Linearly implicit methods for nonlinear evolution equations

Summary.  We construct and analyze combinations of rational implicit and explicit multistep methods for nonlinear evolution equations and extend thus recent results concerning the discretization of nonlinear parabolic equations. The resulting schemes are linearly implicit and include as particular cases implicit–explicit multistep schemes as well as the combination of implicit Runge–Kutta schemes and extrapolation. We establish optimal order error estimates. The abstract results are applied to a third–order evolution equation arising in the modelling of flow in a fluidized bed. We discretize this equation in space by a Petrov–Galerkin method. The resulting fully discrete schemes require solving some linear systems to advance in time with coefficient matrices the same for all time levels. Received October 22, 2001 / Revised version received April 22, 2002 / Published online December 13, 2002 Mathematics Subject Classification (1991): Primary 65M60, 65M12; Secondary 65L06 Correspondence to: G. Akrivis