Uniformly Convergent Numerical Scheme for Singularly Perturbed Parabolic PDEs with Shift Parameters
Mesfin Mekuria Woldaregay, Gemechis File Duressa
Abstract
In this article, singularly perturbed parabolic differential difference equations are considered. The solution of the equations exhibits a boundary layer on the right side of the spatial domain. The terms containing the advance and delay parameters are approximated using Taylor series approximation. The resulting singularly perturbed parabolic PDEs are solved using the Crank–Nicolson method in the temporal discretization and nonstandard finite difference method in the spatial discretization. The existence of a unique discrete solution is guaranteed using the discrete maximum principle. The uniform stability of the scheme is investigated using solution bound. The uniform convergence of the scheme is discussed and proved. The scheme converges uniformly with the order of convergence <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"><a:mi>O</a:mi><a:mfenced open="(" close=")" separators="|"><a:mrow><a:msup><a:mrow><a:mi>N</a:mi></a:mrow><a:mrow><a:mo>−</a:mo><a:mn>1</a:mn></a:mrow></a:msup><a:mo>+</a:mo><a:msup><a:mrow><a:mfenced open="(" close=")" separators="|"><a:mrow><a:mi mathvariant="normal">Δ</a:mi><a:mi>t</a:mi></a:mrow></a:mfenced></a:mrow><a:mrow><a:mn>2</a:mn></a:mrow></a:msup></a:mrow></a:mfenced></a:math> , where <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" id="M2"><j:mi>N</j:mi></j:math> is number of subintervals in spatial discretization and <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" id="M3"><l:mi mathvariant="normal">Δ</l:mi><l:mi>t</l:mi></l:math> is mesh length in temporal discretization. Two test numerical examples are considered to validate the theoretical findings of the scheme.