhD thesis, Oxford University, 2006.
42. L. N. TREFETHEN, Rational Chebyshev approximation on the unit disk, Numer. Math., 37 (1981), pp. 297-320.
43. -, Chebyshev approximation on the unit disk, in Computational aspects of complex analysis, H. Werner et al., ed., D. Reidel Publishing, Dordrecht, 1983, pp. 309-323.
44. L. N. TREFETHEN AND M. H. GUTKNECHT, The Carathéodory-Fejér method for real rational approximation, SIAM J. Numer. Anal., 20 (1983), pp. 420-436.
45. L. N. TREFETHEN, J. WEIDEMAN, AND T. SCHMELZER, Talbot quadratures and rational approximations, BIT, 46 (2006), pp. 653-670.
46. J. A. C. WEIDEMAN AND L. N. TREFETHEN, Parabolic and hyperbolic contours for computing the Bromwich integral. Math. Comp., 76 (2007), pp. 1341-1356.