无界区域上带波动算子的非线性薛定谔方程的局部吸收边界条件

The nonlinear Schrodinger equation with wave operator(NLSW) arises from different physical applications, such as the Langmuir wave envelop approximation in plasma, and the nonrelativistic limit of the Klein-Gordon equation. This project is devoted to studying the numerical solution and simulation of the NLSW on unbounded domain. Since it is impossible to solve directly the problem defined on the unbounded domain, they are often approximated in bounded computational domain by imposing appropriate conditions on the artificial boundary. For unboundedness, one of the most popular approaches is the use of the local absorbing boundary conditions (ABCs) method, which is a powerful method to reduce the problems on an unbounded domain to a bounded domain, for an appropriate bounded computational domain. In order to design efficient local ABCs for this nonlinear problem, we extend the idea of the unified approach. The idea underlying the unified approach is based on the well-known operator splitting method, which distinguish between incoming and outgoing waves along the boundaries for the linear subproblem, and to approximate the corresponding linear operator by using a ‘one-way operator’ (to make the wave outgoing), then unite the approximation operator and the nonlinear subproblem to yield nonlinear boundary conditions. Then the original problem is reduced to an initial-boundary-value problem (IBVP) on a finite computational domain, which can be solved by using standard numerical methods such as the finite difference method. We analyze the stability of the reduced IBVP by constructing an energy functional for the NLSW. Numerical examples are given to demonstrate the effectiveness and tractability of the proposed method, and simulate the propagation of wave on unbounded domain.

带波动算子的非线性薛定谔方程在等离子体物理学中应用广泛，本项目主要研究它在无界区域上的数值解和数值模拟。区域的无界性使得定义在无界区域上的问题很难直接进行数值求解，为了在无界区域上有效地进行数值计算，需引入人工边界，在人工边界上设计合理的局部吸收边界条件；利用算子分裂方法的思想来克服方程非线性带来的困难，结合线性薛定谔方程的局部吸收边界条件，构造带波动算子的非线性薛定谔方程的局部非线性吸收边界条件，将定义在无界区域上的原问题转化为有界区域上的初边值问题，并通过构造能量泛函对含局部吸收边界条件的初边值问题进行稳定性分析。利用成熟的有限差分方法对含局部吸收边界条件的初边值问题进行离散，最后通过对算例的数值计算验证局部吸收边界条件的有效性和精确性，以及模拟波在无界区域上的传播过程。

我们采用由Zhang等提出的基于算子分裂思想的unified approach设计了带波动算子的非线性薛定谔方程在无界区域上的局部非线性吸收边界条件，将定义在无界区域上的原问题转化为有界区域上的初边值问题。为了避免局部吸收边界条件中出现高阶混合偏导数，通过引入辅助变量分析了带有局部非线性吸收边界条件的初边值问题的稳定性。在有界区域上对初边值问题设计了有效的差分格式，最后通过数值算例验证了局部吸收边界条件的精确性和有效性，并模拟了相关的物理现象。