A Fourth-Order Symmetric WENO Scheme with Improved Performance by New Linear and Nonlinear Optimizations

A fourth-order, symmetric, weighted essentially non-oscillatory scheme is proposed with improved linear and nonlinear properties. In order to improve its linear property as the dispersion and dissipation relations, an optimization method is developed by solving a set of equations proposed in the paper. Using this approach, optimization objectives carefully chosen are realized, through which improved linear performance is attained while maintaining stability in the shock-reflection problem. Implementation of a nonlinear scheme is another important issue which usually affects its practical performance in applications. Optimization in this regard is thought to make the scheme work in its linear form effectively and problem-independently when the flow varies relatively smoothly. To fulfill the objectives, four topics were investigated. The first is a new hybrid indicator of smoothness which is based on the concept of total variation, through which good resolution is obtained for resolving structures with short wavelength. The second is a modification specially designed for the most downwind indicator to avoid numerical oscillations. The third is a new transition algorithm to make the scheme work between its linear and nonlinear states by a variable \(p_{wr}\). The new algorithm is so designed as to avoid misjudgment of smooth and oscillatory flow field. The fourth one regards case generality or problem-independence. To address four issues, the so-called rescale functions are presented separately. They are then integrated as one function for easy implementation. Using this integrated function, a fourth-order scheme for solving the Euler/Navier-Stokes equations can work in its optimized linear form in the smooth region and behave nonlinearly at discontinuities to ensure essentially oscillation-free solutions. Numerical examples manifest its capabilities to resolve waves from acoustics to shocks, flows from subsonic to hypersonic speed, and flow patterns from laminar to turbulent.