463 research outputs found
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
Robust Numerical Methods for Singularly Perturbed Differential Equations--Supplements
The second edition of the book "Roos, Stynes, Tobiska -- Robust Numerical
Methods for Singularly Perturbed Differential Equations" appeared many years
ago and was for many years a reliable guide into the world of numerical methods
for singularly perturbed problems. Since then many new results came into the
game, we present some selected ones and the related sources.Comment: arXiv admin note: text overlap with arXiv:1909.0827
An Upwind Finite Difference Method to Singularly Perturbed Convection Diffusion Problems on a Shishkin Mesh
This paper introduces a numerical approach to solve singularly perturbed
convection diffusion boundary value problems for second-order ordinary
differential equations that feature a small positive parameter {\epsilon}
multiplying the highest derivative. We specifically examine Dirichlet boundary
conditions. To solve this differential equation, we propose an upwind finite
difference method and incorporate the Shishkin mesh scheme to capture the
solution near boundary layers. Our solver is both direct and of high accuracy,
with computation time that scales linearly with the number of grid points.
MATLAB code of the numerical recipe is made publicly available. We present
numerical results to validate the theoretical results and assess the accuracy
of our method. The tables and graphs included in this paper demonstrate the
numerical outcomes, which indicate that our proposed method offers a highly
accurate approximation of the exact solution.Comment: 19 pages, 4 figures. arXiv admin note: text overlap with
arXiv:2305.18711 by other author
A numerical approach for a two-parameter singularly perturbed weakly-coupled system of 2-D elliptic convection–reaction–diffusion PDEs
In this work, we consider the numerical approximation of a two dimensional elliptic singularly perturbed weakly-coupled system of convection–reaction–diffusion type, which has two different parameters affecting the diffusion and the convection terms, respectively. The solution of such problems has, in general, exponential boundary layers as well as corner layers. To solve the continuous problem, we construct a numerical method which uses a finite difference scheme defined on an appropriate layer-adapted Bakhvalov–Shishkin mesh. Then, the numerical scheme is a first order uniformly convergent method with respect both convection and diffusion parameters. Numerical results obtained with the algorithm for some test problems are presented, which show the best performance of the proposed method, and they also corroborate in practice the theoretical analysis
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