SCHEMES CONVERGENT Ε-UNIFORMLY FOR PARABOLIC SINGULARLY PERTURBED PROBLEMS WITH A DEGENERATING CONVECTIVE TERM AND A DISCONTINUOUS SOURCE

Carmelo Clavero,Grigorii I. Shishkin,Lidia P. Shishkina,Jose Luis Gracia

doi:10.3846/13926292.2015.1091041

Abstract

We consider the numerical approximation of a 1D singularly perturbed convection-diffusion problem with a multiply degenerating convective term, for which the order of degeneracy is 2p + 1, p is an integer with p ≥ 1, and such that the convective flux is directed into the domain. The solution exhibits an interior layer at the degeneration point if the source term is also a discontinuous function at this point. We give appropriate bounds for the derivatives of the exact solution of the continuous problem, showing its asymptotic behavior with respect to the perturbation parameter ε, which is the diffusion coefficient. We construct a monotone finite difference scheme combining the implicit Euler method, on a uniform mesh, to discretize in time, and the upwind finite difference scheme, constructed on a piecewise uniform Shishkin mesh condensing in a neighborhood of the interior layer region, to discretize in space. We prove that the method is convergent uniformly with respect to the parameter ε, i.e., ε-uniformly convergent, having first order convergence in time and almost first order in space. Some numerical results corroborating the theoretical results are showed.

Highlights

In this paper we construct a numerical method for solving the following 1D parabolic initial-boundary singularly perturbed problem, with a convective term degenerating inside the domain: L u(x, t) = f (x, t), u(x, t) = φ(x, t), l±u(x, t) ≡ ε
A technique to analyze the ε-uniform convergence of numerical schemes defined on piecewise-uniform meshes, for a singularly perturbed elliptic equation when the convective term degenerates on the boundary, was considered in [4, Chapter 7]
In this paper we have considered 1D parabolic singularly perturbed problems having a multiple degenerating convective term at an interior point and a discontinuous source term where the discontinuity is located at the same interior point

Summary

Introduction

In this paper we construct a numerical method for solving the following 1D parabolic initial-boundary singularly perturbed problem, with a convective term degenerating inside the domain:. Problem (1.1) is an initial-boundary value problem for a singularly perturbed parabolic equation with the multiply degenerating convection term, and a source term having discontinuity on the set of degenerating convection. We assume that the data of problem (1.1) satisfy sufficient conditions that guarantee the required smoothness of the solution on the sets G + and G −. In [3] a difference scheme for a singularly perturbed parabolic equation, where the convective term degenerates on the boundary, was considered. A technique to analyze the ε-uniform convergence of numerical schemes defined on piecewise-uniform meshes, for a singularly perturbed elliptic equation when the convective term degenerates on the boundary, was considered in [4, Chapter 7]. By M we denote a generic positive constant independent of both the diffusion parameter ε and the discretization parameters N and N0, where N and N0 are number of mesh intervals in x and t, respectively

Asymptotic behavior of the continuous problem

Numerical experiments

Conclusions