New formulations for constructed polynomials and smoothness indicators of weighted essentially non-oscillatory schemes based on the forward-divided differences

Abstract

This paper introduces new formulations for the constructed polynomials and smoothness indicators within the weighted essentially non-oscillatory (WENO) scheme framework using the forward-divided differences approach. This technique transforms the interpolation polynomial and smoothness indicators into linear combinations of coefficients from different orders of forward-divided differences in the WENO polynomial reconstruction procedure. This approach simplifies the computation of higher-order versions of the global interpolation polynomial and smoothness indicators by adding extra terms to the lower-order version if previously calculated. As a result, this method simplifies the original expressions, reduces computational complexity, and improves computational efficiency. The new expressions are examined using an improved adaptive order WENO scheme, denoted as WENO-D5. This scheme computes a simple smoothness indicator for fifth-order linear reconstruction by linearly combining the existing smoothness indicators for third-order linear reconstructions. The WENO-D5 also employs new compact non-linear weights and global smoothness indicator. Several numerical experiments are performed to demonstrate the efficiency and performance of the considered fifth-order schemes. It is found that the forward-divided differences approach has improved the computational efficiency. The analysis also reveals that WENO-D5 consumed lower computational time than the adaptive order WENO [WENO-AO(5,3)] scheme while retaining the advantageous features of adaptive order schemes.