Boundary Closures for ESWENO Schemes

The systematic methodology for constructing fourth-order finite domain Energy Stable WENO schemes is developed. We prove that for hyperbolic systems, the finite domain ESWENO scheme is stable in the energy norm for both continuous and discontinuous solutions. The eigenvalues of the finite domain ESWENO dissipation operator are located in the left-half plane. Based on the rigorous truncation error analysis, the new weight functions are developed, which drastically improve the accuracy of the ESWENO scheme and provide excellent shock-capturing capabilities. Numerical experiments show that the new finite domain ESWENO scheme with the new weights outperform the conventional WENO schemes in terms of accuracy.