Adaptive decomposition finite difference methods for solving singular problems—A review

Abstract

Decomposition, or splitting, finite difference methods have been playing an important role in the numerical solution of nonsingular differential equation problems due to their remarkable efficiency, simplicity, and flexibility in computations as compared with their peers. Although the numerical strategy is still in its infancy for solving singular differential equation problems arising from many applications, explorations of the next generation decomposition schemes associated with various kinds of adaptations can be found in many recent publications. The novel approaches have been proven to be highly effective and reliable in operations. In this article, we will focus on some of the latest developments in the area. Key comments and discussion will be devoted to two particularly interesting issues in the research, that is, direct solutions of degenerate singular reactiondiffusion equations and nonlinear sine-Gordon wave equations. Numerical experiments with simulated demonstrations will be given.

Key words： Singularity; degeneracy; finite difference approximation; uniform and nonuniform grid; decomposition; adaptation; monotonicity and stability; large system of equations