Anti-diffusive flux corrections for high order finite difference WENO schemes

In this paper, we generalize a technique of anti-diffusive flux corrections, recently introduced by Després and Lagoutière [Journal of Scientific Computing 16 (2001) 479-524] for first-order schemes, to high order finite difference weighted essentially non-oscillatory (WENO) schemes. The objective is to obtain sharp resolution for contact discontinuities, close to the quality of discrete traveling waves which do not smear progressively for longer time, while maintaining high order accuracy in smooth regions and non-oscillatory property for discontinuities. Numerical examples for one and two space dimensional scalar problems and systems demonstrate the good quality of this flux correction. High order accuracy is maintained and contact discontinuities are sharpened significantly compared with the original WENO schemes on the same meshes.