Improved third‐order weighted essentially nonoscillatory scheme

Summary A new third‐order WENO scheme is proposed to achieve the desired order of convergence at the critical points for scalar hyperbolic equations. A new reference smoothness indicator is introduced, which satisfies the sufficient condition on the weights for the third‐order convergence. Following the truncation error analysis, we have shown that the proposed scheme achieves the desired order accurate for smooth solutions with arbitrary number of vanishing derivatives if the parameter ε satisfies certain conditions. We have made a comparative study of the proposed scheme with the existing schemes such as WENO‐JS, WENO‐Z, and WENO‐N3 through different numerical examples. The result shows that the proposed scheme (WENO‐MN3) achieves better performance than these schemes. In this paper, we have analyzed the third‐order WENO‐N3 scheme and observed that the scheme WENO‐N3 does not satisfies the desired order of convergence at critical points. To achieve the required order of convergence in smooth regions with arbitrary number of vanishing derivatives, we have imposed certain conditions on the parameter ε, which is involved in the definition of nonlinear weights, so that the resulted scheme achieves the better performance in comparison to WENO‐N3, WENO‐JS and WENO‐Z schemes and this can be seen in the following figure.