On a Power WENO Scheme with Improved Accuracy Near Discontinuities

International audienceThis paper is devoted to the construction and analysis of a new prediction operator based on weighted essentially nonoscillatory (WENO) interpolation and capable of improving the accuracy near corner discontinuities in the point values and near jump discontinuities in the cell averages. The prediction operator presented is initially developed for the point values and then extended to the cell-average setting of Harten's multiresolution. In the cell averages we use the smoothness index proposed by Jiang and Shu [J. Comput. Phys., 126 (1996), pp. 202-228]. In the point values we use the one that was presented in [S. Amat and J. Ruiz, J. Sci. Comput., 71 (2017), pp. 1265-1302], where the authors show that the new smoothness index that they propose is better adapted for corner discontinuities. The inherent problem of WENO schemes and all the smoothness indexes available in the literature is that it is impossible to avoid the effect of the discontinuity when it is placed at the central interval of the stencil. Thus, our aim is to improve the results obtained by WENO interpolation even in the mentioned case. In order to do so, it will be necessary to detect the presence of the discontinuity in order to apply a nonlinear strategy that allows us to improve the accuracy at the central interval of the stencil. This nonlinear modification will be inspired by the piecewise polynomial harmonic (PPH) strategy introduced in [S. Amat et. al, Found. Comput. Math., 6 (2006), pp. 193-225] and the power means introduced in [S. Serna and A. Marquina, J. Comput. Phys., 194 (2004), pp. 632-658]. The numerical experiments show that the new strategy proposed improves the methods that can be found in the literature and, in particular, WENO schemes