Cell-average WENO with progressive order of accuracy close to discontinuities with applications to signal processing

Abstract: In this paper we translate to the cell-average setting the algorithm for the point-value discretization presented in S. Amat, J. Ruiz, C.-W. Shu, D. F. Y\'a~nez, A new WENO-2r algorithm with progressive order of accuracy close to discontinuities, submitted to SIAM J. Numer. Anal.. This new strategy tries to improve the results of WENO-($2r-1$) algorithm close to the singularities, resulting in an optimal order of accuracy at these zones. The main idea is to modify the optimal weights so that they have a nonlinear expression that depends on the position of the discontinuities. In this paper we study the application of the new algorithm to signal processing using Harten's multiresolution. Several numerical experiments are performed in order to confirm the theoretical results obtained.