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 - <p style="text-align: justify;">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.</p>