Constructing spectral schemes of the immersed interface method via a global description of discontinuous functions

A global description of discontinuous functions is introduced in this paper. By expressing a discontinuous function as the sum of a smooth function and a correction term determined by jump conditions, we turn the unknown function from a discontinuous one into a sufficiently smooth one when solving a differential equation. Spectral schemes are developed based on this concept with the intention of eliminating or reducing the Gibbs oscillation. Finite difference schemes are also constructed as an alternative of the current immersed interface methods. Both spectral and finite difference schemes are tested on one- and two-dimensional cases.