  This paper presents a new class of high order linear ImEx multistep schemes
with large regions of unconditional stability. Unconditional stability is a
desirable property of a time stepping scheme, as it allows the choice of time
step solely based on accuracy considerations. Of particular interest are
problems for which both the implicit and explicit parts of the ImEx splitting
are stiff. Such splittings can arise, for example, in variable-coefficient
problems, or the incompressible Navier-Stokes equations. To characterize the
new ImEx schemes, an unconditional stability region is introduced, which plays
a role analogous to that of the stability region in conventional multistep
methods. Moreover, computable quantities (such as a numerical range) are
provided that guarantee an unconditionally stable scheme for a proposed
implicit-explicit matrix splitting. The new approach is illustrated with
several examples. Coefficients of the new schemes up to fifth order are
provided.
