Journal of Computational Physics ( IF 4.1 ) Pub Date : 2021-04-16 , DOI: 10.1016/j.jcp.2021.110358 H. Carrillo , E. Macca , C. Parés , G. Russo , D. Zorío
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 -point stencils, where p may take values in 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, so that they are first order accurate near discontinuities and have order 2p in smooth regions, where is the size of the biggest stencil in which large gradients are not detected. CAT methods, introduced in [3], 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 [22] 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 conservative laws is given and the performance is checked 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:
守恒律系统的阶自适应紧逼近泰勒方法
我们提出了一系列新的守恒律系统的高阶震荡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,使用居中点模具,其中p可以取根据模板中新的平滑度指标系列。这些方法基于稳健的一阶方案和2 p阶的紧凑近似泰勒(CAT)方法的组合,因此它们在不连续点附近是一阶精确的,在光滑区域中具有2 p阶,其中是最大模板的大小,其中未检测到大的渐变。[3]中介绍的CAT方法是Lax-Wendroff方法的非线性问题的扩展,其中,Cauchy-Kovalesky(CK)过程遵循[22]中介绍的策略,该方法允许人们计算时间中的导数。使用高阶中心微分公式结合泰勒展开式进行时间递归的方法。给出了一维和二维守恒律系统的ACAT方法的表达式,并在多个线性和非线性守恒律系统的测试案例(包括用于气体动力学的欧拉方程)中测试了其性能。