Optimal error estimate of upwind scheme on Shishkin-type meshes for singularly perturbed parabolic problems with discontinuous convection coefficients

Abstract

This paper analyzes the implicit upwind finite difference scheme on Shishkin-type meshes (including the classical piecewise-uniform Shishkin mesh and the Bakhalov-Shishkin mesh) for a class of singularly perturbed parabolic convection-diffusion problems exhibiting strong interior layers. Suitable conditions on the mesh-generating functions are derived and are found to be sufficient for the convergence of the method, uniformly with respect to the perturbation parameter. Utilizing these conditions, it is shown that the method converges uniformly in the discrete supremum norm with an optimal error bound. Numerical results are presented to validate the theoretical results.

Parameter-Uniform Improved Hybrid Numerical Scheme for Singularly Perturbed Problems With Interior Layers

“…Over the last few years, several researchers developed the fitted mesh methods for solving singularly perturbed problems with non-smooth data, one can refer the articles [1,2,4,10,12] for the stationary case and [7,8,9] for the nonstationary case. However, it is worth mentioning that the hybrid numerical scheme proposed by Cen in [1], is an well-known fitted mesh method for solving singularly perturbed BVPs of the form (1.1)-(1.2) with discontinuous convection coefficient and the method is almost second-order accurate throughout the domain [0, 1] provided the perturbation parameter ε satisfies ε N −1 , otherwise the method is at worst first-order uniformly convergent with respect to ε in the discrete supremum norm (see the detailed discussion in Section 6).…”

Uniformly convergent higher-order finite difference scheme for singularly perturbed parabolic problems with non-smooth data

“…Thus, several methods have been established by various authors for different kinds of singularly perturbed parabolic problems with smooth data [8][9][10][11][12]. But, works on problems with non-smooth data are rare.…”

“…Recently, Chandru et al, [1], proposed a numerical treatment of two-parameter singularly perturbed parabolic convection-diffusion problems with non-smooth data. The optimal error estimate of the upwind scheme on Shishkin-type meshes [10] and an -Uniform error estimate of the hybrid numerical scheme [11] for singularly perturbed parabolic problems with interior layers are proposed by Mukherjee and Natesan. These methods are based on piecewise-uniform Shishkin meshes, and most of them are first-order spatial accurate.…”

Robust computational method for singularly perturbed system of parabolic convection‐diffusion problems with interior layers

“…For singularly perturbed parabolic convection‐diffusion IBVPs with nonsmooth data, O'Riordan and Shishkin 11 proposed a robust numerical method. Mukherjee and Natesan 12 obtained optimal error estimate for the upwind difference scheme for singularly perturbed parabolic IBVPs with discontinuous convection coefficients by using Shishkin‐type meshes. Also Mukherjee and Natesan 13 considered the hybrid numerical scheme to improve the accuracy of the numerical approximate solution of parabolic IBVPs with interior layers.…”