A uniformly convergent quadratic B-spline collocation method for singularly perturbed parabolic partial differential equations with two small parameters

Abstract

This paper aims to construct a parameters-uniform numerical scheme to solve the singularly perturbed parabolic partial differential equations whose solution exhibits parabolic (or exponential) boundary layers at both the lateral surfaces of the rectangular domain. The method comprises an implicit Euler scheme on a uniform mesh in the temporal direction and the quadratic B-spline collocation scheme on an exponentially graded mesh in the spatial direction. The exponentially graded mesh is generated by choosing an appropriate mesh generating function which adapts the mesh points in the boundary layers appear in the spatial direction. To establish the error estimates the solution is decomposed into its regular and singular components and the error estimates for these components are obtained separately. We prove the parameters-uniform convergence of the proposed numerical scheme and the method is shown to be of $${\mathcal {O}}(N^{-2}_{x}+\Delta t)$$ where $$N_{x}$$ denotes the number of mesh points in the space direction and $$\Delta t$$ is the mesh step size in the temporal direction. To support the obtained theoretical estimates, two test examples are considered numerically.