A moving mesh refinement based optimal accurate uniformly convergent computational method for a parabolic system of boundary layer originated reaction–diffusion problems with arbitrary small diffusion terms

A graded mesh refinement approach for boundary layer originated singularly perturbed time-delayed parabolic convection diffusion problems

We consider a uniform finite difference method on a Bakhvalov mesh to solve a quasilinear first order parameterized singularly perturbed problem with integral boundary conditions. Uniform first order error estimates in the discrete maximum norm have been established. Numerical results that demonstrate the sharpness of our theoretical analysis are presented.

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A parameter uniform difference scheme for the parameterized singularly perturbed problem with integral boundary condition

In this article, we consider a system of nonlinear singularly perturbed differential equations with two different parameters. To solve this system, we develop a weighted monotone hybrid scheme on a...

Numerical simulation and convergence analysis for a system of nonlinear singularly perturbed differential equations arising in population dynamics

Abstract A nonlinear singularly perturbed boundary value problem depending on a parameter is considered. First, we solve the problem using the backward Euler finite difference scheme on an adaptive grid. The adaptive grid is a special nonuniform mesh generated through equidistribution principle by a positive monitor function depending on the solution. The behavior of the solution, the stability and the error estimates are discussed. Then, the Richardson extrapolation technique is applied to improve the accuracy of the computed solution associated to the backward Euler scheme. The proofs of the uniform convergence for the backward Euler scheme and the Richardson extrapolation are carried out. Numerical experiments validate the theoretical estimates and indicates that the estimates are sharp.

A second order numerical method for a class of parameterized singular perturbation problems on adaptive grid

Layer-adapted Meshes for Parameterized Singular Perturbation Problem

In this article, a weighted finite difference scheme is proposed for solving a class of parameterized singularly perturbed problems (SPPs). Depending upon the choice of the weight parameter, the scheme is automatically transformed fromthe backward Euler scheme to amonotone hybrid scheme. Three kinds of nonuniform grids are considered, especially the standard Shishkin mesh, the Bakhavalov–Shishkinmesh and the adaptive grid. Themethods are shown to be uniformly convergent with respect to the perturbation parameter for all three types of meshes. The rate of convergence is of first order for the backward Euler scheme and second order for themonotone hybrid scheme. Furthermore, the proposed method is extended to a parameterized problem with mixed type boundary conditions and is shown to be uniformly convergent. Numerical experiments are carried out to show the efficiency of the proposed schemes, which indicate that the estimates are optimal.

Parameter-Uniform Numerical Methods for a Class of Parameterized Singular Perturbation Problems

This article deals with a singularly perturbed quasilinear boundary value problem with integral boundary condition which arises in neural network. The problem is discretised by using an upwind finite difference scheme on a non-uniform mesh obtained via equidistribution of a monitor function. We prove that the method is first order convergent in the discrete maximum norm independent of perturbation parameter. The parameter uniform convergence is confirmed by numerical computations.

D. Shakti

Papers

Numerical simulation and convergence analysis for a system of nonlinear singularly perturbed differential equations arising in population dynamics

A second order numerical method for a class of parameterized singular perturbation problems on adaptive grid

Layer-adapted Meshes for Parameterized Singular Perturbation Problem

Parameter-Uniform Numerical Methods for a Class of Parameterized Singular Perturbation Problems

A uniformly convergent numerical scheme for singularly perturbed differential equation with integral boundary condition arising in neural network