### Abstract

We propose a time-exact Krylov-subspace-based method for solving linear ODE (ordinary differential equation) systems of the form $y'=-Ay + g(t)$ and $y''=-Ay + g(t)$, where $y(t)$ is the unknown function. The method consists of two stages. The first stage is an accurate piecewise polynomial approximation of the source term $g(t)$, constructed with the help of the truncated SVD (singular value decomposition). The second stage is a special residual-based block Krylov subspace method.
The accuracy of the method is only restricted by the accuracy of the piecewise polynomial approximation and by the error of the block Krylov process. Since both errors can, in principle, be made arbitrarily small, this yields, at some costs, a time-exact method. Numerical experiments are presented to demonstrate efficiency of the new method, as compared to an exponential time integrator with Krylov subspace matrix function evaluations.

Original language | Undefined |
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Place of Publication | Enschede |

Publisher | University of Twente, Department of Applied Mathematics |

Number of pages | 15 |

Publication status | Published - Jan 2012 |

### Publication series

Name | Memorandum / Department of Applied Mathematics |
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Publisher | University of Twente, Department of Applied Mathematics |

No. | 1973 |

ISSN (Print) | 1874-4850 |

ISSN (Electronic) | 1874-4850 |

### Keywords

- MSC-65F60
- MSC-65F30
- MSC-65M22
- MSC-65M20
- EWI-21277
- Unconditionally stable time integration
- Proper orthogonal decomposition
- Truncated SVD
- Exponential residual
- Matrix exponential
- Block Krylov subspace methods
- Exponential time integration
- IR-79354
- MSC-65L05
- METIS-285233

## Cite this

Bochev, M. A. (2012).

*A block Krylov subspace time-exact solution method for linear ODE systems*. (Memorandum / Department of Applied Mathematics; No. 1973). Enschede: University of Twente, Department of Applied Mathematics.