High‐order numerical methods for one‐dimensional parabolic singularly perturbed problems with regular layers

Abstract

In this work we construct and analyse some finite difference schemes used to solve a class of timedependent one-dimensional convection-diffusion problems, which present only regular layers in their solution. We use the implicit Euler or the Crank-Nicolson method to discretize the time variable and a HODIE finite difference scheme, defined on a piecewise uniform Shishkin mesh, to discretize the spatial variable. In both cases we prove that the numerical method is uniformly convergent with respect to the diffusion parameter, having order near two in space and order one or 3/2, depending on the method used, in time. We show some numerical examples according to the theoretical results, in the case of using the Euler implicit method, and a better numerical behaviour than the predicted theoretically, showing order two in time and order N −2 log 2 N in space, if the Crank-Nicolson scheme is used to discretize the time variable. Finally, we construct a numerical algorithm by combining a third order A-stable SDIRK with two stages and a third order HODIE difference scheme, showing its uniformly convergent behaviour, reaching order three, up to a logarithmic factor. c (Year) John Wiley & Sons, Inc.

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

“…As the exact solutions of the IBVPs (5.1) and (5.2) are not known, we illustrate the numerical results of the upwind scheme (3.9) by using the double mesh principle as in [2] which is described below.…”

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

“…3. As the exact solutions of Examples 6.1 and 6.2 are not known, to show the ε-uniform convergence of the proposed scheme (4.1) and also to obtain the accuracy of the numerical solutions, we use the double mesh principle as in [7] which is described below. with N mesh-intervals in the spatial direction and M mesh-intervals in the t-direction such that t = T/M is the uniform time step.…”

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

“…where L ε u(x, t) ≡ −εu xx (x, t) + a(x)u x (x, t) + b(x, t)u(x, t), (2) with a(x) > α > 0 and b = b(x, t) ≥ 0 onΩ. The diffusion coefficient ε is a small positive parameter.…”

“…In two recent papers (Clavero et al (2005) [2] and Mukherjee and Natesan (2009) [3]), this method has been shown to be convergent, uniformly in the small diffusion parameter, using somewhat elaborate analytical techniques and under a certain mesh restriction. In the present paper, a much simpler argument is used to prove a higher order of convergence (uniformly in the diffusion parameter) than in [2,3] and under a slightly less restrictive condition on the mesh.…”