Towards time-stepping-free solution of large initial value problems by block Krylov projections
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Thu, 17/05/2012 14:00 |
Dr Mike Botchev (University of Twente) |
Computational Mathematics and Applications  |
Gibson Grd floor SR |
| 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. | |||