Nonstandard hybrid numerical scheme for singularly perturbed parabolic differential equations with large delay

Abstract

In this paper, a numerical scheme for a class of singularly perturbed delay parabolic convection–diffusion problems having Dirichlet boundary conditions is proposed. When the perturbation parameter tends to zero, the solution to these problems exhibits a boundary layer at the right side of the domain. The classical methods fail to provide good accuracy for these problems. To overcome this drawback, we proposed the Crank–Nicolson finite difference scheme to discretize in temporal direction on a uniform mesh and a nonstandard midpoint upwind scheme in spatial direction on a piecewise uniform Shishkin mesh. The resulting scheme has been shown to be uniformly stable and convergent with respect to perturbation parameter and mesh size. The scheme converges uniformly with the order of convergence OΔt2+N−1lnN2, where Δt is mesh size in temporal discretization and N is number of mesh elements in spatial discretization. To validate theoretical results, numerical experiments have been carried out and presented in the form of tables and graphs.

Keywords

• Midpoint upwind
• Crank–Nicolson
• Nonstandard hybrid
• Singular perturbation
• Boundary layer
• Shishkin mesh