Abstract
We consider the solution of ordinary differential equations (ODEs) using boundary value methods. These methods require the solution of one or more unsymmetric, large and sparse linear systems. The GMRES method with the Strang-type block-circulant preconditioner is proposed for solving these linear systems. We show that if an Ak1,k2-stable boundary value method is used for an m-by-m system of ODEs, then our preconditioners are invertible and all the eigenvalues of the preconditioned systems are 1 except for at most 2m(k1 + k2) outliers. It follows that when the GMRES method is applied to solving the preconditioned systems, the method will converge in at most 2m(k1 + k2) + 1 iterations. Numerical results are given to illustrate the effectiveness of our methods.
| Original language | English |
|---|---|
| Pages (from-to) | 451-462 |
| Number of pages | 12 |
| Journal | IMA Journal of Numerical Analysis |
| Volume | 21 |
| Issue number | 2 |
| DOIs | |
| Publication status | Published - 1 Apr 2001 |
| Externally published | Yes |
Funding
†Research supported in part by Hong Kong Research Grants Council Grant No. CUHK 4207/97P and CUHK DAG Grant No. 2060143. ‡Research supported in part by Hong Kong Research Grants Council Grant No. HKU 7147/99P and UK/HK Joint Research Scheme Grant No. 20009819. §Research supported in part by UM Research Grant No. RG009/98-99S/JXQ/FST.