An improved third-order weighted essentially non-oscillatory scheme achieving optimal order near critical points

In this paper, we present an improved third-order weighted essentially non-oscillatory scheme to recover its convergence order at critical points. This scheme is constructed with the way of Taylor expansion of the smoothness indicators for third-order convergence. The recovery of design-order is verified by several linear wave tests. The performance of the proposed scheme is demonstrated on a variety of one-dimensional problems like the Shu-Osher problem, Titarev-Toro problem and the Interacting blast wave problem, As well as two-dimensional problems such as Rayleigh-Taylor instability, Two-dimensional Riemann problem and Double Mach reflection of a strong shock problem. Numerical results indicate that the present scheme provides better results in comparison with the WENO-JS3, WENO-Z3 and WENON3 schemes.