The goal of this work is to introduce new families of shock-capturing high-order numerical methods for systems of conservation laws that combine Fast WENO (FWENO) and Optimal WENO (OWENO) reconstructions with Approximate Taylor methods for the time discretization. FWENO reconstructions are based on smoothness indicators that require a lower number of calculations than the standard ones. OWENO reconstructions are based on a definition of the nonlinear weights that allows one to unconditionally attain the optimal order of accuracy regardless of the order of critical points. Approximate Taylor methods update the numerical solutions by using a Taylor expansion in time in which, instead of using the Cauchy–Kovalevskaya procedure, the time derivatives are computed by combining spatial and temporal numerical differentiation with Taylor expansions in a recursive way. These new methods are compared between them and against methods based on standard WENO implementations and/or SSP-RK time discretization. A number of test cases are considered ranging from scalar linear 1d problems to nonlinear systems of conservation laws in 2d.

这项工作的目的是为守恒律系统引入新的系列震荡捕捉高阶数值方法，这些方法将快速WENO（FWENO）和最优WENO（OWENO）重构与近似泰勒方法相结合，以实现时间离散。FWENO重建基于平滑度指标，该指标所需的计算量少于标准的计算量。OWENO重建基于非线性权重的定义，该定义使得无论临界点的顺序如何，都可以无条件地获得最佳的精度顺序。近似泰勒方法通过使用时间上的泰勒展开来更新数值解，其中，而不是使用柯西–科瓦列夫斯卡娅过程，时间导数是通过以递归方式将空间和时间数值微分与泰勒展开式相结合来计算的。将这些新方法与基于标准WENO实现和/或SSP-RK时间离散的方法进行比较。考虑了许多测试案例，从标量线性一维问题到二维守恒律非线性系统。