WENO Reconstructions of Unconditionally Optimal High Order

A modified weighted essentially nonoscillatory (WENO) reconstruction technique preventing accuracy loss near critical points (regardless of their order) of the underlying data is presented. This approach only uses local data from the reconstruction stencil and does not rely on any sort of scaling parameters. The key novel ingredient is a weight design based on a new smoothness indicator, which defines the first WENO reconstruction procedure that never loses accuracy on smooth data, regardless of the presence of critical points of any order, and is therefore addressed as the optimal WENO (OWENO) method. The corresponding weights are nondimensional and scale-independent. The weight designs are supported by theoretical results concerning the accuracy of the smoothness indicators. The method is validated by numerical tests related to algebraic equations, scalar conservation laws, and systems of conservation laws.