Richardson Extrapolation Of Layer-adapted Mesh Method For Some Singularly Perturbed Problems

Singularly perturbed problems arise from physics,biological field and so on.Some effective numerical methods of singularly perturbed problems are very popular in computational mathematics field.These problems are discretized by using classical finite difference scheme on the layer adapted meshes,then the accuracy of numerical method is first-order uniformly convergent.In this paper,the Richardson extrapolation technique will be used to obtain the second-order parameter-uniform accuracy for singularly perturbed problems on the Vulanovi(?)-Bakhvalov(VB)mesh.The rest of the paper is organized as follows.The research background and current situation of the numerical methods for singular perturbed problems are introduced in Chapter I.Then,the brief introductions of the main contents for our work are illustrated in Chapter I.The VB mesh is introduced and analyzed in Chapter II.In Chapter Ⅲ,on the VB mesh,the discretization scheme is constructed to approximate a singularly perturbed Volterra integro-differential equation.We proved the first order uniform convergence of the discretization scheme by using the stability of the scheme.Then,we prove that the Richardson extrapolation increases the orders of convergence of the scheme from first-order to second-order.Finally,the theoretical finds are illustrated by some numerical result.In Chapter Ⅳ,for a system of singularly perturbed convection-diffusion equations,a finite difference scheme is constructed on the VB mesh.Firstly,we proved the first order uniform convergence of the proposed numerical method by using the barrier-function approach.Furthermore,we proved that the accuracy of the scheme is second-order uniform convergence after using the Richardson extrapolation.Finally,the theoretical finds are illustrated by some numerical result.In Chapter Ⅴ,a 2-dimensional singularly perturbed convection-diffusion problem is discretized by using an upwind finite difference scheme on VB mesh.Based on the discrete comparison principle of discrete operator,the appropriate barrier-function is designed to prove that Richardson extrapolation technique can make the order of uniform convergence of the scheme is 2.Finally,the theoretical finds are illustrated by some numerical result.