Exponential integrators for large systems of differential equations.

*(English)*Zbl 0912.65058For the numerical solution of high-dimensional (mildly stiff, stiff, and highly oscillatory) differential equations new integration methods are proposed. Their common feature is that they use matrix-vector multiplications with the exponential of the Jacobian. Rosenbrock-type exponential methods and the computation of the matrix exponential by Krylov subspace methods are studied in much detail. Fourth-order methods are derived which solve exactly linear problems with constant coefficients, and which are optimized with respect to the number of necessary function evaluations and with respect to an application of Krylov subspace techniques. A new code exp4 (available in Matlab and in C) is presented. Numerical experiments on reaction-diffusion equations and on a Schrödinger equation with time-dependent potential show the efficiency of the new methods.

Reviewer: E.Hairer (Genève)

##### MSC:

65L05 | Numerical methods for initial value problems |

34A30 | Linear ordinary differential equations and systems, general |

34A34 | Nonlinear ordinary differential equations and systems, general theory |

34E13 | Multiple scale methods for ordinary differential equations |

65F10 | Iterative numerical methods for linear systems |

65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |

34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |

35K57 | Reaction-diffusion equations |