Linear algorithms for the approximation of smooth functions are stable and convergent, but if the functions are piecewise continuous then we encounter diffusion and the Gibbs effect [1].

The authors extend the idea of the immersed interface method [2] to construct and analyze a nonlinear algorithm for the interpolation or approximation of discontinuous functions in one variable. This requires information on the jump discontinuities of the function and its derivatives. If the information is not available, the algorithm can approximate it using multiresolution representation [3]. The nonlinearity of the algorithm comes only when the jump conditions are not known a priori.

Several examples demonstrate algorithm performance in cases of corners and jump discontinuities. The results are compared to a previously known algorithm, namely the essentially non-oscillatory scheme with subcell resolution (ENO-SR) [4]. The numerical experiments validate the stability of the algorithm and its order of convergence. The authors also prove the full accuracy of the method.