讲座内容：

For solving hyperbolic conservation laws that arise frequently in computational physics, high order finite volume WENO (FV-WENO) schemes and discontinuous Galerkin (DG) methods are more popular. However, when there are smaller scale structures in the flow field, the classic FV-WENO schemes will produce a significant oscillation at small scale discontinuities (or large gradients). This phenomenon also exists in DG methods that use nonlinear WENO limiters (DG-WENO), and this will disrupt the stability of numerical methods. In this study, a simple, robust, and effective affine-invariant finite volume WENO (FV-Ai-WENO) scheme under nonuniform meshes is devised. We prove and validate that for any given sensitivity parameter, the WENO operator and the affine transformation operator in the present schemes are commutable. In the presence of smaller scale discontinuities, the new operator satisfies the ENO property while the classic WENO operator does not. In addition, we investigate using FV-Ai-WENO methodology as limiters for the DG methods. Several classical examples are used to verify the performance of the FV-Ai-WENO schemes and DG methods with the Ai-WENO limiter (DG-Ai-WENO) in terms of accuracy, robustness, and affine invariance.