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Linear multistep methods for stable differential equations $ \ddot y=Ay+B(t)\dot y+c(t)$

Author: Eckart Gekeler
Journal: Math. Comp. 39 (1982), 481-490
MSC: Primary 65L05
MathSciNet review: 669641
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Abstract: The approximation of $ {y^{..}} = Ay + B(t){y^.} + c(t)$ by linear multistep methods is studied. It is supposed that the matrix A is real symmetric and negative semidefinite, that the multistep method has an interval of absolute stability $ [ - s,0]$, and that $ {h^2}\left\Vert A \right\Vert \leqslant s$ where h is the time step. A priori error bounds are derived which show that the exponential multiplication factor is of the form $ \exp \{ {\Gamma _s}\vert\vert\vert B\vert\vert\vert _{n}(nh)\} $, $ \vert\vert\vert B\vert\vert\vert _{n} = {\max _{0 \leqslant t \leqslant nh}}\left\Vert {B(t)} \right\Vert$.

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Article copyright: © Copyright 1982 American Mathematical Society