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Exponential integrators for large systems of differential equations. (English) Zbl 0912.65058
For 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)

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
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