A Parameter-Robust Numerical Method Based on Defect-Correction Technique for Parabolic Singular Perturbation Problems with Discontinuous Convection Coefficient and Source

Abstract

The paper presents a uniformly convergent finite difference method based on the defect correction technique to solve parabolic singular perturbation problems with discontinuous convection coefficient and source. The solution to the problem exhibits interior layer across the discontinuity and demonstrates turning point behavior. The simultaneous presence of perturbation parameters and discontinuity makes the problem stiff. A higher-order method is developed using an implicit difference scheme in time on a uniform mesh and a combination of the upwind difference method and the central difference scheme over an adaptive mesh in space. The method involves iteratively solving increasingly accurate discrete problems by computing and using the defect to correct the approximate solution. Parameter uniform error estimates show uniform convergence of first-order in time and second-order in space. Numerical experiments confirm the accuracy of the proposed scheme and support the theoretical analysis.