A hybrid finite difference scheme for turning-point parabolic functional differential equations with discontinuous coefficients and source

The paper presents a hybrid finite difference scheme to solve a singularly perturbed parabolic functional differential equation with discontinuous coefficients and source. The simultaneous presence of deviating argument with a discontinuous source and coefficients makes the problem stiff. The solution of the problem exhibits turning point behaviour across discontinuities as ε tends to zero. The hybrid scheme presented is a composition of a central difference scheme in the layer region on a specially generated mesh and a midpoint upwind scheme outside the layer region. At the same time, an implicit finite difference scheme is used to discretize the time variable. The proposed numerical method has been analyzed for consistency, stability, and convergence. The proposed method converges uniformly independent of the perturbation parameter. Numerical results have been presented for two test examples that demonstrate the effectiveness of the scheme.