New upwind-WENO difference scheme based on dissipation-relation-preserving (DRP)

Purpose. There exists a difference scheme based on DRP^[7]. We now present a new upwind-WENO (Weighted Essentially Non-Oscillatory) difference scheme based on DRP that, we believe, can offer numerical results good in precision and high-resolution. In the full paper, we explain our new scheme in much detail, here we only give a briefing. WENO schemes are designed for high-resolution of discontinuities such as shock waves, while DRP schemes are designed for optimization of short waves in the wave number space. We seek to unite the advantages of WENO and DRP schemes through the development of optimized upwind-WENO schemes to tackle shocks whose waves are high in frequency relative to the grid spacing. Optimized upwind-WENO scheme is obtained by minimizing the error in the wave number space based on dissipation and dispersion relations. Finally we give three figures showing results of numerical tests of our new upwind-WENO difference scheme based on DRP for representative 1D computational aero-acoustic problems. We now explain briefly only the results concerning the very important shock and sine wave interaction. Our new scheme can capture correctly discontinuity and short wave with high-resolution, and obtain quite correct phase and amplitude positions. In short, these numerical results show preliminarily the good precision and high-resolution of our new scheme.