Fitted computational method for solving singularly perturbed small time lag problem

Abstract

Objectives An accurate exponentially fitted numerical method is developed to solve the singularly perturbed time lag problem. The solution to the problem exhibits a boundary layer as the perturbation parameter approaches zero. A priori bounds and properties of the continuous solution are discussed. Result The backward-Euler method is applied in the time direction and the higher order finite difference method is employed for the spatial derivative approximation. An exponential fitting factor is induced on the difference scheme for stabilizing the computed solution. Using the comparison principle, the stability of the method is examined and analyzed. It is proved that the method converges uniformly with linear order of convergence. To validate the theoretical findings and analysis, two test examples are given. Comparison is made with the results available in the literature. The proposed method has better accuracy than the schemes in the literature.

Hybrid Fitted Numerical Scheme for Singularly Perturbed Convection-Diffusion Problem with a Small Time Lag

“…The proposed hybrid scheme is a combination of the cubic spline method and the midpoint upwind scheme on piecewise Shishkin mesh in the spatial direction and the Crank-Nicolson method in the temporal direction. The advantage of the present method over the other methods in [17][18][19] is that it is a second-order accurate in both time and space variables, as well as its accuracy. In addition, in the paper [17][18][19], Taylor's series expansion is applied at the beginning without any restriction on the domain.…”

“…The advantage of the present method over the other methods in [17][18][19] is that it is a second-order accurate in both time and space variables, as well as its accuracy. In addition, in the paper [17][18][19], Taylor's series expansion is applied at the beginning without any restriction on the domain. In such a case, the advantage of the interval condition for the delay term in the approximation is meaningless.…”

“…In this paper, both small and large time delays are considered, and the developed scheme is first-order convergent. In [19], the scheme is developed using the backward Euler method in the discretization of the time derivative, and a higher-order finite difference method is employed for the approximation of the spatial derivative. In this scheme, an exponential fitting factor is introduced, and the resulting scheme is first-order convergent.…”

Singular Perturbations and Large Time Delays Through Accelerated Spline-Based Compression Technique

“…An almost first order convergent finite difference scheme by using piecewise Shishkin type mesh is presented in [26] and an exponentially fitted finite difference method is suggested in [27] to tackle the problem. The works in [28][29][30][31][32][33][34][35] also give the approximate solution of these kind of problems with different numerical approaches.…”

Fitted computational method for singularly perturbed convection-diffusion equation with time delay

“…Podila and Kumar [20] proposed a new stable finite difference scheme on a uniform mesh and also on an adaptive mesh. The backward Euler scheme in the time direction and exponentially fitted difference method is considered in [21]. The Crank-Nicolson method in the time direction and a novel fitted finite difference scheme in spatial direction are proposed in [22].…”