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

中文翻译：

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守恒定律系统的阶次自适应紧逼逼近泰勒方法

我们提出了一系列新的用于守恒定律系统的高阶冲击捕获有限差分数值方法。这些方法称为自适应紧凑近似泰勒 (ACAT) 方案，使用居中的 $(2p + 1)$ 点模板，其中 $p$ 可能取 $\{1, 2, \dots, P\}$ 中的值，根据模板中的一系列新的平滑度指标。这些方法基于稳健的一阶方案和 $2p$-order, $p=1,2,\dots, P$ 阶的紧凑近似泰勒 (CAT) 方法的组合，因此它们在近不连续性并且在平滑区域中的顺序为 $2p$，其中 $(2p +1)$ 是未检测到大梯度的最大模板的大小。CAT 方法，在 \cite{CP2019} 中介绍，是对 Lax-Wendroff 方法非线性问题的扩展，其中 Cauchy-Kovalesky (CK) 过程遵循 \cite{ZBM2017} 中引入的策略被规避，该策略允许使用高阶中心以递归方式计算时间导数微分公式与时间上的泰勒展开相结合。给出了用于一维和二维平衡定律系统的 ACAT 方法的表达式，并在多个测试用例中测试了几个线性和非线性守恒定律系统的性能，包括气体动力学的欧拉方程。