Adaptive decomposition finite difference methods for solving singular problems -- a review
This survey article presents the latest developments in the field of adaptive decomposition finite difference methods for solving singular problems. The first part of it presents a history of this type of methods that can be traced back to the early fifties. Section 2 is concerned with the basic theory of decomposition methods and approximations with a precise description regarding basic splitting. Section 3 of the paper focuses on adaptive decomposition methods for solving nonlinear reaction-diffusion equations with quenching combustion type nonlinearities. Here the monotonicity of the sequential numerical solutions generated from decomposition procedure is proved. Section 4 is devoted to investigations of solitary wave problems governed by sine-Gordon equation. Locally linearized stability analysis is studied for the variable step decomposed finite difference methods. Various numerical experiments are presented. Finally, last section summarizes the whole presentation and presents some expectations for the forthcoming researches in the field.
quenching combustion type nonlinearities
scientific article; zbMATH DE number 5627219
scientific article