An Asymptotic-Numerical Method for a Class of Weakly Coupled System of Singularly Perturbed Convection-Diffusion Equations. (3rd October 2019)

• Record Type: Journal Article
• Authors: Kaushik, Aditya; Vashishth, Anil K.; Kumar, Vijayant; Sharma, Manju
• Is Part Of: Numerical functional analysis and optimization. Volume 40:Number 13(2019)
• Page Start: 1550
• Page End: 1571
• Publication Date: 2019-10-03
• Subjects: Convection-diffusion equations -- Runge–Kutta method -- singularly perturbed -- weakly coupled; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2019.1615946
• Languages: English
• ISSNs: 0163-0563

Abstract

Abstract: A weakly coupled convection dominated system of m -equations is analyzed. A higher order accurate asymptotic-numerical method is presented. The solutions of convection dominated problem are known to exhibit multi-scale character. There exist narrow region across the boundary of the domain where the solution exhibit steep gradient. This region is termed as boundary layer region and the solution of problem is said to have a boundary layer. Outside of this region, the solution of system behaves smoothly. To capture this multi-scale nature given system is factorized into two explicit systems. The degenerate system of initial value problems (IVPs), obtained by setting ϵ = 0, corresponds to the smooth solution, which lies outside of boundary layers. For solution inside boundary layers, a system of boundary value problems (BVPs) is obtained using stretching transformation. Regardless of this simple factorization, solutions of these systems preserve the key features of the given coupled system. Runge–Kutta method is used to solve the degenerate system of IVPs, whereas the system of BVPs is solved analytically. Stability and consistency of the proposed method is established. A uniform convergence of higher order is obtained. Possible extension to differential difference equations are also brought to attention. A comparative study of the present method with some state of art existing numerical schemes is carried out by means of several test problems. The results so obtained demonstrate the effectiveness and potential of present approach.