Second Order Implicit for Local Effects and Explicit for Nonlocal Effects Is Unconditionally Stable

A family of implicit-explicit second order time-stepping methods is analyzed for a system of ODEs motivated by ones arising from spatial discretizations of evolutionary partial differential equations. The methods we consider are implicit in local and stabilizing terms in the underlying PDE and explicit in nonlocal and unstabilizing terms. Unconditional stability and convergence of the numerical scheme are proved by the energy method and by algebraic techniques. This is the first solution to the problem of finding a scheme for (1.1) that is (provably) unconditionally stable and treats the Cu term explicitly. First order schemes were known in [2, 10] and [10] gives a second order scheme stable provided all operators commute.