The Hybrid Difference Methods On Layer-adapted Meshes For Singularly Perturbed Two-point Boundary Value Problems

Singularly perturbed problems have extensive applications in a variety of fields,such as fluid dynamics,celestial mechanics,engineering technology,even financial modelling.Because of the small perturbation parameter in the singularly perturbed problem,the exact solution produces a drastic change in the boundary layer,which leads to getting unsatisfied results with classical numerical methods.The numerical solution of the singularly perturbed problems becomes a hot research topic.Therefore,this paper studies the finite difference schemes on the layer-adapted meshes for singularly perturbed boundary value problems.In the first part,the hybrid difference scheme on the Shishkin mesh is used for one-dimensional singularly perturbed two-point boundary value problem.The new error estimate of second-order convergence on the interval[0,xphN]and almost second-order convergence on the interval(xph,N 1],where ph=1-1/(2e)?0.8161,are proved by means of truncation error,discrete comparison principle,barrier functions and so on.Furthermore,this method also applies to the midpoint upwind scheme and the simple upwind scheme,and the new better error estimates are also obtained.The numerical examples support theoretical results.In the second part,a new hybrid difference scheme on a modified Bakhvalov-Shishkin mesh is constructed for one-dimensional singularly perturbed two-point boundary value problem and better s-uniform order of convergence are proved.The numerical examples confirm the theoretical results and display the advantage on accuracy for the method in practice.In the third part,a midpoint upwind scheme and a new hybrid difference scheme on tensor-product layer-adapted meshes are proposed for two-dimensional singularly perturbed boundary value problems,and the truncation error estimators are given.The numerical examples demonstrate that the feasibility of the midpoint upwind scheme and the new hybrid difference scheme,and the corresponding convergence orders to one dimension are obtained.