Uniformly convergent numerical method for singularly perturbed convection‐diffusion type problems with nonlocal boundary condition

• Published in: International journal for numerical methods in fluids Vol. 92; no. 12; pp. 1914 - 1926
• Main Authors: Debela, Habtamu Garoma, Duressa, Gemechis File
• Format: Journal Article
• Language: English
• Published: Hoboken, USA John Wiley & Sons, Inc 01.12.2020; Wiley Subscription Services, Inc
• Subjects: 2010 Mathematics Subject Classification: 65 L11, 65 L12, 65 L20; Boundary conditions; Convection; Convergence; Differential equations; Diffusion; Finite difference method; Integration; Mathematical models; nonlocal boundary condition; nonstandard finite difference; Numerical analysis; Numerical integration; Numerical methods; Singular perturbation methods; singularly perturbed problem
• ISSN: 0271-2091; 1097-0363
• DOI: 10.1002/fld.4854

Summary In this article, we consider a class of singularly perturbed differential equations of convection‐diffusion type with nonlocal boundary conditions. A uniformly convergent numerical method is constructed via nonstandard finite difference and numerical integration methods to solve the problem. The nonlocal boundary condition is treated using numerical integration techniques. Maximum absolute errors and rates of convergence for different values of perturbation parameter and mesh sizes are tabulated for the numerical example considered. The method is shown to be ϵ‐uniformly convergent. The introduced method is applicable for the values for which the perturbation parameter ϵ is much less than the mesh size h and where other methods fail to give good results. Furthermore, it is more accurate and gives good result where existing numerical methods fails (That is for the values where the perturbation parameter, ϵ is much less than the mesh size, h). This study also helps to introduce the technique of establishing parameter uniform convergence.