  We consider hyperbolic systems of conservation laws with relaxation source
terms leading to a diffusive asymptotic limit under a parabolic scaling. We
introduce a new class of secondorder in time and space numerical schemes, which
are uniformly asymptotic preserving schemes. The proposed Implicit-Explicit
(ImEx) approach, does not follow the usual path relying on the method of lines,
either with multi-step methods or Runge-Kutta methods, or semi-discretized in
time equations, but is inspired from the Lax-Wendroff approach with the proper
level of implicit treatment of the source term. As a result, it yields a very
compact stencil in space and time and we are able to rigorously show that both
the second-order accuracy and the stability conditions are independent of the
fast scales in the asymptotic regime, including the study of boundary
conditions. We provide an original derivation of l 2 and l $\infty$ stability
conditions of the scheme that do not deteriorate the second order accuracy
without relying on a limiter of any type in the linear case, in particular for
shock solutions, and extend such results to the nonlinear case, showing the
novelty of the method. The prototype system for the linear case is the
hyperbolic heat equation, whereas barotropic Euler equations of gas dynamics
with friction are the one for the nonlinear case. The method is also able to
yield very accurate steady solutions in the nonlinear case when the relaxation
coefficient in the source term depends on space. A thorough numerical
assessment of the proposed strategy is provided by investigating smooth
solutions, solutions with shocks and solutions leading to a steady state with
space dependent relaxation coefficient.
