A higher order uniformly convergent method for singularly perturbed delay parabolic partial differential equations

ABSTRACT

In this article, we aim to introduce a high-order uniformly convergent method to solve singularly perturbed delay parabolic convection diffusion problems exhibiting a regular boundary layer. The domain is discretized by a uniform mesh in the time direction and a piecewise-uniform Shishkin mesh for the spatial direction. We use the Crank–Nicolson method for the time derivative and we develop a fourth-order compact difference method to solve the set of ordinary differential equations at each time level. The stability analysis and the truncation error are discussed. Parameter-uniform error estimates are derived and it is shown that the method is $\epsilon$-uniformly convergent of second-order accurate in time, and in the spatial direction it is of second-order outside region of boundary layer, and of almost fourth-order inside the layer region. Numerical examples are presented to verify the theoretical results and to confirm the efficiency and high accuracy of the proposed method.

KEYWORDS:

• Singularly perturbed parabolic problem
• delay differential equation
• boundary layer
• high-order method
• uniform convergence

2010 AMS SUBJECT CLASSIFICATIONS:

• 65L11
• 65M06
• 65M12

Disclosure statement

No potential conflict of interest was reported by the authors.