Abstract
When a stiff differential systemy'(t)=f(y, t),y εR n, is solved by an implicit multistep method, then in each time step one has to solve a set of nonlinear equations by a modified Newton iteration. A fixed approximate JacobianW=(1/hλ)I − J, J=∂f/∂y is normally used for many time steps.
The cost of factorizingW and of solving the resulting linear systems can be high. For the case that onlyk ≪n eigenvalues ofJ are stiff, we derive an approximation ofJ which is more easily factorized and still often gives almost the same rate of convergence in the Newton iterations. The approximation is based on a block Schur factorization ofJ, which can be efficiently computed by a modified version of theQR algorithm.
Limited numerical experiences indicate that typically just a few iterations in the blockQR algorithm suffice to give a good approximation toJ. It is shown that for sparse Jacobians a similar scheme can be realized by using a slight modification of orthogonal iteration.
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Björck, Å. A blockQR algorithm for partitioning stiff differential systems. BIT 23, 329–345 (1983). https://doi.org/10.1007/BF01934462
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DOI: https://doi.org/10.1007/BF01934462