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Artificial Boundary Conditions For Klein-Gordon Equation And Schr(o_¨)dinger Equation

Posted on:2012-07-25Degree:DoctorType:Dissertation
Country:ChinaCandidate:Z W ZhangFull Text:PDF
GTID:1110330362967978Subject:Mathematics
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In this thesis, we study the artificial boundary condition method for the Klein-Gordon equation (including the one-dimensional linear Klein-Gordon equation, theone-dimensional nonlinear Klein-Gordon equation, and the two-dimensional sine-Gordon equation) and the two-level Schro¨dinger equation system on the unboundeddomain. In addition we develop a surface hopping method (based on the semi-classicallimit theorem) for the two-level Schro¨dinger equation with conical crossing and themultiscale tailored finite point method for the multiscale elliptic equation.The Klein-Gordon equation has been widely applied in relativistic quantum me-chanics, statistical physics, and soliton theory in the nonlinear partial differential equa-tion. Since this equation is defined on the unbounded domain, it cannot be solveddirectly by the traditional numerical methods (for instance the finite difference methodand the finite element method). To overcome this difficulty, we introduce some arti-ficial boundaries and divide the unbounded domain into a bounded domain and someunbounded parts. These artificial boundaries will become the boundaries for the prob-lems on the bounded domain. We systematically study and obtain the artificial bound-ary conditions for the Klein-Gordon equation and reduce the original problems to initialboundary value problems on the bounded domain. By solving these initial boundaryvalue problems, we can obtain the numerical solution of the original problems.The two-level Schro¨dinger equation with conical crossing is a computationalmodel in quantum chemistry. When the potential energy surfaces approach eachother, or even cross, the traditional Born-Oppenheimer approximation will break down.We propose a semi-classical limit method—surface hopping method for the two-levelSchro¨dinger equation with conical crossing and observe the surface hopping phe-nomenon numerically. We also design the artificial boundary condition for the two-level Schro¨dinger equation and reduce the original problem to an initial boundary valueproblem on the bounded domain. This allows us to reduce the computational domain for practical computation and improve the computational efficiency.The multiscale elliptic equation arises in many engineering fields, such as com-posite materials and porous media. To obtain the small scale information of the numer-ical solution, it will take large amount of computational cost and memory consumptionfor the traditional numerical method, or it may sometimes even be impossible. There-fore we propose a multiscale tailored finite point method to overcome this difficulty.Numerical results indicate that our method can capture the small scale information ofthe solution on the coarse grid and reduce the computational cost and memory con-sumption.
Keywords/Search Tags:Artificial boundary condition, Klein-Gordon equation, two-levelSchro(o|¨)dinger equation, multiscale elliptic equation, finite point method
PDF Full Text Request
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