Uniformly Convergent Numerical Approximation for Parabolic Singularly Perturbed Delay Problems with Turning Points

Abstract

We construct and analyze a second-order parameter uniform numerical method for parabolic singularly perturbed space-delay problems with interior turning point. The considered problem’s solution possesses an interior layer in addition to twin boundary layers due to the presence of delay. Some theoretical estimates on derivatives of the analytical solution, which are useful for conducting the error analysis, are given. The proposed technique employs an upwind scheme on a fitted Bakhvalov–Shishkin mesh in the spatial variable and implicit-Euler scheme on a uniform mesh in the time variable. This discretization of the problem is shown to be uniformly convergent of [Formula: see text], where [Formula: see text] is the step size in the temporal direction and [Formula: see text] denotes the number of mesh-intervals in the spatial direction. Further, to improve the accuracy, we make use of Richardson extrapolation and establish parameter-uniform convergence of [Formula: see text] for the resulting scheme. Numerical experiments are performed over two test problems for validation of the theoretical predictions.

Numerical approximation of parabolic singularly perturbed problems with large spatial delay and turning point

“…A comparison of maximum «-uniform error and the corresponding order of convergence obtained using the proposed numerical method with those outlined in Sharma et al (2023) for the Problems (5.1) and (5.2) is presented in Tables 1 and 2, respectively. The numerical solutions obtained for the Problems 5.1 and 5.2 are plotted in Figures 1 and 2, respectively.…”