A new framework to construct third‐order weighted essentially nonoscillatory weights using weight limiter functions

A new simple and generic framework is proposed to construct nonlinear weights for third‐order weighted essentially nonoscillatory scheme (WENO) reconstructions. It is done by imposing necessary conditions on nonlinear weights to get a nonoscillatory WENO scheme. These conditions give further insight into the required structure of nonlinear weights to design third‐order WENO schemes. This new framework for WENO weights is completely different from the existing prevailing approaches. Several nonlinear weights using different functions of a smoothness parameter (termed as weight limiter functions) are proposed and analyzed. These new weights by construction guarantee for third‐order accurate nonoscillatory scheme. Numerical results for various benchmark test problems are given and compared with WENO‐JS3 wnd WENO‐Z3 scheme. Computational results show that WENO schemes using proposed weights achieves third‐order accuracy for smooth solution and resolves discontinuities without spurious oscillations. In this work a simple and generalized framework is given to construct nonlinear weights for third‐order WENO scheme. Necessary conditions are imposed on weights to get nonoscillatory third‐order reconstruction. Several new nonlinear weights are proposed using different functions of smoothness parameter termed as weight limiter functions. Design principle of proposed new WENO weights is simple and by construction guarantee for exactly third‐order accuracy in smooth solution region including smooth extrema. Resulting schemes also capture discontinuities without any spurious oscillations.