A New Parameter-Uniform Discretization of Semilinear Singularly Perturbed Problems

Abstract

In this paper, we present a numerical approach to solving singularly perturbed semilinear convection-diffusion problems. The nonlinear part of the problem is linearized via the quasilinearization technique. We then design and implement a fitted operator finite difference method to solve the sequence of linear singularly perturbed problems that emerges from the quasilinearization process. We carry out a rigorous analysis to attest to the convergence of the proposed procedure and notice that the method is first-order uniformly convergent. Some numerical evaluations are implemented on model examples to confirm the proposed theoretical results and to show the efficiency of the method.

Keywords

• singularly perturbed problems
• semilinear differential equation
• quasilinearization
• boundary layer
• fitted operator finite difference method
• uniform convergence