A parameter uniform fitted mesh method for a weakly coupled system of two singularly perturbed convection-diffusion equations

In this paper, a boundary value problem for a singularly perturbed linear system of two second order ordinary differential equations of convection- diffusion type is considered on the interval [0, 1]. The components of the solution of this system exhibit boundary layers at 0. A numerical method composed of an upwind finite difference scheme applied on a piecewise uniform Shishkin mesh is suggested to solve the problem. The method is proved to be first order convergent in the maximum norm uniformly in the perturbation parameters. Numerical examples are provided in support of the theory

Second order parameter-uniform convergence for a finite difference method for a partially singularly perturbed linear parabolic system

A linear system of $n$ second order differential equations of parabolic reaction-diffusion type with initial and boundary conditions is considered. The first $k$ equations are singularly perturbed. Each of the leading terms of the first $m$ equations, $mleq k$, is multiplied by a small positive parameter and these parameters are assumed to be distinct. The leading terms of the next $k-m$ equations are multiplied by the same perturbation parameter $varepsilon_m$. Since the components of the solution exhibit overlapping layers, Shishkin piecewise-uniform meshes are introduced, which are used in conjunction with a classical finite difference discretisation, to construct a numerical method for solving this problem. It is proved that in the maximum norm the numerical approximations obtained with this method are first order convergent in time and essentially second order convergent in the space variable, uniformly with respect to all of the parameters

Second order parameter-uniform convergence for a finite difference method for a partially singularly perturbed linear parabolic system

A linear system of $n$ second order differential equations of parabolic reaction-diffusion type with initial and boundary conditions is considered. The first $k$ equations are singularly perturbed. Each of the leading terms of the first $m$ equations, $mleq k$, is multiplied by a small positive parameter and these parameters are assumed to be distinct. The leading terms of the next $k-m$ equations are multiplied by the same perturbation parameter $varepsilon_m$. Since the components of the solution exhibit overlapping layers, Shishkin piecewise-uniform meshes are introduced, which are used in conjunction with a classical finite difference discretisation, to construct a numerical method for solving this problem. It is proved that in the maximum norm the numerical approximations obtained with this method are first order convergent in time and essentially second order convergent in the space variable, uniformly with respect to all of the parameters

A parameter uniform fitted mesh method for a weakly coupled system of two singularly perturbed convection-diffusion equations

In this paper, a boundary value problem for a singularly perturbed linear system of two second order ordinary differential equations of convection-diffusion type is considered on the interval [0, 1]. The components of the solution of this system exhibit boundary layers at 0. A numerical method composed of an upwind finite difference scheme applied on a piecewise uniform Shishkin mesh is suggested to solve the problem. The method is proved to be first order convergent in the maximum norm uniformly in the perturbation parameters. Numerical examples are provided in support of the theory

Second order parameter-uniform numerical method for a partially singularly perturbed linear system of reaction-diusion type

A partially singularly perturbed linear system of second order ordinary differential equations of reaction-diffusion type with given boundary conditions is considered. The leading terms of first $m$ equations are multiplied by small positive singular perturbation parameters which are assumed to be distinct. The rest of the equations are not singularly perturbed. The first $m$ components of the solution exhibit overlapping layers and the remaining $n-m$ components have less-severe overlapping layers. Shishkin piecewise-uniform meshes are used in conjunction with a classical finite difference discretisation, to construct a numerical method for solving this problem. It is proved that the numerical approximation obtained by this method is essentially second order convergent uniformly with respect to all the parameters. Numerical illustrations are presented in support of the theory

A parameter uniform almost first order convergent numerical method for non-linear system of singularly perturbed differential equations

In this paper an initial value problem for a non-linear system of two singularly perturbed first order differential equations is considered on the interval (0,1]. The components of the solution of this system exhibit initial layers at 0. A numerical method composed of a classical finite difference scheme on a piecewise uniform Shishkin mesh is suggested. This method is proved to be almost first order convergent in the maximum norm uniformly in the perturbation parameters

Second order parameter-uniform numerical method for a partially singularly perturbed linear system of reaction-diusion type

A partially singularly perturbed linear system of second order ordinary differential equations of reaction-diffusion type with given boundary conditions is considered. The leading terms of first $m$ equations are multiplied by small positive singular perturbation parameters which are assumed to be distinct. The rest of the equations are not singularly perturbed. The first $m$ components of the solution exhibit overlapping layers and the remaining $n-m$ components have less-severe overlapping layers. Shishkin piecewise-uniform meshes are used in conjunction with a classical finite difference discretisation, to construct a numerical method for solving this problem. It is proved that the numerical approximation obtained by this method is essentially second order convergent uniformly with respect to all the parameters. Numerical illustrations are presented in support of the theory

A parameter uniform fitted mesh method for a weakly coupled system of two singularly perturbed convection-diffusion equations

In this paper, a boundary value problem for a singularly perturbed linear system of two second order ordinary differential equations of convection-diffusion type is considered on the interval [0, 1]. The components of the solution of this system exhibit boundary layers at 0. A numerical method composed of an upwind finite difference scheme applied on a piecewise uniform Shishkin mesh is suggested to solve the problem. The method is proved to be first order convergent in the maximum norm uniformly in the perturbation parameters. Numerical examples are provided in support of the theory

International Winter Workshop on Differential Equations and Numerical Analysis

This book offers an ideal introduction to singular perturbation problems, and a valuable guide for researchers in the field of differential equations. It also includes chapters on new contributions to both fields: differential equations and singular perturbation problems. Written by experts who are active researchers in the related fields, the book serves as a comprehensive source of information on the underlying ideas in the construction of numerical methods to address different classes of problems with solutions of different behaviors, which will ultimately help researchers to design and assess numerical methods for solving new problems. All the chapters presented in the volume are complemented by illustrations in the form of tables and graphs