Mesfin Mekuria Woldaregay, Gemechis File Duressa
abstract:
This paper deals with the numerical treatment of singularly perturbed parabolic
differential-difference equations. The considered equations contain a small
perturbation parameter $\varepsilon\in (0,1]$ multiplied by the highest order
derivative term, and shift parameters attached with the nonderivative terms. The
solution of the equations exhibits an exponential boundary layer due to the
presence of the perturbation parameter $\varepsilon$. Classical numerical
methods fail to give relevant approximate solutions when the perturbation
parameter approaches zero. We propose numerical schemes that converge uniformly
irrespective of the parameter $\varepsilon$. The numerical schemes are
formulated by using the Crank Nicolson method in temporal discretization, and
the midpoint upwind non-standard finite difference method on uniform mesh and
Shishkin mesh for spatial discretization. The schemes satisfy the discrete
maximum principle and the uniform stability estimate. The uniform convergence of
the schemes is proved with the second order of convergence in the temporal
direction and with the first order of convergence in the spatial direction.
Numerical test examples are considered for validating the theoretical findings
and analysis of the schemes.
Mathematics Subject Classification: 65M06, 65M12, 65M15
Key words and phrases: Midpoint upwind, singularly perturbed, differential difference, uniform convergence, Shishkin mesh