Affine-Invariant Trigonometric WENO Finite Difference Schemes for Hyperbolic Conservation Laws

Novel affine-invariant trigonometric weighted essentially non-oscillatory (Ai-TWENO) finite difference schemes are proposed to improve the accuracy and robustness of existing TWENO schemes in capturing small-scale discontinuities and highly oscillatory solutions. By systematically developing a fifth-order WENO operator based on trigonometric polynomials, a comprehensive mathematical framework is presented that can be generalized to higher-order schemes. The existence of cell-size dependency of the non-polynomial-based WENO operator is analyzed. A three-step Ai-weighting process–de-averaging, de-scaling, and re-averaging–is introduced to ensure the affine-invariant property for both polynomial and non-polynomial basis functions and the essential non-oscillatory property. Numerical experiments on Euler equations demonstrate the improved performance of Ai-TWENO schemes in resolving small-scale structures and maintaining robustness in the presence of shocks and contact discontinuities. Moreover, the Ai-TWENO schemes for steady-state shallow water equations confirm the effectiveness of achieving a well-balanced property. These findings suggest that the proposed framework is a promising tool for developing high-order shock-capturing schemes with broad applicability in complex flow structures.