A generalized regularization scheme for solving singularly perturbed parabolic PDEs

Abstract

Many problems in science and engineering can be modeled as singularly perturbed partial differential equations. Solutions to such problems are not generally continuous with respect to the perturbation parameter(s) and hence developing stable and efficient numerical schemes to singularly perturbed problems are always more interesting and mathematically challenging to researchers. In this paper, we propose a generalized regularization scheme as an alternate numerical scheme for solving singularly perturbed parabolic PDEs with higher-order derivatives multiplied by a small parameter. Since the involved operators are unbounded, first, we propose a general theoretical framework for unbounded operators with an a posteriori parameter choice rule for selecting a regularization parameter, and then we apply it to singularly perturbed problems. We numerically implement the proposed scheme and compare it with other traditional schemes. The study illustrates that the proposed scheme can be considered as an alternate and competent method for solving singularly perturbed parabolic problems.