A Parameter-Uniform Tailored Finite Point Method for Singularly Perturbed Linear ODE Systems

Abstract

In scientific applications from plasma to chemical kinetics, a wide range of temporal scales can present in a system of differential equations. A major difficulty is encountered due to the stiffness of the system and it is required to develop fast numerical schemes that are able to access previously unattainable parameter regimes. In this work, we consider an initial-final value problem for a multi-scale singularly perturbed system of linear ordinary differential equations with discontinuous coefficients. We construct a tailored finite point method, which yields approximate solutions that converge in the maximum norm, uniformly with respect to the singular perturbation parameters, to the exact solution. A parameter-uniform error estimate in the maximum norm is also proved. The results of numerical experiments, that support the theoretical results, are reported.Mathematics subject classification: 37M05, 65G99.

The Tailored Finite Point Method

“…We also proposed TFPM for the anisotropic di usion problems [22] and the ODEs with di erent parameters [24]. We proved the uniform convergence of our TFPM for those problems.…”

“…After ( , ) is obtained, the corresponding eigenvector ( , ) is given by the following two cases [24]:…”

“…A uniformly convergent semi-discrete TFPM for a class of anisotropic di usion problems was proposed by Han, Huang and Ying [22]. A parameter-uniform TFPM for a singularly perturbed linear ODE system with multiple perturbation parameters was obtained by Han, Miller and Tang [24]. Two parameter-uniform tailored nite point schemes for two-dimensional discrete ordinary di erential equations with boundary layers and interfaces were given by Han, Tang and Ying [25], in which the problem can be treated as a singularly perturbed problem of a system of ordinary di erential equations.…”

An Anisotropic Convection-Diffusion Model Using Tailored Finite Point Method for Image Denoising and Compression

“…Due to this special feature, many applications including convection-dominated convection-diffusion-reaction problems [19], first order wave equation [12] and steady magnetohydrodynamics duct flow problem [11] have been successfully implemented. More recently, TFPM demonstrates the effectiveness in solving singularly perturbed problems [9,10]. Here we will present an anisotropic CD (ACD) filter with the advantage of the TFPM for image denoising.…”

Tailored Finite Point Method for Numerical Solutions of Singular Perturbed Eigenvalue Problems

“…10) and the general solution is expressed by u(r,θ) = a 0 I 0 (τr)+ ∞ ∑ n=1 a k I k (τr)cos(kθ)+b k I k (τr)sin(kθ) ,(3.11) …”