Affiliation:
1. Krasovskii Institute of Mathematics and Mechanics , Ural Branch, Russian Academy of Sciences, 620990 Yekaterinburg , Russia
Abstract
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 ε taking arbitrary values in the interval
(
0
,
1
]
{(0,1]}
.
For small ε, the problem involves a boundary layer of width
𝒪
(
ε
)
{\mathcal{O}(\varepsilon)}
, where the solution changes by a finite value, while its derivative grows unboundedly as ε 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
{
ε
N
}
,
N
0
→
∞
{\{\varepsilon N\},N_{0}\to\infty}
in the maximum norm with first-order accuracy in
{
ε
N
}
{\{\varepsilon N\}}
and
N
0
{N_{0}}
; as
N
,
N
0
→
∞
{N,N_{0}\to\infty}
, the convergence is conditional with respect to
N, where
N
+
1
{N+1}
and
N
0
+
1
{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
ε
{h\leq m\varepsilon}
, 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
(
ε
-
2
N
-
2
+
N
0
-
1
)
{\mathcal{O}(\varepsilon^{-2}N^{-2}+N^{-1}_{0})}
.
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
ε
=
1
{\varepsilon=1}
), as well as the Samarskii scheme, converges in the maximum norm with second-order accuracy in
x and first-order accuracy in
t.
Subject
Applied Mathematics,Computational Mathematics,Numerical Analysis
Reference6 articles.
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Robust Computational Techniques for Boundary Layers,
Chapman & Hall/CRC Press, Boca Raton, 2000.
2. J. J. H. Miller, E. O’Riordan and G. I. Shishkin,
Fitted Numerical Methods for Singular Perturbation Problems. Error Estimates in Maximum Norm for Linear Problems in One and Two Dimentions,
World Scientific, Singapore, 2012.
3. A. A. Samarskii,
Monotonic difference schemes for elliptic and parabolic equations in the case of a non-selfadjoint elliptic operator,
USSR Comput. Math. Math. Phys. 5 (1965), no. 3, 212–217.
4. A. A. Samarskii,
The Theory of Difference Schemes (in Russian), 3rd ed.,
Nauka, Moscow, 1989.
5. G. I. Shishkin,
Discrete Approximations of Singularly Perturbed Elliptic and Parabolic Equations (in Russian),
Russian Academy of Sciences, Ekaterinburg, 1992.
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