A Parameter-Uniform Finite Difference Method for Multiscale Singularly Perturbed Linear Dynamical Systems

2009/03/10 by Valarmathi, S, Miller, John J H
#65L05 #65L12 #65L20 #65L70 #Dynamical Systems (math.DS) #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.0903.1795

Abstract

A system of singularly perturbed ordinary differential equations of first order with given initial conditions is considered. The leading term of each equation is multiplied by a small positive parameter. These parameters are assumed to be distinct and they determine the different scales in the solution to this problem. A Shishkin piecewise--uniform mesh is constructed, which is used, in conjunction with a classical finite difference discretization, to form a new numerical method for solving this problem. It is proved that the numerical approximations obtained from this method are essentially first order convergent uniformly in all of the parameters.

Related

• A parameter--uniform finite difference method for a singularly perturbed linear system of second order ordinary differential equations of reaction-diffusion type
• Second order parameter-uniform convergence for a finite difference method for a singularly perturbed linear reaction-diffusion system
• Numerical approximations to the scaled first derivatives of a two parameter singularly perturbed problem
• Second order parameter-uniform convergence for a finite difference method for a singularly perturbed linear parabolic system
• A parameter uniform fitted mesh method for a weakly coupled system of two singularly perturbed convection-diffusion equations