Construction of an Improved Third-Order WENO Scheme with a New Smoothness Indicator

The aim of this study is to present an improved third-order weighted essentially non-oscillatory (WENO) scheme for solving hyperbolic conservation laws. We first present a novel smoothness indicator by using discrete differential operator which annihilates exponential polynomials. The new smoothness indicator can vanish to zero in smooth regions with higher rates than the classical methods such that it can distinguish the smooth region and discontinuity more efficiently. The proposed scheme achieves the maximal approximation order without loss of accuracy at critical points. A detailed analysis is provided to verify the third-order accuracy. The proposed scheme attains better resolution in smooth regions, while reducing numerical dissipation significantly near singularities. The advantages are more pronounced in two-dimensional model problems. Some numerical experiments are provided to illustrate the performance of the proposed WENO scheme.