Parameter-uniform hybrid numerical scheme for time-dependent convection-dominated initial-boundary-value problems

Uniformly convergent hybrid numerical scheme for singularly perturbed delay parabolic convection–diffusion problems on Shishkin mesh

“…For t ≥ τ , the approach of [6] is not applicable because, the term u(x, t) depends on u(x, t − τ ), which is unknown for t ≥ τ , explicitly. So, we will do the detailed proof to get the error over the interval [τ , 2τ ].…”

“…To cite a few: Natesan and Ramanujam [9] considered a boundary-value problem (BVP) for singularly perturbed ODE, and provided a booster method, which incorporates an asymptotic expansion into a numerical method and give higher-order accuracy. To obtain higher-order ε-uniform convergent numerical solution for singularly perturbed parabolic PDEs, Mukherjee and Natesan proposed the hybrid scheme in [6,7] and applied the Richardson extrapolation technique in [8].…”

ε-Uniform error estimate of hybrid numerical scheme for singularly perturbed parabolic problems with nterior layers

“…Optimal error estimates for an upwind finite difference scheme on Shishkin-type meshes are obtained in [18] for singularly perturbed parabolic PDEs with discontinuous convection coefficients. In this context, it is also be noted that we have developed a similar hybrid scheme in [19] to examine the robustness of the method for timedependent convection-dominated problems with continuous data. Further, in [20], we proposed a Richardson extrapolation method to yield second-order convergent solutions for singularly perturbed parabolic IBVPs.…”

A simpler analysis of a hybrid numerical method for time-dependent convection–diffusion problems

“…In the present paper, we focus on two finite difference methods for (1) that were presented and analysed in recent papers in [2,3]. Convergence, uniformly in ε, is proved for these methods in these papers under the restriction that b = b(x).…”

“…Both papers use the same mesh (equidistant mesh in time with mesh spacing τ , piecewise-equidistant Shishkin mesh in space with N mesh intervals) and backward Euler differencing to approximate the time derivative, but their spatial discretizations seem to be different: Clavero et al use the second-order HODIE scheme from [4] while Mukherjee and Natesan favour the hybrid difference scheme of [5]. In Section 3, we shall show that in fact the methods of [2,3] are essentially identical, despite the claim in [3,Introduction] that the method of [3] is simpler than that of [2]. We do this for the more general case…”