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