基于局部加密纯无网格法非线性Cahn-Hilliard方程的模拟

为数值求解描述不同物质间相位分离现象的高阶非线性Cahn-Hilliard(C-H)方程,发展了一种基于局部加密纯无网格有限点集法(local refinement finite pointset method,LR-FPM).其构造过程为:1)将C-H方程中四阶导数降阶为两个二阶导数,连续应用基于Taylor展开和加权最小二乘法的FPM离散空间导数;2)对区域进行局部加密和采用五次样条核函数以提高数值精度;3)局部线性方程组求解中准确施加含高阶导数Neumann边值条件.随后,运用LR-FPM求解有解析解的一维/二维C-H方程,分析粒子均匀分布/非均匀分布以及局部粒子加密情况的误差和收敛阶,展示了LR-FPM较网格类算法在非均匀布点情况下的优点.最后,采用LR-FPM对无解析解的一维/二维C-H方程进行了数值预测,并与有限差分结果相比较.数值结果表明,LR-FPM方法具有较高的数值精度和收敛阶,比有限差分法更易数值实现,能够准确展现不同类型材料间相位分离非线性扩散现象随时间的演化过程.     

A local refinement purely meshless scheme for time fractional nonlinear Schrodinger equation in irregular geometry region

A local refinement hybrid scheme(LRCSPH-FDM)is proposed to solve the two-dimensional(2D)time fractional nonlinear Schrodinger equation(TF-NLSE)in regularly or irregularly shaped domains,and extends the scheme to predict the quantum mechanical properties governed by the time fractional Gross-Pitaevskii equation(TF-GPE)with the rotating Bose-Einstein condensate.It is the first application of the purely meshless method to the TF-NLSE to the author’s knowledge.The proposed LRCSPH-FDM(which is based on a local refinement corrected SPH method combined with FDM)is derived by using the finite difference scheme(FDM)to discretize the Caputo TF term,followed by using a corrected smoothed particle hydrodynamics(CSPH)scheme continuously without using the kernel derivative to approximate the spatial derivatives.Meanwhile,the local refinement technique is adopted to reduce the numerical error.In numerical simulations,the complex irregular geometry is considered to show the flexibility of the purely meshless particle method and its advantages over the grid-based method.The numerical convergence rate and merits of the proposed LRCSPH-FDM are illustrated by solving several 1D/2D(where 1D stands for one-dimensional)analytical TF-NLSEs in a rectangular region(with regular or irregular particle distribution)or in a region with irregular geometry.The proposed method is then used to predict the complex nonlinear dynamic characters of 2D TF-NLSE/TF-GPE in a complex irregular domain,and the results from the posed method are compared with those from the FDM.All the numerical results show that the present method has a good accuracy and flexible application capacity for the TF-NLSE/GPE in regions of a complex shape.     

基于耦合纯无网格方法时间分数阶下孤立子波碰撞过程的数值模拟研究北大核心CSCD

为数值预测时间分数阶耦合非线性Schrödinger(TF-CNLS)方程描述的孤立子波非弹性碰撞过程,首次发展了一种耦合纯无网格有限点集法(coupled finite pointset method,CFPM).其构造过程为:1)对时间分数阶Caputo导数项采用一种高精度的差分格式;2)对空间导数采用基于Taylor展开和加权最小二乘法的有限粒子法(FPM)离散格式;3)对区域进行局部加密和采用稳定性好的双曲余弦核函数以提高数值精度.数值研究中,首先,运用CFPM对有解析解的一维TF-CNLS方程进行求解,分析了节点均匀分布或局部加密情况下的误差和收敛阶,表明给出的耦合无网格法具有近似二阶精度和易局部加密求解的灵活性;其次,运用CFPM对无解析解一维TF-CNLS方程描述的孤立子波非弹性碰撞过程进行了数值预测,其出现的波塌缩现象与整数阶下出现的多波现象截然不同;最后,与有限差分结果作对比,表明CFPM数值预测时间分数阶下孤立子波非弹性碰撞过程的复杂传播现象是可靠的.