论文标题
全球能源保留模型的模型降低多合成PDE
Global energy preserving model reduction for multi-symplectic PDEs
论文作者
论文摘要
许多哈密顿系统可以以多核形式重铸。我们为保留全球能量的多型哈密顿部分微分方程（PDE）开发了一个减少阶模型（ROM）。全阶溶液是通过空间和全球能量保留平均向量场（AVF）方法的有限差异化获得的。 ROM的构建方式与全阶模型（FOM）相同，该模型（FOM）将适当的正交分解（POD）与Galerkin投影相同。降级系统的结构与FOM具有相同的结构，并保留了离散的减少全球能量。应用离散的经验插值方法（DEIM），在在线阶段有效地计算了降级解决方案。 DEIM近似与非线性哈密顿量的DEIM近似是绑定的。对于多合格形式的Korteweg de Vries（KDV）方程，Zakharov-Kuznetzov（ZK）方程（ZK）方程（Zakharov-Kuznetzov（ZK）方程，ROM的准确性和计算效率已证明。减少能量的保存表明，减少订单的溶液确保了溶液的长期稳定性。
Many Hamiltonian systems can be recast in multi-symplectic form. We develop a reduced-order model (ROM) for multi-symplectic Hamiltonian partial differential equations (PDEs) that preserves the global energy. The full-order solutions are obtained by finite difference discretization in space and the global energy preserving average vector field (AVF) method. The ROM is constructed in the same way as the full-order model (FOM) applying proper orthogonal decomposition (POD) with the Galerkin projection. The reduced-order system has the same structure as the FOM, and preserves the discrete reduced global energy. Applying the discrete empirical interpolation method (DEIM), the reduced-order solutions are computed efficiently in the online stage. A priori error bound is derived for the DEIM approximation to the nonlinear Hamiltonian. The accuracy and computational efficiency of the ROMs are demonstrated for the Korteweg de Vries (KdV) equation, Zakharov-Kuznetzov (ZK) equation, and nonlinear Schr{ö}dinger (NLS) equation in multi-symplectic form. Preservation of the reduced energies shows that the reduced-order solutions ensure the long-term stability of the solutions.
