Exponential time integrators are a powerful tool for numerical solution
of time dependent problems. The actions of the matrix functions on vectors,
necessary for exponential integrators, can be efficiently computed by
different elegant numerical techniques, such as Krylov subspaces.
Unfortunately, in some situations the additional work required by
exponential integrators per time step is not paid off because the time step
can not be increased too much due to the accuracy restrictions.
To get around this problem, we propose the so-called time-stepping-free
approach. This approach works for linear ordinary differential equation (ODE)
systems where the time dependent part forms a small-dimensional subspace.
In this case the time dependence can be projected out by block Krylov
methods onto the small, projected ODE system. Thus, there is just one
block Krylov subspace involved and there are no time steps. We refer to
this method as EBK, exponential block Krylov method. The accuracy of EBK
is determined by the Krylov subspace error and the solution accuracy in the
projected ODE system. EBK works for well for linear systems, its extension
to nonlinear problems is an open problem and we discuss possible ways for
such an extension.