On an New Algorithm for Function Approximation with Full Accuracy in the Presence of Discontinuities Based on the Immersed Interface Method

This paper is devoted to the construction and analysis of an adapted and nonlinear multiresolution algorithm designed for interpolation or approximation of discontinuous univariate functions. The adaption attained allows to avoid numerical artifacts that appear when using linear algorithms and, at the same time, to obtain a high order of accuracy close to the singularities. It is known that linear algorithms are stable and convergent for smooth functions, but diffusion and Gibbs effect appear if the functions are piecewise continuous. Our aim is to develop an algorithm for function approximation with full accuracy that is capable to adapt to corners (kinks) and jump discontinuities, that uses a centered stencil and that does not use extrapolation. In order to reach this goal, we will need some information about the jumps in the function that we want to approximate and its derivatives. If this information is available, the algorithm is the most compact possible in the sense that the stencil is fixed and we do not need a stencil selection procedure as other algorithms do, such as ENO subcell resolution (ENO-SR). If the information about the jumps is not available, we will show a technique to approximate it. The algorithm is based on linear interpolation plus correction terms that provide the desired accuracy close to corners or jump discontinuities.