Deformation and Smoothing of Cusp Singularities

A cusp singularity (CS), is a point at which the slope of a continuous curve changes abruptly in sign and magnitude. A particular type of CS, which is the focus of this paper, is where only the sign of the slope is altered while the magnitude of the slope is unchanged. This type of CSs occur in many natural phenomena such as Kato's cusp and particular plasmonics. Solving such problems numerically can be challenging because of the discontinuity in the derivatives. In this paper, we present an efficient spectral method incorporated with transformation (mapping) to handle the cusp problem. The transformation is based on functions that are locally odd around all the cusp points. The idea is to transform functions from C⁰ continuity to C^N continuity (N < 1), and then implement a spectral method to solve the mapped problem without any domain decomposition. The final solution is obtained with inverse mapping.