A new type of improved third order WENO scheme with finite difference framework

Abstract

In this paper, we present a novel variety of third order finite difference weighted essentially non-oscillatory (WENO) scheme based on WENO-ZQ scheme to approximate the computational solutions of one- and two-dimensional non-linear hyperbolic conservation laws. The new reconstruction process of fluxes is the convex combination of a second degree polynomial and two linear polynomials. We apply a new modified global smoothness indicator into the non-linear weights. We also use the power parameter ‘p’ to balance the accuracy of the proposed scheme. This novel third order WENO scheme applies three points data {xi−1,xi,xi+1} same as WENO-JS3 and WENO-Z3 schemes. In the proposed scheme, the main advantages are that the associated linear weights can be taken as any positive numbers on condition their summation is one. Also, it is easy to be used in multi-dimensional engineering problems. We implement the time discretization by using third order accurate strong stability preserving Runge–Kutta method. The extensive computational experiments of a collection of benchmark test problems are provided to demonstrate high resolution, robustness and absolute truncation errors in L1 and L∞ norms. We observe from the results that the new scheme contains better capability in performance than WENO-PZ3 scheme.