Adaptive Moving Grid Method For Singularly Perturbed Equations With Discontinuous Convection Coefficient

Singularly perturbed problems with discontinuous coefficient commonly exist in kinds of mathematics,physics and engineering areas.Due to the discontinuity coefficient and perturbation parameter,it is difficult or impossible to find the exact solution of this kind of equation.Therefore,the effective numerical method can better solve the singular perturbation problem with discontinuous coefficient in actual application.Although the adaptive moving grid algorithm is broadly applied in singularly perturbed differential equations,and relatively perfect research results have been obtained,the work of adaptive moving grid algorithm for singular perturbation problem with discontinuous coefficient is limited.So this paper mainly studies a adaptive moving grid method algorithm for the singularly perturbed convection-diffusion with discontinuance coefficient or the singularly perturbed Fredholm integral-differential equation with discontinuous convection coefficient.The main contents are as follows:In Chapter Ⅰ,the research background and progress of continuous and discontinuous singular perturbation problems are introduced.In Chapter Ⅱ,we consider an adaptive moving grid method algorithm for singularly perturbed convection-diffusion equation with a discontinuous convection coefficient Firstly,the upwind finite difference scheme is constructed when the discontinuity points coincide with the grid node,and the posteriori error estimate is given.Then the a posteriori error bound is used to design an adaptive gird algorithm,and the algorithm is proved to be a first-order rate of convergence.Numerical results are presented that support our theoretical results.In Chapter Ⅲ,considering the case that the discontinuous point does not coincide with the grid point,the purpose of this chapter is to construct the discrete scheme of the equation in any grid,at the same time,we give a posteriori error estimation and adaptive moving mesh generation algorithm.At last,the numerical results show the effectiveness of the proposed algorithm.In Chapter Ⅳ,for a class of singularly perturbed Fredholm integral-differential equations with discontinuous coefficients,a discretization scheme is constructed on any grid.Then,based on a priori error bound and grid distribution principles,we design a mesh generation algorithm.Finally,and the numerical results are provided to demonstrate the effectiveness of our adaptive moving grid method.