Numerical Study of a Singularly Perturbed Two Parameter Problems on a Modified Bakhvalov Mesh
- PDF / 630,916 Bytes
- 9 Pages / 612 x 792 pts (letter) Page_size
- 53 Downloads / 341 Views
ENERAL NUMERICAL METHODS
Numerical Study of a Singularly Perturbed Two Parameter Problems on a Modified Bakhvalov Mesh P. Pramod Chakravarthya,* and Meenakshi Shivharea,** aDepartment
of Mathematics, Visvesvaraya National Institute of Technology, Nagpur, 440010 India *e-mail: [email protected] **e-mail: [email protected] Received December 12, 2019; revised January 16, 2020; accepted June 9, 2020
Abstract—In this article, we consider a two parameter singular perturbation problem (TPSPP) and the numerical solution is obtained by using tension spline method on a layer adapted Bakhvalov type mesh. The method is analyzed for convergence. Numerical results are presented to support the theory. Keywords: tension spline, singular perturbation, Bakhvalov-type mesh, two-parameter problems DOI: 10.1134/S0965542520110111
1. INTRODUCTION Singular perturbation problem (SPP) arises very frequently in many areas of physics and applied mathematics [1, 2]. SPPs are characterized by the fact of steep gradients in the solution. The finite difference method and finite element method does not work on uniform mesh and leads to oscillatory solution as ε → 0. These methods need very fine mesh inside the boundary layer region which is computationally costly. The mathematical models related to TPSPPs arise in the chemical-reaction processes, lubrication theories [3], transport phenomena in biology, chemistry [4] and flow through unsaturated porous media [5]. In TPSPPs, the diffusion and convection terms are multiplied by the perturbation parameters. The study on TPSPPs is very limited when compared to single parameter singular perturbation problems. O’Malley [6] studied the problem asymptotically and had given the asymptotic solutions depending on the ratios between μ2 and ε . Valarmathi and Ramanujam [7] proposed a fitted operator method to solve the TPSPPs using zeroth order asymptotic solutions. Vulanovic considered a quasilinear TPSPP with a special case μ = ε1/2 + p , p > 0 in which the solution behaviour is similar to the reaction-diffusion problem [8, 9]. Linß and Roos developed uniformly convergent numerical schemes which are of first order on Shishkin mesh [10]. Roos [11] obtained an almost second order convergent scheme by using streamline-diffusion finite element method on a piecewise uniform Shishkin mesh. Kadalbajoo and Yadaw [12] proposed a numerical scheme based on cubic B-Spline collocation method and obtained almost second order convergence. Brdar and Zarin [13, 14] analyzed the problem on layer adapted meshes. Zarin [15] proposed an exponentially graded mesh for solving TPSPPs. One dimensional parabolic TPSPPs are examined on adaptive moving mesh method by Das and Mehrmann [16]. TPSPPs with the discontinuous source term were handled by Prabha et al. [17, 18]. O’Riordan and Pickett [19] had given the numerical approximations to the scaled first derivatives of the solution to a TPSPP. In this paper, we constructed a tension spline method for solving TPSPPs on a modified Bakhvalov type layer adapted m
Data Loading...
