On a Nonlinear Three-Point Subdivision Scheme Reproducing Piecewise Constant Functions

Abstract

In this article, a nonlinear binary three-point non-interpolatory subdivision scheme is presented. It is based on a nonlinear perturbation of the three-point subdivision scheme: A new three point approximating C2 subdivision scheme, where the convergence and the stability of this linear subdivision scheme are analyzed. It is possible to prove that this scheme does not present Gibbs oscillations in the limit functions obtained. The numerical experiments show that the linear scheme is stable even in the presence of jump discontinuities. Even though, close to jump discontinuities, the accuracy is loosed. This order reduction is equivalent to the introduction of some diffusion. Diffusion is a good property for subdivision schemes when the discontinuities are numerical, i.e., they appear when discretizing a continuous function close to high gradients. On the other hand, if the initial control points come from the discretization of a piecewise continuous function, it can be interesting that the subdivision scheme produces a piecewise continuous limit function. For instance, in the approximation of conservation laws, real discontinuities appear as shocks in the solution. The nonlinear modification introduced in this work allows to attain this objective. As far as we know, this is the first subdivision scheme that appears in the literature with these properties.