Analysis and improvement of weights in WENO schemes

In order to improve the computational performance of the WENO scheme, a new WENO scheme, namely WENO-E scheme was constructed, which reduces dissipation close to discontinuities. Based on the analysis of the algorithm for weighted factors in the classical WENO scheme (namely WENO-JS) proposed by Jiang and Shu, the new scheme was constructed by introducing indirect smooth indicator. Theoretical analysis shows that the WENO-E scheme can reach the same convergence order of WENO-JS with the same computational efficiency; while it can obtain smaller truncation errors at smooth parts of the solution and higher resolution close to the discontinuities with the same grids than the WENO-JS. Subsequently, compared with the classical WENO scheme, when numerical experiments with the linear transport equation, the nonlinear Burgers equation and the one dimensional Euler system of equations are conducted, the WENO-E scheme achieves better numerical solutions.