Stability of time-stepping methods for abstract time-dependent parabolic problems

We consider an abstract nonautonomous parabolic problem, u'(t)=A(t)u(t); u(t(0))=u(0); where A(t) : D-t subset of X --> X, t(0) less than or equal to t less than or equal to t(1), is a family of sectorial operators in a Banach space X. This problem is discretized in time by means of either an A(theta)-stable Runge-Kutta or an A(theta)-stable linear multistep method. We prove that the resulting discretization is stable, under some natural assumptions on the relative total variation of A(t) with respect to t. For strongly A(theta)-stable Runge-Kutta methods, stability holds even for variable time step-sizes. Our results are applicable to the analysis of time-dependent parabolic problems in the L-p norms.