Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-06T10:44:46.513Z Has data issue: false hasContentIssue false
  • English
  • Français

Exponential integrators

Published online by Cambridge University Press:  10 May 2010

Marlis Hochbruck  and
Alexander Ostermann
Marlis Hochbruck
Affiliation:
Karlsruher Institut für Technologie, Institut für Angewandte und Numerische Mathematik, D-76128 Karlsruhe, Germany, E-mail: [email protected]
Alexander Ostermann
Affiliation:
Institut für Mathematik, Universität Innsbruck, A-6020 Innsbruck, Austria, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper we consider the construction, analysis, implementation and application of exponential integrators. The focus will be on two types of stiff problems. The first one is characterized by a Jacobian that possesses eigenvalues with large negative real parts. Parabolic partial differential equations and their spatial discretization are typical examples. The second class consists of highly oscillatory problems with purely imaginary eigenvalues of large modulus. Apart from motivating the construction of exponential integrators for various classes of problems, our main intention in this article is to present the mathematics behind these methods. We will derive error bounds that are independent of stiffness or highest frequencies in the system.

Since the implementation of exponential integrators requires the evaluation of the product of a matrix function with a vector, we will briefly discuss some possible approaches as well. The paper concludes with some applications, in which exponential integrators are used.

Type
Research Article
Information
Acta Numerica , Volume 19 , May 2010 , pp. 209 - 286
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Bergamaschi, L. and Vianello, M. (2000), ‘Efficient computation of the exponential operator for large, sparse, symmetric matrices’, Numer. Linear Algebra Appl. 7, 27–45.3.0.CO;2-4>CrossRefGoogle Scholar
Bergamaschi, L., Caliari, M. and Vianello, M. (2004), The ReLPM exponential integrator for FE discretizations of advection–diffusion equations. In Computational Science: ICCS 2004, Vol. 3039 of Lecture Notes in Computer Science, Springer, pp. 434–442.Google Scholar
Bergamaschi, L., Caliari, M., Mart́nez, A. and Vianello, M. (2005), A parallel exponential integrator for large-scale discretizations of advection-diffusion models. In Recent Advances in Parallel Virtual Machine and Message Passing Interface, Vol. 3666 of Lecture Notes in Computer Science, Springer, pp. 483–492.Google Scholar
Bergamaschi, L., Caliari, M., Martínez, A. and Vianello, M. (2006), Comparing Leja and Krylov approximations of large scale matrix exponentials. In Computational Science: ICCS 2006, Vol. 3994 of Lecture Notes in Computer Science, Springer, pp. 685–692.Google Scholar
Berland, H., Islas, A. L. and Schober, C. M. (2007 a), ‘Conservation of phase space properties using exponential integrators on the cubic Schrödinger equation’, J. Comput. Phys. 225, 284–299.CrossRefGoogle Scholar
Berland, H., Owren, B. and Skaflestad, B. (2006), ‘Solving the nonlinear Schrödinger equation using exponential integrators’, Model. Identif. Control 27, 201–217.CrossRefGoogle Scholar
Berland, H., Skaflestad, B. and Wright, W. M. (2007 b), ‘EXPINT: A MATLAB package for exponential integrators’, ACM Trans. Math. Software 33, 4: 1–4: 17.CrossRefGoogle Scholar
Beylkin, G., Keiser, J. M. and Vozovoi, L. (1998), ‘A new class of time discretization schemes for the solution of nonlinear PDEs’, J. Comput. Phys. 147, 362–387.CrossRefGoogle Scholar
Biesiadecki, J. J. and Skeel, R. D. (1993), ‘Dangers of multiple time step methods’, J. Comput. Phys. 109, 318–328.CrossRefGoogle Scholar
Blanes, S., Casas, F. and Ros, J. (2002), ‘High order optimized geometric integrators for linear differential equations’, BIT 42, 262–284.CrossRefGoogle Scholar
Blanes, S., Casas, F., Oteo, J. and Ros, J. (2009), ‘The Magnus expansion and some of its applications’, Physics Reports 470, 151–238.CrossRefGoogle Scholar
Born, M. and Fock, V. (1928), ‘Beweis des Adiabatensatzes’, Z. Phys. A Hadrons and Nuclei 51, 165–180.Google Scholar
Botchev, M. A., Faragó, I. and Horváth, R. (2009), ‘Application of operator splitting to the Maxwell equations including a source term’, Appl. Numer. Math. 59, 522–541.CrossRefGoogle Scholar
Botchev, M. A., Harutyunyan, D. and van der Vegt, J. J. W. (2006), ‘The Gautschi time stepping scheme for edge finite element discretizations of the Maxwell equations’, J. Comput. Phys. 216, 654–686.CrossRefGoogle Scholar
Budd, C. and Iserles, A. (1999), ‘On the solution of linear differential equations in Lie groups’, Philos. Trans. Royal Soc. A 357, 946–956.Google Scholar
Caliari, M. (2007), ‘Accurate evaluation of divided differences for polynomial interpolation of exponential propagators’, Computing 80, 189–201.CrossRefGoogle Scholar
Caliari, M. and Ostermann, A. (2009), ‘Implementation of exponential Rosenbrocktype integrators’, Appl. Numer. Math. 59, 568–581.CrossRefGoogle Scholar
Caliari, M., Vianello, M. and Bergamaschi, L. (2004), ‘Interpolating discrete advection-diffusion propagators at Leja sequences’, J. Comput. Appl. Math. 172, 79–99.CrossRefGoogle Scholar
Calvo, M. P. and Palencia, C. (2006), ‘A class of explicit multistep exponential integrators for semilinear problems’, Numer. Math. 102, 367–381.CrossRefGoogle Scholar
Celledoni, E., Cohen, D. and Owren, B. (2008), ‘Symmetric exponential integrators with an application to the cubic Schrödinger equation’, Found. Comp. Math. 8, 303–317.CrossRefGoogle Scholar
Celledoni, E., Marthinsen, A. and Owren, B. (2003), ‘Commutator-free Lie group methods’, Future Generation Computer Systems 19, 341–352.CrossRefGoogle Scholar
Certaine, J. (1960), The solution of ordinary differential equations with large time constants. In Mathematical Methods for Digital Computers, Wiley, pp. 128–132.Google Scholar
Cohen, D., Jahnke, T., Lorenz, K. and Lubich, C. (2006), Numerical integrators for highly oscillatory Hamiltonian systems: A review. In Analysis, Modeling and Simulation of Multiscale Problems (Mielke, A., ed.), Springer, pp. 553–576.CrossRefGoogle Scholar
Condon, M., Deaño, A. and Iserles, A. (2009), ‘On highly oscillatory problems arising in electronic engineering’, Mathematical Modelling and Numerical Analysis 43, 785–804.CrossRefGoogle Scholar
Cox, S. M. and Matthews, P. C. (2002), ‘Exponential time differencing for stiff systems’, J. Comput. Phys. 176, 430–455.CrossRefGoogle Scholar
De la Cruz, H., Biscay, R. J., Carbonell, F., Ozaki, T. and Jimenez, J. (2007), ‘A higher order local linearization method for solving ordinary differential equations’, Appl. Math. Comput. 185, 197–212.Google Scholar
Deuflhard, P. (1979), ‘A study of extrapolation methods based on multistep schemes without parasitic solutions’, Z. Angew. Math. Phys. 30, 177–189.CrossRefGoogle Scholar
Deuflhard, P., Hermans, J., Leimkuhler, B., Mark, A., Reich, S. and Skeel, R. D., eds (1999), Algorithms for Macromolecular Modelling, Vol. 4 of Lecture Notes in Computational Science and Engineering, Springer.Google Scholar
Dixon, J. and McKee, S. (1986), ‘Weakly singular discrete Gronwall inequalities’, Z. Angew. Math. Mech. 66, 535–544.CrossRefGoogle Scholar
Druskin, V. L. and Knizhnerman, L. A. (1991), ‘Error bounds in the simple Lanczos procedure for computing functions of symmetric matrices and eigenvalues’, Comput. Math. Math. Phys. 31, 20–30.Google Scholar
Druskin, V. L. and Knizhnerman, L. A. (1994), ‘On application of the Lanczos method to solution of some partial differential equations’, J. Comput. Appl. Math. 50, 255’262.CrossRefGoogle Scholar
Druskin, V. L. and Knizhnerman, L. A. (1995), ‘Krylov subspace approximation of eigenpairs and matrix functions in exact and computer arithmetic’, Numer. Linear Algebra Appl. 2, 205–217.CrossRefGoogle Scholar
Ehle, B. L. and Lawson, J. D. (1975), ‘Generalized Runge-Kutta processes for stiff initial-value problems’, J. Inst. Math. Appl. 16, 11–21.CrossRefGoogle Scholar
Eiermann, M. and Ernst, O. G. (2006), ‘A restarted Krylov subspace method for the evaluation of matrix functions’, SIAM J. Numer. Anal. 44, 2481–2504.CrossRefGoogle Scholar
Emmrich, E. (2005), ‘Stability and error of the variable two-step BDF for semilinear parabolic problems’, J. Appl. Math. Comput. 19, 33–55.CrossRefGoogle Scholar
Engel, K.-J. and Nagel, R. (2000), One-Parameter Semigroups for Linear Evolution Equations, Vol. 194 of Graduate Texts in Mathematics, Springer.Google Scholar
Friedli, A. (1978), Verallgemeinerte Runge-Kutta Verfahren zur Lösung steifer Differentialgleichungssysteme. In Numerical Treatment of Differential Equations (Burlirsch, R., Grigorieff, R. and Schröder, J., eds), Vol. 631 of Lecture Notes in Mathematics, Springer, pp. 35–50.CrossRefGoogle Scholar
Friesner, R. A., Tuckerman, L. S., Dornblaser, B. C. and Russo, T. V. (1989), ‘A method for exponential propagation of large systems of stiff nonlinear differential equations’, J. Sci. Comput. 4, 327–354.CrossRefGoogle Scholar
Frommer, A. and Simoncini, V. (2008), Matrix functions. In Model Order Reduction: Theory, Research Aspects and Applications (Schilders, W. H. and van der Vorst, H. A., eds), Mathematics in Industry, Springer, pp. 275–304.CrossRefGoogle Scholar
Gallopoulos, E. and Saad, Y. (1992), ‘Efficient solution of parabolic equations by Krylov approximation methods’, SIAM J. Sci. Statist. Comput. 13, 1236–1264.CrossRefGoogle Scholar
García-Archilla, B., Sanz-Serna, J. M. and Skeel, R. D. (1998), ‘Long-time-step methods for oscillatory differential equations’, SIAM J. Sci. Comput. 20, 930–963.CrossRefGoogle Scholar
Gautschi, W. (1961), ‘Numerical integration of ordinary differential equations based on trigonometric polynomials’, Numer. Math. 3, 381–397.CrossRefGoogle Scholar
Gondal, M. A. (2010), ‘Exponential Rosenbrock integrators for option pricing’, J. Comput. Appl. Math. 234, 1153–1160.CrossRefGoogle Scholar
González, C. and Thalhammer, M. (2007), ‘A second-order Magnus-type integrator for quasi-linear parabolic problems’, Math. Comp. 76, 205–231.CrossRefGoogle Scholar
González, C., Ostermann, A. and Thalhammer, M. (2006), ‘A second-order Magnustype integrator for nonautonomous parabolic problems’, J. Comput. Appl. Math. 189, 142–156.CrossRefGoogle Scholar
Grimm, V. (2002), Exponentielle Integratoren als Lange-Zeitschritt-Verfahren f̈r oszillatorische Differentialgleichungen zweiter Ordnung. Dissertation, Heinrich-Heine Universität Düsseldorf.Google Scholar
Grimm, V. (2005 a), ‘A note on the Gautschi-type method for oscillatory secondorder differential equations’, Numer. Math. 102, 61–66.CrossRefGoogle Scholar
Grimm, V. (2005 b), ‘On error bounds for the Gautschi-type exponential integrator applied to oscillatory second-order differential equations’, Numer. Math. 100, 71–89.CrossRefGoogle Scholar
Grimm, V. and Hochbruck, M. (2006), ‘Error analysis of exponential integrators for oscillatory second-order differential equations’, J. Phys. A 39, 5495–5507.CrossRefGoogle Scholar
Grimm, V. and Hochbruck, M. (2008), ‘Rational approximation to trigonometric operators’, BIT 48, 215–229.CrossRefGoogle Scholar
Grubmüller, H., Heller, H., Windemuth, A. and Schulten, K. (1991), ‘Generalized Verlet algorithm for efficient molecular dynamics simulations with long-range interactions’, Molecular Simulation 6, 121–142.CrossRefGoogle Scholar
Hairer, E. and Lubich, C. (2000), ‘Long-time energy conservation of numerical methods for oscillatory differential equations’, SIAM J. Numer. Anal. 38, 414–441.CrossRefGoogle Scholar
Hairer, E. and Lubich, C. (2009), Oscillations over long times in numerical Hamiltonian systems. In Highly Oscillatory Problems (Engquist, E. H. B., Fokas, A. and Iserles, A., eds), Vol. 366 of London Mathematical Society Lecture Notes, Cambridge University Press, pp. 1–24.Google Scholar
Hairer, E. and Wanner, G. (1996), Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Vol. 14 of Springer Series in Computational Mathematics, 2nd edn, Springer.CrossRefGoogle Scholar
Hairer, E., Lubich, C. and Wanner, G. (2006), Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations, Vol. 31 of Springer Series in Computational Mathematics, Springer.Google Scholar
Hairer, E., Nørsett, S. P. and Wanner, G. (1993), Solving Ordinary Differential Equations I: Nonstiff Problems, Vol. 8 of Springer Series in Computational Mathematics, 2nd edn, Springer.Google Scholar
Henry, D. (1981), Geometric Theory of Semilinear Parabolic Equations, Vol. 840 of Lecture Notes in Mathematics, Springer.CrossRefGoogle Scholar
Hersch, J. (1958), ‘Contribution à la méthode des équations aux différences’, Z. Angew. Math. Phys. 9, 129–180.CrossRefGoogle Scholar
Higham, N. J. (2008), Functions of Matrices: Theory and Computation, SIAM.CrossRefGoogle Scholar
Higham, N. J. and Al-Mohy, A. H. (2010), Computing matrix functions. In Acta Numerica, Vol. 19, Cambridge University Press, pp. 159–208.Google Scholar
Hochbruck, M. and Lubich, C. (1997), ‘;On Krylov subspace approximations to the matrix exponential operator’, SIAM J. Numer. Anal. 34, 1911–1925.CrossRefGoogle Scholar
Hochbruck, M. and Lubich, C. (1999 a), A bunch of time integrators for quantum/classical molecular dynamics. In Deuflhard et al. (1999), pp. 421–432.Google Scholar
Hochbruck, M. and Lubich, C. (1999 b), ‘Exponential integrators for quantumclassical molecular dynamics’, BIT 39, 620–645.CrossRefGoogle Scholar
Hochbruck, M. and Lubich, C. (1999 c), ‘A Gautschi-type method for oscillatory second-order differential equations’, Numer. Math. 83, 403–426.CrossRefGoogle Scholar
Hochbruck, M. and Lubich, C. (2003), ‘On Magnus integrators for time-dependent Schrödinger equations’, SIAM J. Numer. Anal. 41, 945–963.CrossRefGoogle Scholar
Hochbruck, M. and Ostermann, A. (2005 a), ‘Explicit exponential Runge-Kutta methods for semilinear parabolic problems’, SIAM J. Numer. Anal. 43, 1069–1090.CrossRefGoogle Scholar
Hochbruck, M. and Ostermann, A. (2005 b), ‘Exponential Runge-Kutta methods for parabolic problems’, Appl. Numer. Math. 53, 323–339.CrossRefGoogle Scholar
Hochbruck, M. and Ostermann, A. (2006), ‘Explicit integrators of Rosenbrock-type’, Oberwolfach Reports 3, 1107–1110.Google Scholar
Hochbruck, M. and van den Eshof, J. (2006), ‘Preconditioning Lanczos approximations to the matrix exponential’, SIAM J. Sci. Comput. 27, 1438–1457.Google Scholar
Hochbruck, M., Hönig, M. and Ostermann, A. (2009 a), ‘A convergence analysis of the exponential Euler iteration for nonlinear ill-posed problems’, Inverse Problems 25, 075009.CrossRefGoogle Scholar
Hochbruck, M., Hönig, M. and Ostermann, A. (2009 b), ‘Regularization of nonlinear ill-posed problems by exponential integrators’, Mathematical Modelling and Numerical Analysis 43, 709–720.CrossRefGoogle Scholar
Hochbruck, M., Lubich, C. and Selhofer, H. (1998), ‘Exponential integrators for large systems of differential equations’, SIAM J. Sci. Comput. 19, 1552–1574.CrossRefGoogle Scholar
Hochbruck, M., Ostermann, A. and Schweitzer, J. (2009 c), ‘Exponential Rosenbrock-type methods’, SIAM J. Numer. Anal. 47, 786–803.CrossRefGoogle Scholar
Hundsdorfer, W. and Verwer, J. G. (2007), Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Vol. 33 of Springer Series in Computational Mathematics, corrected 2nd printing, Springer.Google Scholar
't Hout, K. J. and Weideman, J. A. C. (2009), Appraisal of a contour integral method for the Black-Scholes and Heston equations. Technical report, Department of Mathematics and Computer Science, University of Antwerp.Google Scholar
Iserles, A. (2002 a), ‘On the global error of discretization methods for highlyoscillatory ordinary differential equations’, BIT 42, 561–599.CrossRefGoogle Scholar
Iserles, A. (2002 b), ‘Think globally, act locally: Solving highly-oscillatory ordinary differential equations’, Appl. Numer. Math. 43, 145–160.CrossRefGoogle Scholar
Iserles, A. and Nørsett, S. P. (1999), ‘On the solution of linear differential equations in Lie groups’, Philos. Trans. Royal Soc. A 357, 983–1019.CrossRefGoogle Scholar
Iserles, A. and Nørsett, S. (2004), ‘On quadrature methods for highly oscillatory integrals and their implementation’, BIT 44, 755–772.CrossRefGoogle Scholar
Iserles, A., Munthe-Kaas, H. Z., Nørsett, S. P. and Zanna, A. (2000), Lie-group methods. In Acta Numerica, Vol. 9, Cambridge University Press, pp. 215–365.Google Scholar
Izaguirre, J. A., Reich, S. and Skeel, R. D. (1999), ‘Longer time steps for molecular dynamics’, J. Chem. Phys. 110, 9853–9864.CrossRefGoogle Scholar
Jahnke, T. (2003), Numerische Verfahren für fast adiabatische Quantendynamik. PhD thesis, Eberhard-Karls-Universität, Tübingen, Germany.Google Scholar
Jahnke, T. (2004), ‘Long-time-step integrators for almost-adiabatic quantum dynamics’, SIAM J. Sci. Comput. 25, 2145–2164.CrossRefGoogle Scholar
Jahnke, T. and Lubich, C. (2000), ‘Error bounds for exponential operator splittings’, BIT 40, 735–744.CrossRefGoogle Scholar
Jahnke, T. and Lubich, C. (2003), ‘Numerical integrators for quantum dynamics close to the adiabatic limit’, Numer. Math. 94, 289–314.CrossRefGoogle Scholar
Karle, C., Schweitzer, J., Hochbruck, M. and Spatschek, K.-H. (2008), ‘A parallel implementation of a two-dimensional fluid laser-plasma integrator for stratified plasma-vacuum systems’, J. Comput. Phys. 227, 7701–7719.CrossRefGoogle Scholar
Karle, C., Schweitzer, J., Hochbruck, M., Laedke, E. W. and Spatschek, K.-H. (2006), ‘Numerical solution of nonlinear wave equations in stratified dispersive media’, J. Comput. Phys. 216, 138–152.CrossRefGoogle Scholar
Kassam, A.-K. and Trefethen, L. N. (2005), ‘Fourth-order time-stepping for stiff PDEs’, SIAM J. Sci. Comput. 26, 1214–1233.CrossRefGoogle Scholar
Klein, C. (2008), ‘Fourth order time-stepping for low dispersion Korteweg-de Vries and nonlinear Schrödinger equations’, Electron. Trans. Numer. Anal. 29, 116–135.Google Scholar
Knizhnerman, L. A. (1991), ‘Computation of functions of unsymmetric matrices by means of Arnoldi's method’, J. Comput. Math. Math. Phys. 31, 5–16 (in the Russian issue).Google Scholar
Knizhnerman, L. A. (1992), ‘Error bounds in Arnoldi's method: The case of a normal matrix’, Comput. Math. Math. Phys. 32, 1199–1211.Google Scholar
Knizhnerman, L. and Simoncini, V. (2009), ‘A new investigation of the extended Krylov subspace method for matrix function evaluations’, Numer. Linear Algebra Appl. In press.Google Scholar
Kosloff, R. (1994), ‘Propagation methods for quantum molecular dynamics’, Annu. Rev. Phys. Chem. 45, 145–178.CrossRefGoogle Scholar
Krogstad, S. (2005), ‘Generalized integrating factor methods for stiff PDEs’, J. Comput. Phys. 203, 72–88.CrossRefGoogle Scholar
Lambert, J. D. and Sigurdsson, S. T. (1972), ‘Multistep methods with variable matrix coefficients’, SIAM J. Numer. Anal. 9, 715–733.CrossRefGoogle Scholar
Lawson, J. D. (1967), ‘Generalized Runge-Kutta processes for stable systems with large Lipschitz constants’, SIAM J. Numer. Anal. 4, 372–380.CrossRefGoogle Scholar
Lee, H. and Sheen, D. (2009), ‘Laplace transformation method for the Black-Scholes equations’, Int. J. Numer. Anal. Model. 6, 642–659.Google Scholar
López-Fernández, M. (2009), On the implementation of exponential methods for semilinear parabolic equations. Technical report, Instituto de Ciencias Matemáaticas, Madrid, Spain.Google Scholar
López-Fernández, M., Palencia, C. and Schädle, A. (2006), ‘A spectral order method for inverting sectorial Laplace transforms’, SIAM J. Numer. Anal. 44, 1332–1350.CrossRefGoogle Scholar
Lorenz, K., Jahnke, T. and Lubich, C. (2005), ‘Adiabatic integrators for highly oscillatory second-order linear differential equations with time-varying eigendecomposition’, BIT 45, 91–115.CrossRefGoogle Scholar
Lubich, C. (2008), From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS).CrossRefGoogle Scholar
Lunardi, A. (1995), Analytic Semigroups and Optimal Regularity in Parabolic Problems, Vol. 16 of Progress in Nonlinear Differential Equations and their Applications, Birkhäuser.Google Scholar
Ma, Q. and Izaguirre, J. A. (2003 a), Long time step molecular dynamics using targeted Langevin stabilization. In SAC '03: Proc. 2003 ACM Symposium on Applied Computing, ACM, New York, pp. 178–182.CrossRefGoogle Scholar
Ma, Q. and Izaguirre, J. A. (2003 b), ‘Targeted mollified impulse: A multiscale stochastic integrator for long molecular dynamics simulations’, Multiscale Model. Simul. 2, 1–21.CrossRefGoogle Scholar
Ma, Q., Izaguirre, J. A. and Skeel, R. D. (2003), ‘Verlet-I/R-RESPA/impulse is limited by nonlinear instabilities’, SIAM J. Sci. Comput. 24, 1951–1973.CrossRefGoogle Scholar
Magnus, W. (1954), ‘On the exponential solution of differential equations for a linear operator’, Comm. Pure Appl. Math. 7, 649–673.CrossRefGoogle Scholar
Martínez, A., Bergamaschi, L., Caliari, M. and Vianello, M. (2009), ‘A massively parallel exponential integrator for advection-diffusion models’, J. Comput. Appl. Math. 231, 82–91.CrossRefGoogle Scholar
McLachlan, R. I. and Quispel, G. R. W. (2002), Splitting methods. In Acta Numerica, Vol. 11, Cambridge University Press, pp. 341–434.Google Scholar
Minchev, B.V. and Wright, W. (2005), A review of exponential integrators for first order semi-linear problems. Preprint, NTNU Trondheim.Google Scholar
Moan, P. C. and Niesen, J. (2008), ‘Convergence of the Magnus series’, Found. Comput. Math. 8, 291–301.CrossRefGoogle Scholar
Moler, C. and Van Loan, C. (2003), ‘Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later’, SIAM Rev. 45, 3–49.CrossRefGoogle Scholar
Moret, I. and Novati, P. (2001), ‘An interpolatory approximation of the matrix exponential based on Faber polynomials’, J. Comput. Appl. Math. 131, 361–380.CrossRefGoogle Scholar
Moret, I. and Novati, P. (2004), ‘RD-rational approximations of the matrix exponential’, BIT 44, 595–615.CrossRefGoogle Scholar
Nauts, A. and Wyatt, R. E. (1983), ‘New approach to many-state quantum dynamics: The recursive-residue-generation method’, Phys. Rev. Lett. 51, 2238–2241.CrossRefGoogle Scholar
Nettesheim, P. and Schütte, C. (1999), Numerical integrators for quantum-classical molecular dynamics. In Deuflhard et al. (1999), pp. 396–411.Google Scholar
Nettesheim, P., Bornemann, F. A., Schmidt, B. and Schütte, C. (1996), ‘An explicit and symplectic integrator for quantum-classical molecular dynamics’, Chem. Phys. Lett. 256, 581–588.CrossRefGoogle Scholar
Niegemann, J., Tkeshelashvili, L. and Busch, K. (2007), ‘Higher-order time-domain simulations of Maxwell's equations using Krylov-subspace methods’, J. Comput. Theor. Nanoscience 4, 627–634.CrossRefGoogle Scholar
Niehoff, J. (2007), Projektionsverfahren zur Approximation von Matrixfunktionen mit Anwendungen auf die Implementierung exponentieller Integratoren. Dissertation, Heinrich-Heine Universität Düsseldorf, Mathematisches Institut.Google Scholar
Niesen, J. and Wright, W. (2009), A Krylov subspace algorithm for evaluating the φ-functions appearing in exponential integrators. Preprint: arXiv:0907.4631vl.Google Scholar
Nørsett, S. P. (1969), An A-stable modification of the Adams-Bashforth methods. In Conference on the Numerical Solution of Differential Equations, Vol. 109 of Lecture Notes in Mathematics, Springer, pp. 214–219.CrossRefGoogle Scholar
Ostermann, A., Thalhammer, M. and Wright, W. M. (2006), ‘A class of explicit exponential general linear methods’, BIT 46, 409–431.CrossRefGoogle Scholar
Park, T. J. and Light, J. C. (1986), ‘Unitary quantum time evolution by iterative Lanczos reduction’, J. Chem. Phys. 85, 5870–5876.CrossRefGoogle Scholar
Pazy, A. (1992), Semigroups of Linear Operators and Applications to Partial Differential Equations, Vol. 44 of Applied Mathematical Sciences, corrected 2nd printing, Springer.Google Scholar
Peskin, U., Kosloff, R. and Moiseyev, N. (1994), ‘The solution of the time dependent Schrödinger equation by the (t, t′) method: The use of global polynomial propagators for time dependent Hamiltonians’, J. Chem. Phys. 100, 8849–8855.CrossRefGoogle Scholar
Pope, D. A. (1963), ‘An exponential method of numerical integration of ordinary differential equations’, Comm. Assoc. Comput. Mach. 6, 491–493.Google Scholar
Pototschnig, M., Niegemann, J., Tkeshelashvili, L. and Busch, K. (2009), ‘Timedomain simulations of nonlinear Maxwell equations using operator-exponential methods’, IEEE Trans. Antenn. Propag. 57, 475–483.CrossRefGoogle Scholar
Rambeerich, N., Tangman, D. Y., Gopaul, A. and Bhuruth, M. (2009), ‘Exponential time integration for fast finite element solutions of some financial engineering problems’, J. Comput. Appl. Math. 224, 668–678.CrossRefGoogle Scholar
Ramos, J. I. and García-López, C. M. (1997), ‘Piecewise-linearized methods for initial-value problems’, Appl. Math. Comput. 82, 273–302.Google Scholar
Saad, Y. (1992), ‘Analysis of some Krylov subspace approximations to the matrix exponential operator’, SIAM J. Numer. Anal. 29, 209–228.CrossRefGoogle Scholar
Saad, Y. (1994), SPARSKIT: A basic tool kit for sparse matrix computations, version 2. Technical report, Department of Computer Science and Engineering, University of Minnesota.Google Scholar
Saad, Y. (2003), Iterative Methods for Sparse Linear Systems, 2nd edn, SIAM.CrossRefGoogle Scholar
Schädle, A., López-Fernández, M. and Lubich, C. (2006), ‘Fast and oblivious convolution quadrature’, SIAM J. Sci. Comput. 28, 421–438.CrossRefGoogle Scholar
Schlick, T., Skeel, R. D., Brunger, A. T., Kalé, L. V., Board, J. A., Hermans, J. and Schulten, K. (1999), ‘Algorithmic challenges in computational molecular biophysics’, J. Comput. Phys. 151, 9–48.CrossRefGoogle Scholar
Schmelzer, T. and Trefethen, L. N. (2007), ‘Evaluating matrix functions for exponential integrators via Carathéodory-Fejér approximation and contour integrals’, Electron. Trans. Numer. Anal. 29, 1–18.Google Scholar
Schütte, C. and Bornemann, F. A. (1999), ‘On the singular limit of the quantumclassical molecular dynamics model’, SIAM J. Appl. Math. 59, 1208–1224.CrossRefGoogle Scholar
Sidje, R. B. (1998), ‘Expokit: A software package for computing matrix exponentials’, ACM Trans. Math. Software 24, 130–156.CrossRefGoogle Scholar
Stewart, D.E. and Leyk, T. S. (1996), ‘Error estimates for Krylov subspace approximations of matrix exponentials’, J. Comput. Appl. Math. 72, 359–369.CrossRefGoogle Scholar
Strehmel, K. and Weiner, R. (1987), ‘B-convergence results for linearly implicit one step methods’, BIT 27, 264–281.CrossRefGoogle Scholar
Strehmel, K. and Weiner, R. (1992), Linear-implizite Runge-Kutta Methoden und ihre Anwendungen, Vol. 127 of Teubner-Texte zur Mathematik, Teubner.CrossRefGoogle Scholar
Tal-Ezer, H. and Kosloff, R. (1984), ‘An accurate and efficient scheme for propagating the time-dependent Schrödinger equation’, J. Chem. Phys. 81, 3967–3971.CrossRefGoogle Scholar
Tal-Ezer, H., Kosloff, R. and Cerjan, C. (1992), ‘Low-order polynomial approximation of propagators for the time-dependent Schrödinger equation’, J. Comput. Phys. 100, 179–187.CrossRefGoogle Scholar
Tangman, D. Y., Gopaul, A. and Bhuruth, M. (2008), ‘Exponential time integration and Chebychev discretisation schemes for fast pricing of options’, Appl. Numer. Math. 58, 1309–1319.CrossRefGoogle Scholar
Tautenhahn, U. (1994), ‘On the asymptotical regularization of nonlinear ill-posed problems’, Inverse Problems 10, 1405–1418.CrossRefGoogle Scholar
Teufel, S. (2003), Adiabatic Perturbation Theory in Quantum Dynamics, Vol. 1821 of Lecture Notes in Mathematics, Springer.Google Scholar
Thalhammer, M. (2006), ‘A fourth-order commutator-free exponential integrator for nonautonomous differential equations’, SIAM J. Numer. Anal. 44, 851–864.CrossRefGoogle Scholar
Tokman, M. (2006), ‘Efficient integration of large stiff systems of ODEs with exponential propagation iterative (EPI) methods’, J. Comput. Phys. 213, 748–776.CrossRefGoogle Scholar
Tokman, M. and Bellan, P. M. (2002), ‘Three-dimensional model of the structure and evolution of coronal mass ejections’, Astrophys. J. 567, 1202–1210.CrossRefGoogle Scholar
Trefethen, L. N., Weideman, J. A. C. and Schmelzer, T. (2006), ‘Talbot quadratures and rational approximations’, BIT 46, 653–670.CrossRefGoogle Scholar
Tuckerman, M., Berne, B. J. and Martyna, G. J. (1992), ‘Reversible multiple time scale molecular dynamics’, J. Chem. Phys. 97, 1990–2001.CrossRefGoogle Scholar
Verwer, J. (1976), ‘On generalized linear multistep methods with zero-parasitic roots and an adaptive principal root’, Numer. Math. 27, 143–155.CrossRefGoogle Scholar
Verwer, J. G. and Botchev, M. A. (2009), ‘Unconditionally stable integration of Maxwell's equations’, Linear Algebra Appl. 431, 300–317.CrossRefGoogle Scholar