Hybrid Fitted Mesh Strategy for Singularly Perturbed Time-Dependent Convection-Diffusion Problems Featuring Boundary Turning Points

This work investigates the solution of convection-diffusion parabolic partial-differential problems with boundary turning points that are singularly perturbed. These types of problems are stiff for the following reason: the small parameter multiplying coefficient of the diffusion term and the presence of boundary turning points. The solution to the problem under consideration in the spatial domain displays a left boundary layer. Analytical or classical numerical approaches confront computing challenges in the rapidly changing solution behaviour in the layer region. To handle this effect, we developed parameter-uniform numerical method comprised of a hybridized approach that combines central difference and midpoint upwind schemes in space with nonuniform mesh and the Crank–Nicolson method in time with uniform mesh. This scheme is parameter-uniformly convergent in the maximum norm with second-order accuracy. Stability is investigated and assessed using the discrete minimum principle and the bounds of truncation error. The numerical solutions of the three model examples considered are aligned with the theoretical conclusions.