Paper Info
Title
Difference Schemes On Uniform Grids For An Initial-Boundary Value Problem For A Singularly Perturbed Parabolic Convection-Diffusion Equation
Abstract
The convergence of difference schemes on uniform grids for an initial-boundary value problem for a singularly perturbed parabolic convection-diffusion equation is studied; the highest x-derivative in the equation is multiplied by a perturbation parameter epsilon taking arbitrary values in the interval (0, 1]. For small epsilon, the problem involves a boundary layer of width O(epsilon), where the solution changes by a finite value, while its derivative grows unboundedly as epsilon tends to zero. We construct a standard difference scheme on uniform meshes based on the classical monotone grid approximation (upwind approximation of the first-order derivatives). Using a priori estimates, we show that such a scheme converges as (epsilon N), N-0 -> infinity in the maximum norm with first-order accuracy in (epsilon N) and N-0; as N, N-0 -> infinity, the convergence is conditional with respect to N, where N + 1 and N-0 + 1 are the numbers of mesh points in x and t, respectively. We develop an improved difference scheme on uniform meshes using the grid approximation of the first x-derivative in the convective term by the central difference operator under the condition h <= m epsilon, which ensures the monotonicity of the scheme; here m is some rather small positive constant. It is proved that this scheme converges in the maximum norm at a rate of O(epsilon N--2(-2) + N-0(-1)) We compare the convergence rate of the developed scheme with the known Samarskii scheme for a regular problem. It is found that the improved scheme (for epsilon = 1), as well as the Samarskii scheme, converges in the maximum norm with second-order accuracy in x and first-order accuracy in t.
Year
DOI
Venue
2020
10.1515/cmam-2019-0023
COMPUTATIONAL METHODS IN APPLIED MATHEMATICS
Keywords
DocType
Volume
Singularly Perturbed Initial-Boundary Value Problem, Perturbation Parameter, Parabolic Convection-Diffusion Equation, Boundary Layer, Finite Difference Scheme on Uniform Meshes, Improved Scheme, Conditional Monotonicity and Convergence, Maximum Norm, Samarskii Scheme
Journal
20
Issue
ISSN
Citations 
4
1609-4840
0
PageRank 
References 
Authors
0.34
0
2
Name
Order
Citations
PageRank
Grigorii I. Shishkin15215.80
Lidia P. Shishkina212.12