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Difference scheme for an initial-boundary value problem for a singularly perturbed transport equation.

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Title: Difference scheme for an initial-boundary value problem for a singularly perturbed transport equation.
Authors: Shishkin, G.1 shishkin@imm.uran.ru
Source: Computational Mathematics & Mathematical Physics. Nov2017, Vol. 57 Issue 11, p1789-1795. 7p.
Subjects: Initial value problems, Difference algebra, Transport theory, Perturbation theory, Stochastic convergence, Interval analysis
Abstract: 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 ∪ S, and S and S 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 ; 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 ≪ ε and N ≪ 1, where N and N 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 + N ) rate. [ABSTRACT FROM AUTHOR]
Copyright of Computational Mathematics & Mathematical Physics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: 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 ∪ S, and S and S 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 ; 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 ≪ ε and N ≪ 1, where N and N 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 + N ) rate. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Computational Mathematics & Mathematical Physics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1134/S0965542517110136
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        Text: English
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