  The analysis of strong-stability-preserving (SSP) linear multistep methods is
extended to semi-discretized problems for which different terms on the
right-hand side satisfy different forward Euler (or circle) conditions. Optimal
additive and perturbed monotonicity-preserving linear multistep methods are
studied in the context of such problems. Optimal perturbed methods attain
larger monotonicity-preserving step sizes when the different forward Euler
conditions are taken into account. On the other hand, we show that optimal SSP
additive methods achieve a monotonicity-preserving step-size restriction no
better than that of the corresponding non-additive SSP linear multistep
methods.
