基于分裂格式有限点集法对孤立波二维非线性问题的模拟

Simulation of two-dimensional nonlinear problem with solitary wave based on split-step finite pointset method

在提出一种基于时间分裂格式的纯无网格有限点集(split-step finite pointset method, SS-FPM)法的基础上,数值模拟了含孤立波的二维非线性薛定谔(nonlinear Schr?dinger, NLS)/(Gross-Pitaevskii, GP)方程.SS-FPM的构造过程为:1)基于时间分裂的思想将非线性薛定谔方程分成线性导数项和非线性项;2)采用基于Taylor展开和加权最小二乘法的有限点集法,借助Wendland权函数,对线性导数项进行数值离散.随后,模拟了带有Dirichlet和周期性边界条件的NLS方程,将所得结果与解析解做对比.数值结果表明:给出的SS-FPM粒子法的优点是在粒子分布非均匀情况下仍具有近似二阶精度,且较网格类有限差分算法实施容易,较已有改进的光滑粒子动力学方法计算误差小.最后,运用SS-FPM对无解析解的二维周期性边界NLS方程和Dirichlet边界玻色-爱因斯坦凝聚二分量GP方程进行了数值预测,并与其他数值结果进行对比,准确展现了非线性孤立波奇异性现象和量子化涡旋过程.

In this paper,a split-step finite pointset method(SS-FPM)is proposed and applied to the simulation of thenonlinear Schr dinger/Gross-Pitaevskii equation(NLSE/GPE)with solitary wave solution.The motivation andmain idea of SS-FPMisas follows.1)The nonlinear Schr dinger equation is first divided into the linearderivative term and the nonlinear term based on the time-splitting method.2)The finite pointset method(FPM)based on Taylor expansion and weighted least square method is adopted,and the linear derivative termis numerically discretized with the help of Wendland weight function.Then the two-dimensional(2D)nonlinearSchr dinger equation with Dirichlet and periodic boundary conditions is simulated,and the numerical solutionis compared with the analytical one.The numerical results show that the presented SS-FPM has second-orderaccuracy even if in the case of non-uniform particle distribution,and is easily implemented compared with theFDM,and its computational error is smaller than those in the existed corrected SPH methods.Finally,the 2DNLS equation with periodic boundary and the two-component GP equation with Dirichlet boundary and outerrotation BEC,neither of which has an analytical solution,are numerically predicted by the proposed SS-FPM.Compared with other numerical results,our numerical results show that the SS-FPM can accurately display thenonlinear solitary wave singularity phenomenon and quantized vortex process.