An order-adaptive compact approximation Taylor method for systems of conservation laws

We present a new family of high-order shock-capturing finite difference numerical methods for systems conservation laws. These methods, called Adaptive Compact Approximation Taylor (ACAT) schemes, use centered $(2p + 1)$-point stencils, where $p$ may take values in $\{1, 2, \dots, P\}$ according to smoothness indicators the stencils. The are based on combination robust first order scheme and Approximate (CAT) $2p$-order, $p=1,2,\dots, P$ so that they accurate near discontinuities have $2p$ smooth regions, +1)$ is size biggest stencil which large gradients not detected. CAT introduced \cite{CP2019}, an extension nonlinear problems Lax-Wendroff Cauchy-Kovalesky (CK) procedure circumvented following strategy \cite{ZBM2017} allows one compute time derivatives recursive way using differentiation formulas combined with expansions time. expression ACAT 1D 2D balance laws given performance tested number test cases several linear laws, including Euler equations gas dynamics.