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We consider hyperbolic systems of conservation laws with relaxation sourceterms leading to a diffusive asymptotic limit under a parabolic scaling. Weintroduce a new class of secondorder in time and space numerical schemes, whichare 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 intime equations, but is inspired from the Lax-Wendroff approach with the properlevel of implicit treatment of the source term. As a result, it yields a verycompact stencil in space and time and we are able to rigorously show that boththe second-order accuracy and the stability conditions are independent of thefast scales in the asymptotic regime, including the study of boundaryconditions. We provide an original derivation of l 2 and l $\infty$ stabilityconditions of the scheme that do not deteriorate the second order accuracywithout relying on a limiter of any type in the linear case, in particular forshock solutions, and extend such results to the nonlinear case, showing thenovelty of the method. The prototype system for the linear case is thehyperbolic heat equation, whereas barotropic Euler equations of gas dynamicswith friction are the one for the nonlinear case. The method is also able toyield very accurate steady solutions in the nonlinear case when the relaxationcoefficient in the source term depends on space. A thorough numericalassessment of the proposed strategy is provided by investigating smoothsolutions, solutions with shocks and solutions leading to a steady state withspace dependent relaxation coefficient.

Second-order uniformly asymptotic-preserving space-time-ImEx schemes for hyperbolic balance laws with stiff relaxation