The High-order Numerical Algorithms For Several Singular Perturbation Problems

Because of its spectral accuracy,the spectral method has been widely used in the numerical solution of various fields.However,a large number of collocation points is required to describe variation ruler of the boundary layers.In order to improve the efficiency of the numerical simulation of the singular perturbation problem,some researchers decomposed the solution of the problem into regular components and singular components to weaken the singularity,to meet the needs of solving the singular perturbation problem.For describing the boundary layer region,several modified numerical scheme have been proposed,and some transformations that the transformed points are more located in the boundary layer region are introduced.Combining the advantages of these two types of methods,a rational spectral collocation method combined with singularity-separated technology is proposed in this paper.In Chapter One,the research background and research progress of several kinds of singular perturbation problems are introduced,as well as the research issues and main work of this thesis.In Chapter Two,the parameters in the sinh transformation,that is the position and width of the boundary layer are determined by the asymptotic analysis.The original Chebyshev-Gauss-Lobatto points are mapped onto the transformed ones clustered near the singular points of the problem.Then,using Singular Separation Techniques,the singular perturbation problem is decomposed into a weakened singular auxiliary boundary value problem and the boundary layer correction function.The regular component is obtained by solving the auxiliary equation by the rational spectral method with sinh transform.The explicit expressions of singular correct functions are determined by boundary conditions and interface conditions.The error estimation is also given.In the case of variable coefficients,the corrector functions are designed to capture the sharpness of the layers.Then the spectral method is used to solve the regular and the undetermined parameters of singular components,and the numerical solution is obtained.The numerical results are given at the end of this chapter to confirm the theoretical analysis.In Chapter Three,A coupled system of singularly perturbed boundary value problems are considered.The weakly coupled reaction diffusion type and the strongly coupled convection diffusion type are solved by the rational spectral collocation in barycentric form with singularly-separated method.The general solution is deduced and proved,and the special solution of the original problem is represented by the solution of the weakened singular boundary value problem.Explicit expressions of singular correction functions are given by boundary conditions and eigenvalues of homogeneous problems,and proved that this method almost reaches the spectral accuracy when the parameters are very small.The numerical experiments are shown the high accuracy and efficiency of our method.In chapter Four,the singularly perturbed problems with non-smooth data are considered.The singular perturbation problem is divided into left and right subproblems.Then rational spectral method is used to solve the corresponding weak singularity problem to determine the regular components.The parameters in the singular correction function expression are determined by boundary conditions and interface conditions.The solution of the original problem is obtained by using the seam method.In Chapter five,we propose a numerical method to solve singularly perturbed parabolic problems and a time-fractional singularly perturbed problems.The linear time differential equations are transformed into the boundary value problem of ordinary differential equations with spatial variables in the transformed domain using the Laplace transformation.The numerical solution of the problem is obtained by solving the inverse Laplace transform numerically.The numerical solution of inverse Laplace transform problem is obtained by the Talbot method.The use of the Laplace transform circumvents the need for time marching in the temporal domain.Some numerical simulations are given to confirm the effectiveness of this method.