  Many hyperbolic and kinetic equations contain a non-stiff
convection/transport part and a stiff relaxation/collision part (characterized
by the relaxation or mean free time $\varepsilon$). To solve this type of
problems, implicit-explicit (IMEX) multistep methods have been widely used and
their performance is understood well in the non-stiff regime
($\varepsilon=O(1)$) and limiting regime ($\varepsilon\rightarrow 0$). However,
in the intermediate regime (say, $\varepsilon=O(\Delta t)$), uniform accuracy
has been reported numerically without a complete theoretical justification
(except some asymptotic or stability analysis). In this work, we prove the
uniform accuracy -- an optimal {\it a priori} error bound -- of a class of IMEX
multistep methods, IMEX backward differentiation formulas (IMEX-BDF), for
linear hyperbolic systems with stiff relaxation. The proof is based on the
energy estimate with a new multiplier technique. For nonlinear hyperbolic and
kinetic equations, we numerically verify the same property using a series of
examples.
