  Compact Approximate Taylor (CAT) methods for systems of conservation laws
were introduced by Carrillo and Pares in 2019. These methods, based on a
strategy that allows one to extend high-order Lax-Wendroff methods to nonlinear
systems without using the Cauchy-Kovalevskaya procedure, have arbitrary even
order of accuracy 2p and use (2p + 1)-point stencils, where p is an arbitrary
positive integer. More recently in 2021 Carrillo, Macca, Pares, Russo and Zorio
introduced a strategy to get rid of the spurious oscillations close to
discontinuities produced by CAT methods. This strategy led to the so-called
Adaptive CAT (ACAT) methods, in which the order of accuracy, and thus the width
of the stencils, is adapted to the local smoothness of the solution. The goal
of this paper is to extend CAT and ACAT methods to systems of balance laws. To
do this, the source term is written as the derivative of its indefinite
integral that is formally treated as a flux function. The well-balanced
property of the methods is discussed and a variant that allows in principle to
preserve any stationary solution is presented. The resulting methods are then
applied to a number of systems going from a linear scalar conservation law to
the 2D Euler equations with gravity, passing by the Burgers equations with
source term and the 1D shallow water equations: the order and well-balanced
properties are checked in several numerical tests.
