Lax-Wendroff methods for linear systems of conservation laws are based on Taylor expansions in time in which the time derivatives are transformed into spatial derivatives using the governing equations. The main difficulty to extend Lax-Wendroff methods to nonlinear problems comes from the transformation of time derivatives into spatial derivatives through the Cauchy-Kovalesky (CK) procedure: this approach may indeed be impractical from the computational point of view because it often requires extended symbolic calculus, ended up into inefficient codes. For this reason, a new family of high-order shock-capturing finite difference numerical methods for systems of conservation laws, based on the numerically CK procedure, are presented. These methods, called Adaptive Compact Approximation Taylor (ACAT) schemes, use centered (2p + 1)-point stencils, where p may take values in {1, 2,..., P}, and a new family of smoothness indicators defined in the stencils.