Numerical Simulation Of Singular Perturbation Problems Based On Nelder-Mead Simplex Algorithm

The singular perturbation problem exits in many engineering fields,and this kind of problem corresponds to the highest differential equation derivative of the derivative with a small parameter called phi.In recent years,the layer adaptive grid method has been widely applied to various singular perturbations and the numerical solution of the reaction-diffusion equation.The most common method is the Shishkin grid.At the same time,the spectral collocation method is used as a numerical method for numerically solving differential equations,and more and more scholars pay attention to it.However,almost all the literature is to artificially select Shishkin grid parameters and boundary layer width related parameters.Therefore,this thesis is mainly based on the simplex algorithm,studies the mesh parameter optimization of numerical method for singular perturbation problem.The specific contents are as follows:Chapter one is conclusion,introduces the current research situation of layer adaptive grid method,rational spectral collocation method of singular perturbation problem,the structure arrangement and the main work of this dissertation.Chapter two is preliminary knowledge,mainly introduces the rational center interpolation,differential matrix and Nelder-Mead simplex algorithm.Chapter three studies a semi-discrete method for the time-dependent reactiondiffusion equation for a class of singular perturbation problems.First,the upwind difference schemes for spatial and temporal directions of such problems are constructed separately.Secondly,it gives the analysis of the stability.In order to get better Shishkin grid parameters,an unconstrained nonlinear optimization problem with the minimum error as the objective function is designed,and a simplex algorithm for solving the optimization problem is given.Finally,a numerical example is used,and the effectiveness of the algorithm is proved.In chapter four,we study the simplex algorithm for the singular perturbation convection-diffusion problem.First we give the ration of the center of gravity rational spectrum with the sinh transform to discretize the problems we consider in this chapter,and then construct the minimum absolute error norm.For the unconstrained optimization problem of the target,the simplex algorithm is used to solve the optimization problem,and the optimal boundary layer width is related parameters and the numerical solution of the problem is obtained.Finally,numerical examples are used to demonstrate the effectiveness of the algorithm.In chapter five,studies the numerical simulation of the two-differential perturbation convection-diffusion equation based on the simplex algorithm.First,the finite difference method is used to discretize the problems that we consider,and then the construction is based on the minimum error norm,and the optimization problem is solved by using simplex algorithm.Finally,in the optimal Shishkin grid,we use the finite difference method to obtain the numerical solution of the problem,and give a numerical example to verify the effectiveness of the algorithm proposed.