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In this study we present a method for approximating the solution of a Singularly Perturbed Boundary Value Problem (SPBVP) containing two parameters (ε 1 ,ε 2 ), which multiply the diffusion coefficient and the convection term, respectively. Moreover, we consider that the convection coefficient and the source term present a discontinuity at an intermediate point. Theoretical bounds for the solution and its derivatives are derived for two complementary cases. A parameter uniform numerical scheme is constructed, which involves an upwind finite difference method with an appropriate piecewise uniform mesh. The error estimation and convergence analysis are presented, which show that the scheme provides a parameter uniform convergence of almost first order. Some numerical examples are discussed to illustrate the performance of the present method. © 2019

World University root node

Department of Mathematics

School of Advanced Sciences

Vellore Campus

Journal of Computational and Applied Mathematics

Discrete approximation for a two-parameter singularly perturbed boundary value problem having discontinuity in convection coefficient and source term

In this article, a parameter uniform numerical method is developed for a two-parameter singularly perturbed parabolic partial differential equation with discontinuous convection coefficient and source term. The presence of perturbation parameter and the discontinuity in the convection coefficient and source term lead to the boundary and interior layers in the solution. On the spatial domain, an adaptive mesh is introduced before discretizing the continuous problem. The present method observes a uniform convergence in maximum norm which is almost first-order in space and time irrespective of the relation between convection and diffusion parameters. Numerical experiment is carried out to validate the present scheme. © 2017, Foundation for Scientific Research and Technological Innovation.

Differential Equations and Dynamical Systems

A Numerical Method for Solving Boundary and Interior Layers Dominated Parabolic Problems with Discontinuous Convection Coefficient and Source Terms

In the present work, we consider a parabolic convection-diffusion-reaction problem where the diffusion and convection terms are multiplied by two small parameters, respectively. In addition, we assume that the convection coefficient and the source term of the partial differential equation have a jump discontinuity. The presence of perturbation parameters leads to the boundary and interior layers phenomena whose appropriate numerical approximation is the main goal of this paper. We have developed a uniform numerical method, which converges almost linearly in space and time on a piecewise uniform space adaptive Shishkin-type mesh and uniform mesh in time. Error tables based on several examples show the convergence of the numerical solutions. In addition, several numerical simulations are presented to show the effectiveness of resolving layer behavior and their locations. Copyright © 2018 John Wiley & Sons, Ltd.

Mathematical Methods in the Applied Sciences

Numerical treatment of two-parameter singularly perturbed parabolic convection diffusion problems with non-smooth data

A singularly perturbed second order ordinary differential equation having two parameters with a discontinuous source term is presented for numerical analysis. Theoretical bounds on the derivatives, regular and singular components of the solution are derived. A hybrid monotone difference scheme with the method of averaging at the discontinuous point is constructed on Shishkin mesh. Parameter-uniform error bounds for the numerical approximation are established. Numerical results are presented which support the theoretical results. © 2016 Elsevier B.V.

Fulltext

A parameter robust higher order numerical method for singularly perturbed two parameter problems with non-smooth data

A singularly perturbed second order ordinary differential equation having two small parameters with a discontinuous source term is considered. The presence of two parameters gives rise to boundary layers of different widths and the discontinuous source term generates interior layers on both sides of the discontinuous point. Theoretical bounds are derived. The problem is solved numerically with finite difference methods on a Shishkin mesh. The discretization combines a five point second order scheme at the interior layer together with the standard central, mid-point and upwind difference scheme for other regions. This combination is used in order to obtain almost second order convergence for the considered problem. Parameter uniform error bounds for the numerical approximation are established. Numerical results are presented to illustrate the convergence of the numerical approximations. © 2017 Elsevier Inc.

Applied Mathematics and Computation

Hybrid difference scheme for singularly perturbed reaction-convection-diffusion problem with boundary and interior layers

In this paper, we consider a singularly perturbed turning point problem of convection-diffusion of fourth-order ordinary differential equations with a small positive parameter(e). The given fourth-order boundary value problem is transformed into a system of weakly coupled systems of two second-order ordinary differential equations, one without parameter and other with parameter e multiplying highest derivatives with suitable boundary conditions. A computational method is presented for solving the system of both linear and non-linear problems. In the linear case, we first find the zero-order asymptotic approximation expansion of the solution in the second equation. Then, the second equation of the system is solved by the numerical method which is constructed for this problem which involves Shishkin mesh. As in the case of non-linear problem, Newton's method of quasilinearization is applied for the second equation of the system. Numerical results are presented which support the theoretical results. ©Dynamic Publishers, Inc.

Neural, Parallel and Scientific Computations

An asymptotic numerical method for singularly perturbed fourth order ODE of convection-diffusion type turning point problem

In this paper, second order singularly perturbed convection-diffusion Robin type problem with a discontinuous source term is considered. Due to the discontinuity interior layers appears in the solution. A numerical method is constructed for this problem which involves an appropriate piecewise—uniform mesh for the boundary and interior layers. The method is shown to be parameter uniformly convergent with respect to the singular perturbation parameter. Numerical examples are presented to illustrate the theoretical results. © 2015, Springer India Pvt. Ltd.

International Journal of Applied and Computational Mathematics

Fitted Mesh Method for Singularly Perturbed Robin Type Boundary Value Problem with Discontinuous Source Term

A singularly perturbed reaction-diffusion problem with a discontinuous source term is considered. In Miller et al. (J Appl Numer Math 35(4):323–337, 2000) the authors discussed problems that arises naturally in the context of models of simple semiconductor devices. Due to the discontinuity, interior layers appear in the solution. The problem is solved using a hybrid difference scheme on a Shishkin mesh. We prove that the method is second order convergent in the maximum norm, independently of the diffusion parameter. Numerical experiments support these theoretical results and indicate that the estimates are sharp. © 2014, Springer India Pvt. Ltd.

A Hybrid Difference Scheme for a Second-Order Singularly Perturbed Reaction-Diffusion Problem with Non-smooth Data

In this paper a second-order Singularly Perturbed Ordinary Differential Equation (ODE) of Reaction-Diffusion type Boundary Value Problems (BVPs) with discontinuous source term is considered. A numerical method is suggested in this paper to solve such problems. The domain of definition of the differential equation (a closed interval) is divided into five non-overlapping subintervals, which we call "Inner Region" (Boundary Layers) and "Outer Region". Then, the Differential Equation is solved in these intervals separately. The solutions obtained in this region are combined to give a solution in the entire interval. To obtain terminal boundary conditions (boundary values inside this interval), we mostly use zero-order asymptotic expansion of the solution of the BVPs. Error estimates of the solution and numerical examples are provided. © School of Engineering, Taylor’s University.