Optimized Weighted Essentially Nonoscillatory Schemes for Linear Waves with Discontinuity

ENO (essentially nonoscillatory) and weighted ENO (WENO) schemes were designed for high resolution of discontinuities, such as shock waves, while optimized schemes such as the DRP (dispersion–relation–preserving) schemes were optimized for short waves (with respect to the grid spacing Δx, e.g., waves that are 6–8Δx in wavelength) in the wavenumber space. In this paper, we seek to unite the advantages of WENO and optimized schemes through the development of Optimized WENO (OWENO) schemes to tackle shock/broadband acoustic wave interactions and small-scale flow turbulences relative to the grid spacing. OWENO schemes are optimized in two levels. In the first level, optimized schemes are constructed for all candidate stencils by minimizing the error in the wavenumber space. In the second level, these optimized schemes are convexly combined using weights constructed to achieve not only higher order of accuracy but also high resolution for short waves. In addition, a new definition of smoothness indicators is presented for the OWENO schemes. These smoothness indicators are shown to have better resolution for short waves. A third-order OWENO scheme and a seventh-order WENO scheme are compared against each other for performance on the scalar model equation. It has been shown that the OWENO scheme indeed gives much better results in resolving short waves than the WENO scheme while yielding nonoscillatory solutions for discontinuities. Finally the OWENO scheme is extended to the linearized Euler equations to solve two computational aeroacoustics (CAA) benchmark problems to demonstrate its capability.