| Contents |
| We consider one-step discretization methods for nonlinear evolution equations and
address the issues of local error estimation, adaptive time step control,
and global error estimation. Concerning local error estimation we consider a technique which is
generally applicable, on the basis of evaluating the defect of the numerical solution.
We explain the approach and its concrete realization for exponential splitting methods
and partitioned Runge-Kutta methods.
This way of estimating the local error can also be interpreted as a
quadrature approximation of Hermite type for an integral representation of the local error.
The order of this approximation is such that the error estimate is asymptotically correct,
i.e., is deviation from the true error is of a higher order than the error itself.
We present numerical results obtained with adaptive codes based on spectral discretization
in space for a nonlinear Schr\"odinger equation and nonlinear wave equations
as for instance the Klein-Gordon equation. The precise design of the stepsize controller
and its impact on the behavior of the adaptive integrator are discussed
in the pre-asymptotic and asymptotic regimes.
Moreover, we demonstrate how a global error estimate can be computed in parallel at minimal extra
computational cost. |
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