A novel, simple, robust, and effective modification in the nonlinear weights of the scale-invariant WENO operator is proposed that achieves an optimal order of accuracy with smooth function regardless of the critical point (Cp-property), a scale-invariant with an arbitrary scaling of a function (Si-property), an essentially non-oscillatory approximation of a discontinuous function (ENO-property), and, in some cases, a well-balanced WENO finite difference/volume scheme (WB-property) (up to machine rounding error numerically). The classical WENO-JS/Z/D operators do not satisfy the Si-property intrinsically due to a loss of sub-stencils' adaptivity in the WENO reconstruction of a discontinuous function when scaled by a small scaling factor. By introducing the descaling function, an average of the function values in the weights to build the scale-invariant WENO-JSm/Zm/Dm operators, the operators are independent of both the scaling factor and sensitivity parameter. The Si-property and Cp-property of the WENO operators are validated theoretically and numerically in quadruple-precision with small and large scaling factors and sensitivity parameters. The results show that the WENO-JSm/Zm/Dm operators satisfy the Si-property and the WENO-D/Dm operators satisfy the Cp-property. Furthermore, the ENO-property of the WENO-Zm/Dm schemes is illustrated via several one- and two-dimensional shock-tube problems. In solving the Euler equations under gravitational fields, the well-balanced scale-invariant WENO schemes achieve the WB-property intrinsically without imposing the stringent homogenization condition.

提出了一种对尺度不变 WENO 算子的非线性权重的新颖、简单、鲁棒且有效的修改，无论临界点（Cp 属性）如何，都可以实现具有平滑函数的最佳精度顺序，具有函数的任意缩放（Si-property），不连续函数的基本非振荡近似（ENO-property），以及在某些情况下，平衡良好的 WENO 有限差分/体积方案（WB-property）（向上以数字方式加工舍入误差）。经典的 WENO-JS/Z/D 算子本质上不满足 Si 属性，这是由于在不连续函数的 WENO 重建中子模板的适应性损失，当按小比例因子缩放时。通过引入除垢功能，权重函数值的平均值以构建尺度不变的 WENO-JSm/Zm/Dm 算子，算子独立于比例因子和灵敏度参数。WENO 算子的 Si 属性和 Cp 属性在四倍精度下进行了理论和数值验证，具有大小比例因子和灵敏度参数。结果表明，WENO-JSm/Zm/Dm算子满足Si-property，WENO-D/Dm算子满足Cp-property。此外，通过几个一维和二维激波管问题说明了 WENO-Zm/Dm 方案的 ENO 特性。在求解引力场下的欧拉方程时，平衡良好的尺度不变 WENO 方案在不施加严格的均匀化条件的情况下本质上实现了 WB 特性。