Third-order Energy Stable WENO scheme

A new third-order Energy Stable Weighted Essentially Non-Oscillatory (ESWENO) finite difference scheme for scalar and vector hyperbolic equations with piecewise continuous initial conditions is developed. The new scheme is proven to be linearly stable in the energy norm for both continuous and discontinuous solutions. In contrast to the existing high-resolution shock-capturing schemes, no assumption that the reconstruction should be total variation bounded (TVB) is explicitly required to prove stability of the new scheme. We also present new weight functions which drastically improve the accuracy of the third-order ESWENO scheme. Based on a truncation error analysis, we show that the ESWENO scheme is design-order accurate for smooth solutions with any number of vanishing derivatives, if its tuning parameters satisfy certain constraints. Numerical results show that the new ESWENO scheme is stable and significantly outperforms the conventional third-order WENO scheme of Jiang and Shu in terms of accuracy, while providing essentially non-oscillatory solutions near strong discontinuities.