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Title: Numerical studies of the Klein-Gordan-Schrodinger equations
Authors: YANG LI
Keywords: Klein-Gordon-Schrodinger equations, Klein-Gordon equation, Schrodinger-Yukawa equation, time splitting, plane wave, solitary wave
Issue Date: 16-Nov-2006
Source: YANG LI (2006-11-16). Numerical studies of the Klein-Gordan-Schrodinger equations. ScholarBank@NUS Repository.
Abstract: In this thesis, we present a numerical method for the nonlinear Klein-Gordon equation and two numerical methods for studying solutions of the Klein-Gordon-Schrodinger equations.We begin with the derivation of the Klein-Gordon equation (KG) which describes scalar (or pseudoscalar) spinless particles, analyze its properties and present Crank-Nicolson leap-frog spectral method (CN-LF-SP) for numerical discretization of the nonlinear Klein-Gordon equation. Numerical results for the Klein-Gordon equation demonstrat that the method is of spectral-order accuracy in space and second-order accuracy in time and it is much better than the other numerical methods proposed in the literature. It also preserves the system energy, linear momentum and angular momentum very well in the discretized level. We continue with the derivation of the Klein-Gordon-Schrodinger equations (KGS) which describes a system of conserved scalar nucleons interacting with neutral scalar mesons coupled through the Yukawa interaction and analyze its properties. Two efficient and accurate numerical methods are proposed for numerical discretization of the Klein-Gordon-Schrodinger equations. They are phase space analytical solver+time-splitting spectral method (PSAS-TSSP) and Crank-Nicolson leap-frog time-splitting spectral method (CN-LF-TSSP). These methods are explicit, unconditionally stable, of spectral accuracy in space and second order accuracy in time, easy to extend to high dimensions, easy to program, less memory-demanding, and time reversible and time transverse invariant. Furthermore, they conserve (or keep the same decay rate of) the wave energy in KGS when there is no damping (or a linear damping) term, give exact results for plane-wave solutions of KGS, and keep the same dynamics of the mean value of the meson field in discretized level. We also apply our new numerical methods to study numerically soliton-soliton interaction of KGS in 1D and dynamics of KGS in 2D. We numerically find that, when a large damping term is added to the Klein-Gordon equation, bound state of KGS can be obtained from the dynamics of KGS when time goes to inA?nity. Finally, we extend our numerical method, time-splitting spectral method (TSSP) to the Schrodinger-Yukawa equations and present the numerical results of the Schrodinger-Yukawa equations in 1D and 2D cases.
Appears in Collections:Master's Theses (Open)

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