Parameter uniform numerical method for a system of two coupled singularly perturbed parabolic convection-diffusion equations

Li-Bin Liu,Yong Zhang,Guangqing Long

doi:10.1186/s13662-018-1907-1

Li-Bin Liu, Yong Zhang + Show 1 more

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https://doi.org/10.1186/s13662-018-1907-1

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Abstract

In this paper, we propose a numerical scheme for a system of two linear singularly perturbed parabolic convection-diffusion equations. The presented numerical scheme consists of a classical backward-Euler scheme on a uniform mesh for the time discretization and an upwind finite difference scheme on an arbitrary nonuniform mesh for the spatial discretization. Then, for the time semidiscretization scheme, an a priori and an a posteriori error estimations in the maximum norm are obtained. It should be pointed out that the a posteriori error bound is suitable to design an adaptive algorithm, which is used to generate an adaptive spatial grid. It is proved that the method converges uniformly in the discrete maximum norm with first-order time and spatial accuracy, respectively, for the fully discrete scheme. At last, some numerical results are given to validate the theoretical results.

Highlights

In [13,14,15], the authors considered some special strongly coupled system of singularly perturbed convection-diffusion problems and constructed some corresponding layer-adapted mesh approaches
In this paper, we consider the following system of two linear singularly perturbed parabolic convection-diffusion equations: ⎧ ⎪⎪⎨Lε u ≡ ∂u ∂t + Lx,ε u = f(x, t),(x, t) ∈ Ω × (0, T] ≡ (0, 1) × (0, T],⎪⎪⎩uu((0x, 0) t)
5 Numerical examples and discussion we show the numerical results of two examples to verify the theoretical results

Summary

Introduction

In [13,14,15], the authors considered some special strongly coupled system of singularly perturbed convection-diffusion problems and constructed some corresponding layer-adapted mesh approaches. Liu and Chen [19] constructed an adaptive moving grid method for a special strongly coupled system of two singularly perturbed convection-diffusion problems and obtained an a posteriori error estimation in maximum norm.

Results

Conclusion