Interior Layers in a Singularly Perturbed Time Dependent Convection–diffusion Problem

A linear singularly perturbed time dependent convection–diffusion problem is examined. The initial condition is designed to have steep gradients in the vicinity of the inflow point, which are transported in time, thus creating a moving interior shock layer. The location of this interior layer is tracked by the characteristics of the reduced first order problem. A numerical method is designed and analysed, which consists of a monotone finite difference operator and a piecewise-uniform Shishkin mesh, which is aligned to the characteristic curve emanating from the initial shock location. Parameter explicit error bounds are established and numerical results are presented to illustrate the performance of the numerical method.