A note on a parameterized singular perturbation problem

Applied Mathematics and Computation, 2005

We consider a uniform finite difference method on a B-mesh is applied to solve a singularly perturbed quasilinear boundary value problem (BVP) depending on a parameter. We give a uniform first-order error estimates in a discrete maximum norm. Numerical results are presented that demonstrate the sharpness of our theoretical analysis.

Turkish journal of mathematics & computer science, 2022

In this paper, singularly perturbed quasilinear boundary value problems are taken into account. With this purpose, a finite difference scheme is proposed on Shishkin-type mesh (S-mesh). Quasilinearization technique and interpolating quadrature rules are used to establish the numerical scheme. Then, an error estimate is derived. A numerical experiment is demonstrated to verify the theory.

Journal of Applied Mathematics, 2004

We study uniform finite-difference method for solving first-order singularly perturbed boundary value problem (BVP) depending on a parameter. Uniform error estimates in the discrete maximum norm are obtained for the numerical solution. Numerical results support the theoretical analysis.

Applied Mathematics and Computation, 2007

We consider a uniform finite difference method on Shishkin mesh for a quasilinear first order singularly perturbed boundary value problem (BVP) with integral boundary condition. We prove that the method is first order convergent except for a logarithmic factor, in the discrete maximum norm, independently of the perturbation parameter. The parameter uniform convergence is confirmed by numerical computations.

In this paper we are considering a semilinear singular perturbation reaction -diffusion boundary value problem, which contains a small perturbation parameter that acts on the highest order derivative. We construct a difference scheme on an arbitrary nonequidistant mesh using a collocation method and Green's function. We show that the constructed difference scheme has a unique solution and that the scheme is stable. The central result of the paper is ε-uniform convergence of almost second order for the discrete approximate solution on a modified Shishkin mesh. We finally provide two numerical examples which illustrate the theoretical results on the uniform accuracy of the discrete problem, as well as the robustness of the method.

Journal of Modern Methods in Numerical Mathematics, 2015

We are considering a semilinear singular perturbation reaction-diffusion boundary value problem which contains a small perturbation parameter that acts on the highest order derivative. We construct a difference scheme on an arbitrary nonequidistant mesh using a collocation method and Green's function. We show that the constructed difference scheme has a unique solution and that the scheme is stable. The central result of the paper is ϵ-uniform convergence of almost second order for the discrete approximate solution on a modified Shishkin mesh. We finally provide two numerical examples which illustrate the theoretical results on the uniform accuracy of the discrete problem, as well as the robustness of the method.

1999

In this paper we construct and analyze two compact monotone finite difference methods to solve singularly perturbed problems of convection-diffusion type. They are defined as HODIE methods of order two and three, i.e., the coefficients are determined by imposing that the local error be null on a polynomial space. For arbitrary meshes, these methods are not adequate for singularly perturbed problems, but using a Shishkin mesh we can prove that the methods are uniformly convergent of order two and three except for a logarithmic factor. Numerical examples support the theoretical results.

International Journal of Computer Mathematics, 2012

We propose a fully discrete ε-uniform finite-difference method on an equidistant mesh for a singularly perturbed two-point boundary-value problem (BVP). We start with a fitted operator method reflecting the singular perturbation nature of the problem through a local BVP. However, to solve the local BVP, we employ an upwind method on a Shishkin mesh in local domain, instead of solving

Numerische Mathematik, 1986

We examine the problem: Eu"+a(x)u '-b(x)u=f(x) for 0<x<l, a(x)>e>0, b(x)>fi, :t 2+4~fl>0, a, b and f in C2[0,1], e in (0,1], u(0) and u(1) given. Using finite elements and a discretized Green's function, we show that the E1-Mistikawy and Werle difference scheme on an equidistant mesh of width h is uniformly second order accurate for this problem (i.e., the nodal errors are bounded by Ch 2, where C is independent of h and e). With a natural choice of trial functions, uniform first order accuracy is obtained in the U~ 1) norm. On choosing piecewise linear trial functions ("hat" functions), uniform first order accuracy is obtained in the LI(0, 1) norm.

International Journal of Computer Mathematics, 2018

In the present paper, a parameter-uniform numerical method is constructed and analyzed for solving one-dimensional singularly perturbed parabolic problems with two small parameters. The solution of this class of problems may exhibit exponential (or parabolic) boundary layers at both the left and right part of the lateral surface of the domain, depending on the size of the parameters. The asymptotic behavior of the solution and its partial derivatives is given. A decomposition of the solution in its regular and singular parts has been used for the asymptotic analysis of the spatial derivatives. To approximate the solution, we consider the implicit Euler method for time stepping on a uniform mesh and a special hybrid monotone difference operator for spatial discretization on a specially designed piecewise uniform Shishkin mesh. The resulting scheme is shown to be first-order convergent in temporal direction and almost second-order convergent in spatial direction. We then improve the order of convergence in time by means of the Richardson extrapolation technique used in temporal variable only. The resulting scheme is proved to be uniformly convergent of order two in both the spatial and temporal variables. Numerical experiments support the theoretically proved higher order of convergence and show that the present scheme gives better accuracy and convergence compared of other existing methods in the literature.