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Volume 10, Issue 4
Grid Approximation of a Singularly Perturbed Parabolic Equation with Degenerating Convective Term and Discontinuous Right-Hand Side

C. Clavero, J. L. Gracia, G. I. Shishkin & L. P. Shishkina

Int. J. Numer. Anal. Mod., 10 (2013), pp. 795-814.

Published online: 2013-10

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  • Abstract

The grid approximation of an initial-boundary value problem is considered for a singularly perturbed parabolic convection-diffusion equation with a convective flux directed from the lateral boundary inside the domain in the case when the convective flux degenerates inside the domain and the right-hand side has the first kind discontinuity on the degeneration line. The high-order derivative in the equation is multiplied by $\varepsilon^2$, where $\varepsilon$ is the perturbation parameter, $\varepsilon\in (0,1]$. For small values of $\varepsilon$, an interior layer appears in a neighbourhood of the set where the right-hand side has the discontinuity. A finite difference scheme based on the standard monotone approximation of the differential equation in the case of uniform grids converges only under the condition $N^{-1} = o(\varepsilon)$, $N^{-1}_0 = o(1)$, where $N +1$ and $N_0+1$ are the numbers of nodes in the space and time meshes, respectively. A finite difference scheme is constructed on a piecewise-uniform grid condensing in a neighbourhood of the interior layer. The solution of this scheme converges $\varepsilon$-uniformly at the rate $\mathcal{O}(N^{-1}lnN+N^{-1}_0)$. Numerical experiments confirm the theoretical results.

  • Keywords

parabolic convection-diffusion equation, perturbation parameter, degenerating convective term, discontinuous right-hand side, interior layer, technique of derivation to a priori estimates, piecewise-uniform grids , finite difference scheme, $\varepsilon$-uniform convergence, maximum norm.

  • AMS Subject Headings

65M06, 65N06, 65N12

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-10-795, author = {Clavero , C.Gracia , J. L.Shishkin , G. I. and Shishkina , L. P.}, title = {Grid Approximation of a Singularly Perturbed Parabolic Equation with Degenerating Convective Term and Discontinuous Right-Hand Side}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2013}, volume = {10}, number = {4}, pages = {795--814}, abstract = {

The grid approximation of an initial-boundary value problem is considered for a singularly perturbed parabolic convection-diffusion equation with a convective flux directed from the lateral boundary inside the domain in the case when the convective flux degenerates inside the domain and the right-hand side has the first kind discontinuity on the degeneration line. The high-order derivative in the equation is multiplied by $\varepsilon^2$, where $\varepsilon$ is the perturbation parameter, $\varepsilon\in (0,1]$. For small values of $\varepsilon$, an interior layer appears in a neighbourhood of the set where the right-hand side has the discontinuity. A finite difference scheme based on the standard monotone approximation of the differential equation in the case of uniform grids converges only under the condition $N^{-1} = o(\varepsilon)$, $N^{-1}_0 = o(1)$, where $N +1$ and $N_0+1$ are the numbers of nodes in the space and time meshes, respectively. A finite difference scheme is constructed on a piecewise-uniform grid condensing in a neighbourhood of the interior layer. The solution of this scheme converges $\varepsilon$-uniformly at the rate $\mathcal{O}(N^{-1}lnN+N^{-1}_0)$. Numerical experiments confirm the theoretical results.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/596.html} }
TY - JOUR T1 - Grid Approximation of a Singularly Perturbed Parabolic Equation with Degenerating Convective Term and Discontinuous Right-Hand Side AU - Clavero , C. AU - Gracia , J. L. AU - Shishkin , G. I. AU - Shishkina , L. P. JO - International Journal of Numerical Analysis and Modeling VL - 4 SP - 795 EP - 814 PY - 2013 DA - 2013/10 SN - 10 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/596.html KW - parabolic convection-diffusion equation, perturbation parameter, degenerating convective term, discontinuous right-hand side, interior layer, technique of derivation to a priori estimates, piecewise-uniform grids KW - , finite difference scheme, $\varepsilon$-uniform convergence, maximum norm. AB -

The grid approximation of an initial-boundary value problem is considered for a singularly perturbed parabolic convection-diffusion equation with a convective flux directed from the lateral boundary inside the domain in the case when the convective flux degenerates inside the domain and the right-hand side has the first kind discontinuity on the degeneration line. The high-order derivative in the equation is multiplied by $\varepsilon^2$, where $\varepsilon$ is the perturbation parameter, $\varepsilon\in (0,1]$. For small values of $\varepsilon$, an interior layer appears in a neighbourhood of the set where the right-hand side has the discontinuity. A finite difference scheme based on the standard monotone approximation of the differential equation in the case of uniform grids converges only under the condition $N^{-1} = o(\varepsilon)$, $N^{-1}_0 = o(1)$, where $N +1$ and $N_0+1$ are the numbers of nodes in the space and time meshes, respectively. A finite difference scheme is constructed on a piecewise-uniform grid condensing in a neighbourhood of the interior layer. The solution of this scheme converges $\varepsilon$-uniformly at the rate $\mathcal{O}(N^{-1}lnN+N^{-1}_0)$. Numerical experiments confirm the theoretical results.

C. Clavero, J. L. Gracia, G. I. Shishkin & L. P. Shishkina. (1970). Grid Approximation of a Singularly Perturbed Parabolic Equation with Degenerating Convective Term and Discontinuous Right-Hand Side. International Journal of Numerical Analysis and Modeling. 10 (4). 795-814. doi:
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