Affine-invariant WENO weights and operator

The novel and simple nonlinear affine-invariant weights (Ai-weights) are devised for the Ai-WENO operator to handle the case when the function being reconstructed undergoes an affine transformation (Ai-operator) with a constant scaling and translation (Ai-coefficients) within a global (WENO) stencil. The Ai-weights essentially decouple the inter-dependencies of the Ai-coefficients and sensitivity parameter effectively. For any given sensitivity parameter, the Ai-WENO operator guarantees that the WENO-reconstructed affine-transformed-function remains unchanged as the affine-transformed WENO-reconstructed-function. In other words, the Ai-operator commutes with the nonlinear WENO operator (Ai-property) as proven theoretically and validated numerically. With a small scaling and a non-zero translation, the Ai-WENO scheme with a typical sensitivity parameter satisfies the ENO-property even when the corresponding classical and scale-invariant WENO scheme does not. In solving the shallow water wave equations and the Euler equations under gravitational fields, the characteristic-wise well-balanced Ai-WENO scheme satisfies the well-balanced property intrinsically without imposing the WENO linearization technique. Any Ai-weights-based WENO operator enhances the robustness and reliability of the WENO scheme for solving hyperbolic conservation laws.