Uniformly Convergent Numerical Scheme for Solving Singularly Perturbed Parabolic Convection-Diffusion Equations with Integral Boundary Condition

• Published in: Differential equations and dynamical systems Vol. 33; no. 3; pp. 783 - 809
• Main Authors: Hailu, Wondimagegnehu Simon, Duressa, Gemechis File
• Format: Journal Article
• Language: English
• Published: New Delhi Springer India 01.07.2025; Springer Nature B.V
• Subjects: Boundary conditions; Computer Science; Convection-diffusion equation; Convergence; Engineering; Finite difference method; Mathematical analysis; Mathematics; Mathematics and Statistics; Numerical methods; Original Research; Singular perturbation; Singularly perturbed problem; Large negative shift; 65M15; Parabolic convection-diffusion equations; 65M06; Integral boundary condition; Non-standard finite difference method; 65M22; 65M12
• ISSN: 0971-3514; 0974-6870
• DOI: 10.1007/s12591-023-00645-y

The singularly perturbed parabolic convection-diffusion equations with integral boundary conditions and a large negative shift are studied in this paper. The Crank-Nicolson finite difference scheme for the temporal direction and the non-standard 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 spatial variable. The stability and uniform convergence analysis of the proposed method are studied. The method is shown to be uniformly convergent with a quadratic order of convergence in both temporal and spatial directions. Two test examples are considered to verify the validity of the proposed numerical scheme. The obtained numerical results confirm the theoretical estimates, also provide more accurate results and a higher order of convergence than methods available in the literature.