  This paper focuses on the question of how unconditional stability can be
achieved via multistep ImEx schemes, in practice problems where both the
implicit and explicit terms are allowed to be stiff. For a class of new ImEx
multistep schemes that involve a free parameter, strategies are presented on
how to choose the ImEx splitting and the time stepping parameter, so that
unconditional stability is achieved under the smallest approximation errors.
These strategies are based on recently developed stability concepts, which also
provide novel insights into the limitations of existing semi-implicit backward
differentiation formulas (SBDF). For instance, the new strategies enable higher
order time stepping that is not otherwise possible with SBDF. With specific
applications in nonlinear diffusion problems and incompressible channel flows,
it is demonstrated how the unconditional stability property can be leveraged to
efficiently solve stiff nonlinear or nonlocal problems without the need to
solve nonlinear or nonlocal problems implicitly.
