321 research outputs found
Error analysis of trigonometric integrators for semilinear wave equations
An error analysis of trigonometric integrators (or exponential integrators)
applied to spatial semi-discretizations of semilinear wave equations with
periodic boundary conditions in one space dimension is given. In particular,
optimal second-order convergence is shown requiring only that the exact
solution is of finite energy. The analysis is uniform in the spatial
discretization parameter. It covers the impulse method which coincides with the
method of Deuflhard and the mollified impulse method of Garc\'ia-Archilla,
Sanz-Serna & Skeel as well as the trigonometric methods proposed by Hairer &
Lubich and by Grimm & Hochbruck. The analysis can also be used to explain the
convergence behaviour of the St\"ormer-Verlet/leapfrog discretization in time.Comment: 25 page
On the exact discretization of the classical harmonic oscillator equation
We discuss the exact discretization of the classical harmonic oscillator
equation (including the inhomogeneous case and multidimensional
generalizations) with a special stress on the energy integral. We present and
suggest some numerical applications.Comment: 29 page
Numerical Integrators for Highly Oscillatory Hamiltonian Systems: A Review
Numerical methods for oscillatory, multi-scale Hamiltonian systems are reviewed. The construction principles are described, and the algorithmic and analytical distinction between problems with nearly constant high frequencies and with time- or state-dependent frequencies is emphasized. Trigonometric integrators for the first case and adiabatic integrators for the second case are discussed in more detail
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