Although multistep methods with variable step sizes are widely used in numerical computations, their analysis is still not complete. Because of the non-uniform grid, non-constant coefficients appear in the resulting scheme. Theoretical tools developed for difference equations with constant coefficients are therefore not applicable.
Among the abundance of methods, the backward differentiation formulae (BDF) seem to be of particular interest. In this talk, different stability properties of the two-step BDF will be discussed. It turns out that the ratios of adjacent step sizes need to be suitably bounded in order to prove stability.
New results can be presented for the time discretisation of abstract linear and semilinear parabolic equations with a moderate nonlinearity: Stability as well as optimal smooth-data error estimates can be derived for step size ratios bounded from above by 1.91, cf. also [1].