Linear multistep methods for stable differential equations 𝑦̈=𝐴𝑦+𝐵(𝑡)𝑦̇+𝑐(𝑡)

Gekeler, Eckart

doi:10.1090/S0025-5718-1982-0669641-8

Linear multistep methods for stable differential equations $\ddot y=Ay+B(t)\dot y+c(t)$
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Math. Comp. 39 (1982), 481-490 Request permission

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 ${h^2}\left \| A \right \| \leqslant s$ where h is the time step. A priori error bounds are derived which show that the exponential multiplication factor is of the form $\exp \{ {\Gamma _s}|||B|||_{n}(nh)\}$, $|||B|||_{n} = {\max _{0 \leqslant t \leqslant nh}}\left \| {B(t)} \right \|$.

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