Parameter-uniform approximations for a singularly perturbed convection-diffusion problem with a discontinuous initial condition

Jose Luis Gracia,Eugene O'Riordan

doi:10.1016/j.apnum.2020.12.013

Abstract

A singularly perturbed parabolic problem of convection-diffusion type with a discontinuous initial condition is examined. A particular complimentary error function is identified which matches the discontinuity in the initial condition. The difference between this analytical function and the solution of the parabolic problem is approximated numerically. A coordinate transformation is used so that a layer-adapted mesh can be aligned to the interior layer present in the solution. Numerical analysis is presented for the associated numerical method, which establishes that the numerical method is a parameter-uniform numerical method. Numerical results are presented to illustrate the pointwise error bounds established in the paper.

Highlights

We examine a singularly perturbed convection-diffusion problem with a discontinuous initial condition of the form: Find usuch that
−εuss + aus + ˆbu + ut = f, (s, t) ∈ Q := (0, 1) × (0, T ]; (1a) u(s, 0) = φ(s) ∈ C0(0, 1); a > 0; ˆb ≥ 0, (1b) with Dirichlet boundary conditions. As this is a parabolic problem, an interior layer emerges from the initial discontinuity, which is diffused over time if ε = O(1)
In [8], we examined a related singularly perturbed reaction-diffusion problem (set a ≡ 0 in (1)) with a discontinuous initial condition and we used an idea from [3] to first identify an analytical function which matched the discontinuity in the initial condition and satisfied a constant coefficient version of the differential equation

Summary

Introduction

We examine a singularly perturbed convection-diffusion problem with a discontinuous initial condition of the form: Find usuch that. When the convective coefficient depends solely on time (a(s, t) ≡ a(t) > 0), the main singularity generated by the discontinuous initial condition can be explicitly identified by a particular complimentary error function This error function tracks the location of the interior layer emanating from the discontinuity in the initial condition and it satisfies the homogenous partial differential equation (1a) exactly. When this discontinuous error function is subtracted from the solution uof (1), the remaining function (denoted below by y) contains no interior layer and it can be adequately approximated numerically by designing a numerical method which incorporates a Shishkin mesh in the vicinity of the boundary layer [6]. Functions defined in the computational domain will be denoted by f (x, t) and functions defined in the untransformed domain will be denoted by f(s, t)

Continuous problem

Numerical method in the transformed domain and associated error analysis

Modifications when source term is present

Numerical results