Accelerated parameter-uniform numerical method for singularly perturbed parabolic convection-diffusion problems with a large negative shift and integral boundary condition

Abstract

The singularly perturbed parabolic convection–diffusion equations with integral boundary conditions and a large negative shift are studied in this paper. The implicit Euler method for the temporal direction and the exponentially fitted finite difference scheme for the spatial direction are applied to formulate a parameter-uniform numerical method. The Simpson’s integration rule is used to handle the integral boundary condition. The Richardson extrapolation technique is applied to enhance the order of convergence of the method. The stability and uniform convergence analysis of the proposed method are studied. It is shown that the method is uniformly convergent with a convergence order of two in both temporal and spatial direction after Richardson extrapolation. Two test examples are considered to verify the validity of the proposed numerical scheme. The obtained numerical results confirm the theoretical estimates. The proposed method provide more accurate results and a higher order of convergence than methods available in the literature.

Keywords

• Singularly perturbed problem
• Parabolic convection–diffusion equations
• Exponentially fitted finite difference method
• Integral boundary condition
• Large negative shift