  When applying the classical multistep schemes for solving differential
equations, one often faces the dilemma that smaller time steps are needed with
higher-order schemes, making it impractical to use high-order schemes for stiff
problems. We construct in this paper a new class of BDF and implicit-explicit
(IMEX) schemes for parabolic type equations based on the Taylor expansions at
time $t^{n+\beta}$ with $\beta > 1$ being a tunable parameter. These new
schemes, with a suitable $\beta$, allow larger time steps at higher-order for
stiff problems than that is allowed with a usual higher-order scheme. For
parabolic type equations, we identify an explicit uniform multiplier for the
new second- to fourth-order schemes, and conduct rigorously stability and error
analysis by using the energy argument. We also present ample numerical examples
to validate our findings.
