Richardson Extrapolation Method on an Adaptive Grid for Singularly Perturbed Volterra Integro-Differential Equations. (1st July 2021)

• Record Type: Journal Article
• Authors: Long, Guangqing; Liu, Li-Bin; Huang, Zaitang
• Is Part Of: Numerical functional analysis and optimization. Volume 42:Number 7(2021)
• Page Start: 739
• Page End: 757
• Publication Date: 2021-07-01
• Subjects: Adaptive grid -- mesh equidistribution principle -- singularly perturbed -- uniform convergent -- Volterra integro-differential equation; 65L10 -- 65L12 -- 65N30; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2021.1928698
• Languages: English
• ISSNs: 0163-0563

Abstract

Abstract: In this paper, an adaptive grid method for singularly perturbed Volterra integro-differential equations exhibiting a boundary layer is studied. At first, based on an arbitrary nonuniform mesh, we use the classical backward Euler formula to discrete the first-order derivative part and the left rectangle formula to approximate the integral part, respectively. Meanwhile, it is shown from the mesh equidistribution principle and the truncation error analysis that the presented adaptive grid method is first-order uniformly convergent with respect to the perturbation parameter. Furthermore, the Richardson extrapolation technique is utilized to improve the uniform accuracy of the presented adaptive grid method in the discrete supremum norm form O ( N − 1 ) to O ( N − 2 ), where N is the mesh parameter. Finally, some numerical results are given to support the theoretical results.