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result(s) for
"Shishkin, G. I"
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A Difference Scheme of the Decomposition Method for an Initial Boundary Value Problem for the Singularly Perturbed Transport Equation
AbstractAn initial boundary value problem for the singularly perturbed transport equation is considered. A new approach to constructing the difference scheme based on a special decomposition of solution into the sum of a regular and a singular components is proposed. A difference scheme is constructed based on the solution decomposition method in which the regular and singular components of the solution are considered on uniform grids, and their \\( \\)-uniform convergence in the maximum norm with the first order of the convergence rate is proved. Given the grid solutions for the components, a continual solution that approximates the solution of the initial boundary value problem for the singularly perturbed transport equation is constructed, and its \\( \\)-uniform convergence in the maximum norm with the first order of the convergence rate is proved. The proposed approach will make it possible to use the technique of improving the convergence rate of grid solutions on embedded grids for constructing difference schemes that converge \\( \\)-uniformly with the second-order rate and higher for the initial boundary value problem for the singularly perturbed transport equation.
Journal Article
Difference scheme for an initial–boundary value problem for a singularly perturbed transport equation
An initial–boundary value problem for a singularly perturbed transport equation with a perturbation parameter ε multiplying the spatial derivative is considered on the set
Ḡ
=
G
∪
S
, where
Ḡ
=
D̅
× [0 ≤
t
≤
T
],
D̅
= {0 ≤
x
≤
d
},
S
=
S
l
∪
S
, and
S
l
and
S
0
are the lateral and lower boundaries. The parameter ε takes arbitrary values from the half-open interval (0,1]. In contrast to the well-known problem for the regular transport equation, for small values of ε, this problem involves a boundary layer of width
O
(ε) appearing in the neighborhood of
S
l
; in the layer, the solution of the problem varies by a finite value. For this singularly perturbed problem, the solution of a standard difference scheme on a uniform grid does not converge ε-uniformly in the maximum norm. Convergence occurs only if
h
=
dN
-1
≪ ε and
N
0
-1
≪ 1, where
N
and
N
0
are the numbers of grid intervals in
x
and
t
, respectively, and
h
is the mesh size in
x
. The solution of the considered problem is decomposed into the sum of regular and singular components. With the behavior of the singular component taken into account, a special difference scheme is constructed on a Shishkin mesh, i.e., on a mesh that is piecewise uniform in
x
and uniform in
t
. On such a grid, a monotone difference scheme for the initial–boundary value problem for the singularly perturbed transport equation converges ε-uniformly in the maximum norm at an Ϭ(
N
−1
+
N
0
−1
) rate.
Journal Article
Monotone Decomposition of the Cauchy Problem for a Hyperbolic Equation Based on Transport Equations
AbstractFor the Cauchy problem for a hyperbolic equation, a multiplicative approach is developed: a monotone decomposition of the problem is constructed since the hyperbolic operator can be represented by a product of transport operators. The problem for the hyperbolic equation is reduced to a system of problems for transport equations—transport in the direction of the axis \\(x\\) and transport in the opposite direction of the axis \\(x\\). Conditions for the monotonicity of each problem for the transport equations and for the entire multiplicative problem are found. Such a decomposition of the Cauchy problem based on transport problems solved one after the other significantly simplifies the solution of the hyperbolic equation, and the problems for the transport equations are monotone thus ensuring the monotonicity of the decomposition of the Cauchy problem for the hyperbolic equation.
Journal Article
Computer difference scheme for a singularly perturbed elliptic convection–diffusion equation in the presence of perturbations
A grid approximation of a boundary value problem for a singularly perturbed elliptic convection–diffusion equation with a perturbation parameter ε, ε ∈ (0,1], multiplying the highest order derivatives is considered on a rectangle. The stability of a standard difference scheme based on monotone approximations of the problem on a uniform grid is analyzed, and the behavior of discrete solutions in the presence of perturbations is examined. With an increase in the number of grid nodes, this scheme does not converge -uniformly in the maximum norm, but only conditional convergence takes place. When the solution of the difference scheme converges, which occurs if N
1
-1
N
2
-1
≪ ε, where
N
1
and
N
2
are the numbers of grid intervals in
x
and
y
, respectively, the scheme is not -uniformly well-conditioned or ε-uniformly stable to data perturbations in the grid problem and to computer perturbations. For the standard difference scheme in the presence of data perturbations in the grid problem and/or computer perturbations, conditions imposed on the “parameters” of the difference scheme and of the computer (namely, on ε,
N
1
,
N
2
, admissible data perturbations in the grid problem, and admissible computer perturbations) are obtained that ensure the convergence of the perturbed solutions as
N
1
,
N
2
→ ∞, ε ∈ (0,1]. The difference schemes constructed in the presence of the indicated perturbations that converges as
N
1
,
N
2
→ ∞ for fixed ε, ε ∈ (0,1, is called a
computer difference scheme
. Schemes converging ε-uniformly and conditionally converging computer schemes are referred to as
reliable schemes
. Conditions on the data perturbations in the standard difference scheme and on computer perturbations are also obtained under which the convergence rate of the solution to the computer difference scheme has the same order as the solution of the standard difference scheme in the absence of perturbations. Due to this property of its solutions, the computer difference scheme can be effectively used in practical computations.
Journal Article
Difference scheme for a singularly perturbed parabolic convection–diffusion equation in the presence of perturbations
An initial–boundary value problem is considered for a singularly perturbed parabolic convection–diffusion equation with a perturbation parameter ε (ε ∈ (0, 1]) multiplying the highest order derivative. The stability of a standard difference scheme based on monotone approximations of the problem on a uniform mesh is analyzed, and the behavior of discrete solutions in the presence of perturbations is examined. The scheme does not converge ε-uniformly in the maximum norm as the number of its grid nodes is increased. When the solution of the difference scheme converges, which occurs if
N
–1
≪ ε and
N
-1
0
≪ 1, where
N
and
N
0
are the numbers of grid intervals in
x
and
t
, respectively, the scheme is not ε-uniformly well conditioned or stable to data perturbations in the grid problem and to computer perturbations. For the standard difference scheme in the presence of data perturbations in the grid problem and/or computer perturbations, conditions on the “parameters” of the difference scheme and of the computer (namely, on ε,
N
,
N
0
, admissible data perturbations in the grid problem, and admissible computer perturbations) are obtained that ensure the convergence of the perturbed solutions. Additionally, the conditions are obtained under which the perturbed numerical solution has the same order of convergence as the solution of the unperturbed standard difference scheme.
Journal Article
Computer difference scheme for a singularly perturbed convection-diffusion equation
The Dirichlet problem for a singularly perturbed ordinary differential convection-diffusion equation with a perturbation parameter ɛ (that takes arbitrary values from the half-open interval (0, 1]) is considered. For this problem, an approach to the construction of a numerical method based on a standard difference scheme on uniform meshes is developed in the case when the data of the grid problem include perturbations and additional perturbations are introduced in the course of the computations on a computer. In the absence of perturbations, the standard difference scheme converges at an
(δ
st
) rate, where δ
st
= (ɛ +
N
−1
)
−1
N
−1
and
N
+ 1 is the number of grid nodes; the scheme is not ɛ-uniformly well conditioned or stable to perturbations of the data. Even if the convergence of the standard scheme is theoretically proved, the actual accuracy of the computed solution in the presence of perturbations degrades with decreasing ɛ down to its complete loss for small ɛ (namely, for ɛ =
(δ
−2
max
i
,
j
|δ
a
i
j
| + δ
−1
max
i
,
j
|δ
b
i
j
|), where δ = δ
st
and δ
a
i
j
, δ
b
i
j
are the perturbations in the coefficients multiplying the second and first derivatives). For the boundary value problem, we construct a computer difference scheme, i.e., a computing system that consists of a standard scheme on a uniform mesh in the presence of controlled perturbations in the grid problem data and a hypothetical computer with controlled computer perturbations. The conditions on admissible perturbations in the grid problem data and on admissible computer perturbations are obtained under which the computer difference scheme converges in the maximum norm for ɛ ∈ (0, 1] at the same rate as the standard scheme in the absence of perturbations.
Journal Article
Richardson’s Third-Order Difference Scheme for the Cauchy Problem in the Case of Transport Equation
The Cauchy problem for the regular transport equation is considered. Using Richardson’s technique, a difference scheme of improved accuracy order on three embedded grids is constructed for this problem. This scheme converges in the maximum norm with the third order of convergence rate.
Journal Article
Conditioning and stability of finite difference schemes on uniform meshes for a singularly perturbed parabolic convection-diffusion equation
For a singularly perturbed parabolic convection-diffusion equation, the conditioning and stability of finite difference schemes on uniform meshes are analyzed. It is shown that a convergent standard monotone finite difference scheme on a uniform mesh is not ɛ-uniformly well conditioned or ɛ-uniformly stable to perturbations of the data of the grid problem (here, ɛ is a perturbation parameter, ɛ ∈ (0, 1]). An alternative finite difference scheme is proposed, namely, a scheme in which the discrete solution is decomposed into regular and singular components that solve grid subproblems considered on uniform meshes. It is shown that this solution decomposition scheme converges ɛ-uniformly in the maximum norm at an
O
(
N
−1
ln
N
+
N
0
−1
) rate, where
N
+ 1 and
N
0
+ 1 are the numbers of grid nodes in
x
and
t
, respectively. This scheme is ɛ-uniformly well conditioned and ɛ-uniformly stable to perturbations of the data of the grid problem. The condition number of the solution decomposition scheme is of order
O
(δ
−2
lnδ
−1
+ δ
0
−1
); i.e., up to a logarithmic factor, it is the same as that of a classical scheme on uniform meshes in the case of a regular problem. Here, δ =
N
−1
ln
N
and δ
0
=
N
0
−1
are the accuracies of the discrete solution in
x
and
t
, respectively.
Journal Article