On the efficient application of weighted essentially nonoscillatory scheme


Kazuhiro Nakahashi, Department of Aeronautical Engineering, Tohoku University, Aoba 6-6-01, Sendai, 980-8579, Japan.

E-mail: naka@ad.mech.tohoku.ac.jp


In this paper, the efficient application of high-order weighted essentially nonoscillatory (WENO) reconstruction to the subsonic and transonic engineering problems is studied. On the basis of the physical considerations, two techniques are proposed to enhance the accuracy and efficiency of the WENO reconstruction. First, it is observed that the WENO scheme using characteristic variable has better accuracy and convergence speed than the scheme using primitive variable. For engineering problems with shock of moderate amplitude, on the basis of the Rankine–Hugoniot conditions, a simplified characteristic-variable-based WENO is developed. The simplified version significantly reduces the cost overhead without sacrificing the shock-capturing capability. Second, in this work, it is found for viscous case that it is better to include the viscous effect. On the basis of a simple analysis, the viscous correction to the parameter ε in the WENO reconstruction is proposed. Numerical results indicate, with the proposed simplified characteristic-variable-based reconstruction and the viscous correction, that the nonlinear WENO interpolation is sharply activated in the region of shock jump, whereas in the shockless area, the WENO interpolation weights are tuned towards the designed optimal value for better accuracy. Compared with the original characteristic-variable-based WENO, the current implementation has similar accuracy and reduced cost. At the same time, compared with the primitive variable-based WENO, better accuracy and convergence speed are obtained at marginal cost overhead. Several practical cases are calculated to demonstrate the accuracy and efficiency of the current methodology. Copyright © 2012 John Wiley & Sons, Ltd.