Development and Numerical Study of Robust Difference Schemes for a Singularly Perturbed Transport Equation

摘要

On the set G = G U 5, G = (0,d] × (0,T] with the boundary S = S_o ∪ S~ℓ, we consider an initial-boundary value problem for the singularly perturbed transport equation with a perturbation parameter ε multiplying the spatial derivative, ε ∈ (0,1]. For small values of the perturbation parameter e, the solution of such a problem has a singularity of the boundary layer type, which makes standard difference schemes unsuitable for practical computations. To solve this problem numerically, an approach to the development of a robust difference scheme is proposed, similar to that used for constructing special ε-uniformly convergent difference schemes for singularly perturbed elliptic and parabolic equations. In this paper, we give a technique for constructing a robust difference scheme and justifying its ε-uniform convergence, and we study numerically solutions of standard and special robust difference schemes for a model initial-boundary value problem for a singularly perturbed transport equation. The results of numerical experiments confirm theoretical results.