Novel Symplectic Discrete Singular Convolution Method for Hamiltonian PDEs
Novel Symplectic Discrete Singular Convolution Method for Hamiltonian PDEs
复制标题
哈密顿偏微分方程的新辛离散奇异卷积方法
DOI：
10.4208/cicp.scpde14.32s
复制
发表时间：
2016
影响因子：
3.7
通讯作者：
Wang Yushun
中科院分区：
文献类型：
--
作者：
Cai Wenjun;Zhang Huai;Wang Yushun
本文研究了求解Hamilton偏微分方程的离散奇异卷积方法。解析地给出了离散奇异卷积的两个Delta型核所对应的微分矩阵，它们具有高阶精度和带限结构的性质，因此可以作为Hamilton偏微分方程空间离散化的优良候选者.以非线性薛定谔方程和耦合薛定谔方程为例，将这类微分矩阵与适当的辛时间积分相结合，构造了两个辛积分器，并证明了这两个辛积分器都满足平方守恒律.通过与中心有限差分法、Fourier拟谱法、小波配置法等数值算例的比较，证明了新的辛积分方法的优越性。
This paper explores the discrete singular convolution method for Hamiltonian PDEs. The differential matrices corresponding to two delta type kernels of the discrete singular convolution are presented analytically, which have the properties of high-order accuracy, bandlimited structure and thus can be excellent candidates for the spatial discretizations for Hamiltonian PDEs. Taking the nonlinear Schrodinger equation and the coupled Schrodinger equations for example, we construct two symplectic integrators combining this kind of differential matrices and appropriate symplectic time integrations, which both have been proved to satisfy the square conservation laws. Comprehensive numerical experiments including comparisons with the central finite difference method, the Fourier pseudospectral method, the wavelet collocation method are given to show the advantages of the new type of symplectic integrators.