Well-balanced adaptive compact approximate Taylor methods for systems of balance laws

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