A New Third-Order Finite Difference WENO Scheme to Improve Convergence Rate at Critical Points

In this work, a new, improved third-order finite difference weighted essentially non-oscillatory scheme is presented for one- and two-dimensional hyperbolic conservation laws and associated problems. The parameter p which is regulate dissipation is introduced in the nonlinear weights in the framework of the conventional WENO-Z scheme, and the higher-order global smoothness indicator is obtained by the idea of Wang [Wang, Y. H., Y. L. Du, K. L. Zhao and L. Yuan. 2020. ‘A Low-dissipation Third-order Weighted Essentially Nonoscillatory Scheme with a New Reference Smoothness Indicator’. International Journal for Numerical Methods in Fluids. 92 (9): 1212–1234.], the sufficient condition of nonlinear weights is proved by using Taylor expansions. Finally, the value range of parameter p is obtained. The proposed scheme is verified to achieve the optimal order near critical points by linear convection equations with different initial values, and the high-resolution characteristic of the present scheme is proved on a variety of one- and two- dimensional standard numerical examples. Numerical results demonstrate that the proposed scheme gives better performance in comparison with the other third-order WENO schemes.