Abstract
We consider the approximation of trigonometric operator functions that arise in the numerical solution of wave equations by trigonometric integrators. It is well known that Krylov subspace methods for matrix functions without exponential decay show superlinear convergence behavior if the number of steps is larger than the norm of the operator. Thus, Krylov approximations may fail to converge for unbounded operators. In this paper, we propose and analyze a rational Krylov subspace method which converges not only for finite element or finite difference approximations to differential operators but even for abstract, unbounded operators. In contrast to standard Krylov methods, the convergence will be independent of the norm of the operator and thus of its spatial discretization. We will discuss efficient implementations for finite element discretizations and illustrate our analysis with numerical experiments.
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References
L. Bergamaschi and M. Vianello, Efficient computation of the exponential operator for large, sparse, symmetric matrices, Numer. Linear Algebra Appl., 7 (2000), pp. 27–45.
D. Cohen, E. Hairer, and C. Lubich, Long-time analysis of nonlinearly perturbed wave equations via modulated Fourier expansions, Arch. Ration. Mech. Anal., 187 (2008), pp. 341–368.
D. Cohen, E. Hairer, and C. Lubich, Conservation of energy, momentum and actions in numerical discretizations of nonlinear wave equations, to appear in Numer. Math. (2008).
D. Cohen, E. Hairer, and C. Lubich, Energy conservation over long times of numerical discretizations for nonlinear wave equations, to appear in Proceedings of Equadiff07, (2008).
V. A. Dougalis and S. M. Serbin, Remarks on a class of rational approximations to the cosine, BIT, 20 (1980), pp. 204–211.
V. L. Druskin and L. A. Knizhnerman, Extended Krylov subspaces: approximation of the matrix square root and related functions, SIAM J. Matrix Anal. Appl., 19 (1998), pp. 755–771.
M. Eiermann and O. Ernst, A restarted Krylov subspace method for the evaluation of matrix functions, SIAM J. Numer. Anal., 44 (2006), pp. 2481–2504.
A. Frommer and V. Simoncini, Stopping criteria for rational matrix functions of Hermitian and symmetric matrices, SIAM J. Sci. Comput., 30 (2008), pp. 1387–1412.
B. García-Archilla, J. Sanz-Serna, and R. Skeel, Long-time-step methods for oscillatory differential equations, SIAM J. Sci. Comput., 30(3) (1998), pp. 930–963.
G. H. Golub and C. F. van Loan, Matrix Computations, 2nd edn., Johns Hopkins University Press, Baltimore, MD, 1989.
V. Grimm, A note on the Gautschi-type method for oscillatory second-order differential equations, Numer. Math., 102 (2005), pp. 61–66.
V. Grimm, On error bounds for the Gautschi-type exponential integrator applied to oscillatory second-order differential equations, Numer. Math., 100 (2005), pp. 71–89.
V. Grimm, On the use of the Gautschi-type exponential integrator for wave equations, in Numerical Mathematics and Advanced Applications, ENUMATH 2005, A. Bermúdez de Castro, D. Gómez, P. Quintela, and P. Salgado, eds., Springer, Berlin, Heidelberg, 2006, pp. 557–563.
V. Grimm and M. Hochbruck, Error analysis of exponential integrators for oscillatory second-order differential equations, J. Phys. A, Math. Gen., 39 (2006), pp. 5495–5507.
E. Hairer and C. Lubich, Long-time energy conservation of numerical methods for oscillatory differential equations, SIAM J. Numer. Anal., 38 (2000), pp. 414–441.
E. Hairer and C. Lubich, Spectral semi-discretisations of weakly nonlinear wave equations over long times, Found. Comput. Math., 8 (2008), pp. 319–334.
E. Hairer and C. Lubich, Oscillations over long times in numerical Hamiltonian systems, to appear in Proceedings of the HOP programme, Isaac Newton Institute, 2008.
M. Hochbruck and C. Lubich, On Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal., 34 (1997), pp. 1911–1925.
M. Hochbruck and C. Lubich, A Gautschi-type method for oscillatory second-order differential equations, Numer. Math., 83 (1999), pp. 403–426.
L. Lopez and V. Simoncini, Analysis of projection methods for rational function approximation to the matrix exponential, SIAM J. Numer. Anal., 44(2) (2006), pp. 613–635.
I. Moret and P. Novati, RD-rational approximations of the matrix exponential, BIT, 44 (2004), pp. 595–615.
I. Moret and P. Novati, Interpolating functions of matrices on zeros of quasi-kernel polynomials, Numer. Linear Algebra Appl., 11 (2005), pp. 337–353.
O. Nevanlinna, Convergence of Iterations for Linear Equations, Birkhäuser, Basel, 1993.
P. P. Petrushev and V. A. Popov, Rational Approximation of Real Functions, Cambridge University Press, Cambridge, 1987.
M. Popolizio and V. Simoncini, Accerleration techniques for approximating the matrix exponential operator, SIAM J. Matrix Anal. Appl., 30 (2008), pp. 657–683.
A. Ruhe, Rational Krylov sequence methods for eigenvalue computation, Linear Algebra Appl., 58 (1984), pp. 391–405.
Y. Saad, Analysis of some Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal., 29 (1992), pp. 209–228.
J. M. Sanz-Serna, Mollified impulse methods for highly-oscillatory differential equations, SIAM J. Numer. Anal., 46 (2008), pp. 1040–1059.
G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer, New York, Berlin, Heidelberg, 2002.
A. F. Timan, Theory of Approximation of Functions of a Real Variable, Pergamon Press, Oxford, 1963.
J. van den Eshof and M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential, SIAM J. Sci. Comput., 24(4) (2006), pp. 1438–1457.
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AMS subject classification (2000)
65F10, 65L60, 65M60, 65N22
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Grimm, V., Hochbruck, M. Rational approximation to trigonometric operators . Bit Numer Math 48, 215–229 (2008). https://doi.org/10.1007/s10543-008-0185-9
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DOI: https://doi.org/10.1007/s10543-008-0185-9