A stabilization of multistep linearly implicit schemes for dissipative systems

We consider numerical integration of dissipative gradient systems. For such systems, a class of special, stable integrators that strictly maintain dissipation is known, but they generally yield expensive fully implicit schemes, and when the system is large, linearization is indispensable for practical efficiency. However, this can in turn destroy the originally expected stability, and so far no effective principle has been formulated for a stable linearization. In this note, we point out that the behavior of the linearized schemes can be understood from a dynamical systems theory viewpoint and propose a simple principle for a stable linearization.