A uniformly convergent second order difference scheme for a singularly perturbed self-adjoint ordinary differential equation in conservation form

1989 ◽  
Vol 10 (3) ◽  
pp. 231-241 ◽  
Author(s):  
Guo Wen ◽  
Lin Peng-cheng
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Difference Scheme ◽  
Second Order ◽  
Singularly Perturbed ◽  
Order Difference ◽  
Uniformly Convergent

scholarly journals Uniform Hybrid Difference Scheme for Singularly Perturbed Differential-Difference Turning Point Problems Exhibiting Boundary Layers

2020 ◽  
Vol 2020 ◽  
pp. 1-14 ◽  
Author(s):  
Wondwosen Gebeyaw Melesse ◽  
Awoke Andargie Tiruneh ◽  
Getachew Adamu Derese
Keyword(s):  
Difference Scheme ◽  
Boundary Layers ◽  
Turning Point ◽  
Original Problem ◽  
Second Order ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problem ◽  
Uniformly Convergent ◽  
Theoretical Results ◽  
Taylor’S Series

In this paper, a class of linear second-order singularly perturbed differential-difference turning point problems with mixed shifts exhibiting two exponential boundary layers is considered. For the numerical treatment of these problems, first we employ a second-order Taylor’s series approximation on the terms containing shift parameters and obtain a modified singularly perturbed problem which approximates the original problem. Then a hybrid finite difference scheme on an appropriate piecewise-uniform Shishkin mesh is constructed to discretize the modified problem. Further, we proved that the method is almost second-order ɛ-uniformly convergent in the maximum norm. Numerical experiments are considered to illustrate the theoretical results. In addition, the effect of the shift parameters on the layer behavior of the solution is also examined.

scholarly journals ROBUST DIFFERENCE SCHEME FOR THE CAUCHY PROBLEM FOR A SINGULARLY PERTURBED ORDINARY DIFFERENTIAL EQUATION

2018 ◽  
Vol 23 (4) ◽  
pp. 527-537 ◽  
Author(s):  
Lidia Pavlovna Shishkina ◽  
Grigorii Ivanovich Shishkin
Keyword(s):  
Differential Equation ◽  
Boundary Layer ◽  
Ordinary Differential Equation ◽  
Cauchy Problem ◽  
Difference Scheme ◽  
Singularly Perturbed ◽  
Uniform Grid ◽  
Piecewise Uniform Grid ◽  
The Cauchy Problem ◽  
Standard Difference

Grid approximation of the Cauchy problem on the interval D = {0 ≤ x ≤ d} is first studied for a linear singularly perturbed ordinary differential equation of the first order with a perturbation parameter ε multiplying the derivative in the equation where the parameter ε takes arbitrary values in the half-open interval (0, 1]. In the Cauchy problem under consideration, for small values of the parameter ε, a boundary layer of width O(ε) appears on which the solution varies by a finite value. It is shown that, for such a Cauchy problem, the solution of the standard difference scheme on a uniform grid does not converge ε-uniformly in the maximum norm; convergence occurs only under the condition h ε, where h = d N −1 , N is the number of grid intervals, h is the grid step-size. Taking into account the behavior of the singular component in the solution, a special piecewise-uniform grid is constructed that condenses in a neighborhood of the boundary layer. It is established that the standard difference scheme on such a special grid converges ε-uniformly in the maximum norm at the rate O(N −1 lnN). Such a scheme is called a robust one. For a model Cauchy problem for a singularly perturbed ordinary differential equation, standard difference schemes on a uniform grid (a classical difference scheme) and on a piecewise-uniform grid (a special difference scheme) are constructed and investigated. The results of numerical experiments are given, which are consistent with theoretical results.

Sign in / Sign up

Export Citation Format

Share Document