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

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 O(Delta tau+kappa(-1)), where Delta tau is the step size in the temporal direction and K 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 O((Delta tau)(2) + kappa(-2)) for the resulting scheme. Numerical experiments are performed over two test problems for validation of the theoretical predictions.