Efficient WENO schemes for nonuniform grids with optimal accuracy

Abstract Non-linear interpolation techniques have been extensively used in several settings such as image processing, data analysis, finance, economy, and overall in solving numerically partial differential equations. In this context, weighted essentially non-oscillatory (WENO) methods have been developed for different problems. The idea of the non-linear methods is to use the data information to avoid the data points where discontinuities or steep gradients appear. The classic WENO combines some polynomial interpolators with a certain order of accuracy to obtain a new interpolator that is more accurate in smooth zones and, at least, as accurate as the original ones when the data present a discontinuity. The construction of the classic WENO method has been formulated for uniform grids and to approximate a particular point of the grid. Recently, Martí et al. in [ J. Sci. Comput. 100 (2024), 6 ] introduced a WENO version for non-uniform grids and used it for the solution of conservation laws and hyperbolic systems in the context of finite volume methods. In this paper, this method is extended and improved, focusing on three aspects. Its efficiency: a more efficient procedure to iteratively compute the reconstruction polynomials for the different types of reconstructions considered in this work is proposed. Its versatility: the designed method can be used in the context of a non-uniform stencil with the reconstruction point located at an arbitrary position. Its order of accuracy: some theoretical results showing that the order of the proposed scheme is optimal on smooth data and as high as possible near discontinuities are presented. Finally, some numerical experiments are performed to illustrate the theoretical results that are presented.