The Barycentric Rational Spectral Collocation Method For Singularly Perturbated Problems And Adaptive Moving Grid Method For Burger-huxley Equation

The numerical methods of the singularly perturbed problems are a hot issue in the current scientific computing research.Based on this,this article will study the numerical methods of several types of singularly perturbed problems from two aspects.On the one hand,based on the barycentric rational spectral collocation method and related intelligent algorithms,respectively the singularly perturbed reaction diffusion equations and a class of high-precision numerical methods for singularly perturbed nonlinear equations with parameters are studied.On the other hand,the adaptive moving grid algorithm for the singularly perturbed Burger-Huxley equations are studied.The specific contents are as follows:In Chapter I,introduced the research background of the singularly perturbed problems and the related research progress.Meanwhile,introduced the main work of this article.In Chapter II,the barycentric rational spectral collocation method of the singularly perturbed reaction-diffusion equations are discussed.First,by using sinh transformation,the Chebyshev collocation points are subjected,more collocation points are gathered at the boundary layers on both ends.Then,in order to calculate the boundary the width of the layer,we constructed an unconstrained nonlinear optimization problem with the minimum absolute error as the objective function,which is solved by using an adaptive differential evolution algorithm for the optimization problem.Finally,getting the corresponding parameters and numerical solutions.Based on the Chapter II,the Chapter III systematically discusses a class of barycentric rational spectral collocation method for nonlinear singularly perturbed problems with parameters.Then,in order to calculate the grid parameters of the boundary layer width after the sinh transformation and the problem with parameters to be solved,a nonlinear optimization with the minimum absolute error as the objective function is designed,and the particle swarm optimization algorithm is used to solve.In Chapter IV,an adaptive grid method for solving the one-dimensional unsteady singularly perturbed Burger-Huxley equations are proposed.First,the backward Euler formula is used to discrete time derivatives under a uniform grid.Then,using the Newton-Raphson-Kantorovich method linearizes the obtained nonlinear singularly perturbed semi-discrete problems,and usesan upwind finite difference scheme to discretize the spatial derivatives under an arbitrary grid.Finally,the corresponding convergence analysis is given.Based on above problems and the proposed numerical methods,we conducted a large number of numerical experiments.Numerical experiments show that the numerical methods proposed in this paper are effective.It is worth mentioning that the numerical methods proposed in this paper can be further extended to solve other singularly perturbed problems.