The Asymptotic Analysis And Efficient Numerical Methods For The Singularly Perturbed Problems

With the development of science and technology,the mathematical models used to describe practical problems are becoming more and more complex,and generally contain multiple scales.Therefore,multi-scale modeling and multi-scale calculation methods have become one of the most important research directions in science and engineering computing,and singularity The asymptotic analysis and numerical solution of the perturbation problem is an important research topic in multi-scale modeling.Although the singular perturbation problem will will show some atypicality,it can help us better understand the qualitative and approximate quantitative understanding of physical problems.This paper mainly studies the asymptotic analysis and numerical solution of the singularly perturbed eigenvalue problems and the singularly perturbed telegraph equations.The singularly perturbed eigenvalue problems?SPEPs? are derived from the steadystate Schr?dinger equation under the semi-classical limit and have extremely important applications in the fields of theoretical physics and physical chemistry.We study the asymptotic analysis and efficient numerical methods for singularly perturbed eigenvalue problems?SPEPs?.We first make a close asymptotic analysis on the singularly perturbed eigenvalue problem,and prove that,for piecewise smooth potential functions V?x?,the eigenvalues converge to the minimum value of V?x? and the eigenfunctions are concentrated in the immediate vicinity of the minimal points of V?x? as ? ? 0⁺.Then we propose some new schemes based on tailored finite point method?TFPM?for numerical solutions of singularly perturbed eigenvalue problems with higher accuracy.Our numerical examples verify our theory and show the feasibility and efficiency of our TFPM.The singularly perturbed telegraph equation?SPTE?is derived from the hyperbolic heat conduction equation and can also be used to modify the traditional diffusion equation and the reaction diffusion equation,which can better explain the experimental results.We study the numerical solutions for the singularly perturbed telegraph equation?SPTE?on unbounded domain.Firstly,we investigate the first consistent effective asymptotic expansion for the solution of SPTE by the asymptotic analysis and obtain that the solutions of SPTE have an initial layer near t = 0.Next,we introduce the artificial boundaries ?_± to get a finite computational domain ?₀ and derive the exact artificial boundary conditions on ?_± for SPTE.Hence,we can reduce the original problem to an initial-boundary value problem on the bounded domain ?₀,and then the equivalence between the original problem and the reduced problem on ?₀ is proved.In addition,we propose a semi-discrete Galerkin finite element scheme and a Crank-Nicolson Galerkin scheme to solve the reduced problem.Specially,we use the exponential wave integrator method to deal with the initial layer.We also analyze the stability and uniform convergence of the semi-discrete finite element scheme and Crank-Nicolson Galerkin scheme.Finally,some numerical examples validate our theoretical results and show the efficiency and reliability of the artificial boundary conditions.