  A new family of high order methods for systems of conservation laws are
introduced: the Compact Approximate Taylor (CAT) methods. These methods are
based on centered (2p + 1)-point stencils where p is an arbitrary integer. We
prove that the order of accuracy is 2p and that CAT methods are an extension of
high-order Lax-Wendroff methods for linear problems. Due to this, they are
linearly L2-stable under a CFL<1 condition. In order to prevent the spurious
oscillations that appear close to discontinuities two shock-capturing
techniques have been considered: a fux-limiter technique (FL-CAT methods) and
WENO reconstruction for the frst time derivative (WENO-CAT methods). We follow
the WENO-Lax Wendroff Approximate Taylor method of Zorio, Baeza and Mullet
(2017) in the second approach. A number of test cases are considered to compare
these methods with other WENO-based schemes: the linear transport equation,
Burgers equation, and the 1D compressible Euler system are considered. Although
CAT methods present an extra computational cost due to the local character,
this extra cost is compensated by the fact that they still give good solutions
with CFL values close to 1.
