65L06 Multistep, Runge-Kutta and extrapolation methods
Spectral deferred correction methods for solving stiff ODEs are known to
converge rapidly towards the collocation limit solution on equidistant grids,
but show a much less favourable contraction on non-equidistant grids such as
Radau-IIa points. We interprete SDC methods as fixed point iterations for the
collocation system and propose new DIRK-type sweeps for stiff problems based
on purely linear algebraic considerations. Good convergence is recovered also
on non-equidistant grids. The properties of different variants are explored on
a couple of numerical examples.
The aim of this paper is to devise an adaptive timestep control in the contact--stabilized Newmark method (CONTACX) for dynamical contact problems between two viscoelastic bodies in the framework of Signorini's condition. In order to construct a comparative scheme of higher order accuracy, we extend extrapolation techniques. This approach demands a subtle theoretical investigation of an asymptotic error expansion of the contact--stabilized Newmark scheme. On the basis of theoretical insight and numerical observations, we suggest an error estimator and a timestep selection which also cover the presence of contact. Finally, we give a numerical example.