【作者】 王秀凤；

【导师】 张传林；

【作者基本信息】 暨南大学 ， 基础数学， 2006， 硕士

【摘要】 本文给出了非线性Schr(?)dinger方程的两类数值解法：有限差分方法和辛算法，并且把非线性Schr(?)dinger方程的辛算法推广到高维。 首先，给出了非线性Schr(?)dinger方程的七种差分格式：二阶两层格式、二阶两层格式的交替显隐格式、四阶两层格式、四阶两层格式的交替显隐格式、二阶蛙跳格式、四阶蛙跳格式和三层隐式格式。对这七种差分格式的局部截断误差阶进行了分析和比较，并且利用“冻结系数法”分析了它们的稳定性进而分析了它们的收敛性。接下来通过数值实验验证了它们的稳定性，比较了它们的运算时间和精度。 然后，证明了非线性哈密顿系统的Euler中点格式以及蛙跳格式是辛格式，具体给出了非线性Schr(?)dinger方程的二阶Euler中点格式、四阶Euler中点格式、二阶蛙跳格式和四阶蛙跳格式，并且作了数值实验验证这些格式的可行性并比较其误差。接下来对同样截断误差阶的一种辛格式和一种非辛的差分格式进行比较。我们选取二阶蛙跳格式和二阶两层格式作了数值实验并对它们的运行结果作了比较。 最后，把非线性Schr(?)dinger方程的辛格式推广到了高维，并给出了一种特殊的非线性Schr(?)dinger方程——非线性双曲Schr(?)dinger方程的二阶蛙跳格式并做了数值实验验证了它的可行性。更多还原

【Abstract】 Finite difference methods and symplectic methods of nonlinear Schrodinger equation have been given out, and symplectic methods of nonlinear Schrodinger equation have been extended to higher dimension.Firstly, seven kinds of difference schemes of nonlinear Schrodinger equation have been established. They are two-order bi-level scheme, the alternating explicit-implicit scheme of the two-order bi-level scheme, four-order bi-level scheme, the alternating explicit-implicit scheme of the four-order bi-level scheme, two-order leap-frog scheme, four-order leap-frog scheme and two-order three levels scheme. The analyses of local truncation error of these seven kinds of difference schemes have been presented. The stability of these seven kinds of difference schemes have been analyze with frozen coefficient method, furthermore their convergence have been analyze. Their stability have been confirmed by the numerical experiments. Their truncation error and speeds have also been compared by numerical experiments.Then it has been proved that Euler midpoint schemes and leap-frog schemes of Hamiltonian system are symplectic schemes. Two-order Euler midpoint scheme, four-order Euler midpoint scheme, two-order leap-frog scheme and four-order leap-frog scheme of Hamiltonian system for nonlinear Schrodinger equation have been given out. And their numerical experiments have been made. Their feasibility have been confirmed by the numerical experiments. Then the results of two-order leap-frog scheme (one of the symplectic schemes) and two-order bi-level scheme (one of the non-symplectic schemes) have been compared by the numerical experiments.At last symplectic schemes of Hamiltonian system for nonlinear Schrodinger equation have been extended to higher dimension. Then two-order leap-frog schemes for nonlinear hyperbolic Schrodinger equation are given out and the numerical experiments are made to confirm it’s feasibility.更多还原