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

Abstract

The approximation of y . . = A y + B ( t ) y . + c ( t ) {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 ] [ - s,0] , and that h 2 ‖ A ‖ ⩽ s {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 ⁡ { Γ s | | | B | | | n ( n h ) } \exp \{ {\Gamma _s}|||B|||_{n}(nh)\} , | | | B | | | n = max 0 ⩽ t ⩽ n h ‖ B ( t ) ‖ |||B|||_{n} = {\max _{0 \leqslant t \leqslant nh}}\left \| {B(t)} \right \| .