无界区域上带波动算子的非线性Schrödinger方程基于人工边界方法的数值解法

Based on the artificial boundary method,the objective of this project is to study the numerical solution and simulation of the nonlinear Schrödinger equation with wave operator (NLSW) with a perturbation strength described by a dimensionless parameter on unbounded domain. The NLSW arises in a large variety of applications in physics,such as the Langmuir wave envelop approximation in plasma.It is impossible to solve directly the problem defined on unbounded domain,since the physical computational domain is unbounded.To deal with the unboundedness,one of the most powerful approaches is the use of absorbing boundary conditions (ABCs) for an appropriate bounded computational domain, which is a powerful method to reduce the problems on unbounded domain to a bounded domain.In order to design efficient local ABCs for this nonlinear problem,we apply the novel unified approach,which based on the well-known operator splitting method.The method distinguish between incoming and outgoing waves along the boundaries for the linear kinetic subproblem, and to approximate the corresponding linear operator by using a ‘one-way operator',then unite the approximate operator and the potential energy subproblem to yield nonlinear local ABCs.Then reduce the original problem into an initial boundary value problem on the bounded computational domain,which can be solved by applying standard numerical methods, such as the finite difference method.We analyze the stability of the reduced initial boundary value problem by constructing an energy functional for the NLSW.In order to solve the high oscillation of the nonlinear equation,we design the efficient algrithm,such as the asymptotic method based on the frozen Gaussian approximation method.Numerical examples are given to illustrate the effectiveness and tractability of the proposed local ABCs method,and simulate the propagation of wave on unbounded domain.

本项目研究带波动算子的非线性Schrödinger方程在无界区域上基于人工边界方法的数值解，此方程在物理中应用广泛，如描述等离子体物理学中的Langmuir波。物理区域的无界性使得原问题很难直接进行数值计算，需引入人工边界，并在人工边界上设计合理的边界条件；利用基于算子分裂思想的unified方法克服非线性方程边界条件设计带来的困难，结合线性Schrödinger方程的局部吸收边界条件，设计带波动算子的非线性Schrödinger方程的局部吸收边界条件。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，构造能量泛函分析初边值问题的稳定性。利用有限差分法对初边值问题进行离散，分析非线性方程的高震荡特性并设计相应有效的算法(基于frozen Gaussian approximation思想的渐进方法)，最后通过数值算例验证局部吸收边界条件的精确性和有效性，并模拟波在无界区域上的传播过程。

本项目研究带波动算子的非线性Schrödinger方程在无界区域上基于人工边界方法的数值解，此方程在物理学中应用广泛，如描述等离子体物理学中的Langmuir波、非相对极限的Klein-Gordon方程及Sine-Gordon方程对光子弹的平面调制近似。. 自项目实施以来，我们根据项目工作计划，借鉴国内外相关研究成果，围绕无界区域上带波动算子的非线性Schrödinger方程开展了深入研究。物理区域的无界性使得原问题很难直接进行数值计算，通过引入人工边界克服区域的无界性，并在人工边界上设计合理的边界条件；利用基于算子分裂思想的unified方法克服非线性方程边界条件设计带来的困难，通过引入线性算子和非线性算子将原问题分裂为两部分，首先针对线性部分设计one-way算子对线性算子进行逼近使得外传波无反射的传出有界计算区域，然后结合线性Schrödinger方程的局部吸收边界条件，设计带波动算子的非线性Schrödinger方程的局部吸收边界条件。利用设计的局部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；因带波动算子的非线性Schrödinger方程在无界区域上质量和能量守恒，构造质量泛函分析初边值问题的稳定性。利用有限差分法对简化的初边值问题进行离散，分析非线性方程的高震荡特性并设计相应有效的算法，证明了离散格式下初边值问题的稳定性。最后通过数值算例计算了有界计算区域上的误差和误差阶，比较了设计的局部吸收边界条件和Dirichlet边界条件对精确解的逼近效果，以及观察了带波动算子的非线性Schrödinger方程在有界区域上不同时刻的质量，验证局部吸收边界条件的准确性和有效性，并模拟波在无界区域上不同时刻的传播过程。. 项目执行期间，项目组在Physical Review E、Journal of Scientific Computing、Journal of Computational and Applied Mathematics、Computer and Mathematics with Applications、East Asian Journal on Applied Mathematics等国内外期刊上发表论文6篇，培养研究生5人。