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首页> 外文期刊>Journal of Computational Physics >POD-based model order reduction with an adaptive snapshot selection for a discontinuous Galerkin approximation of the time-domain Maxwell's equations
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POD-based model order reduction with an adaptive snapshot selection for a discontinuous Galerkin approximation of the time-domain Maxwell's equations

机译：基于POD的模型顺序减少了一种用于时域Maxwell等式的不连续Galerkin近似的自适应快照选择

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摘要

In this work we report on a reduced-order model (ROM) for the system of time-domain Maxwell's equations discretized by a discontinuous Galerkin (DG) method. We leverage previous results on proper orthogonal decomposition (POD) [1,2], in particular for the wave equation [3], to propose a POD-based ROM with an adaptive snapshot selection strategy where the snapshots are produced by a high order discontinuous Galerkin time-domain (DGTD) solver. The latter is formulated on an unstructured simplicial mesh, and combines a centered scheme for the definition of the numerical fluxes of the electric and magnetic fields at element interfaces with a second order leap-frog (LF2) time scheme for the time integration of the associated semi-discrete equations. The POD-based ROM is established by projecting (Galerkin projection) the global semi-discrete DG scheme onto a low-dimensional space generated by the POD basis vectors. Inspired from the approach followed in [2,3], we derive error bounds for the POD-based ROM that is adapted to our particular modeling and discretization settings. The adaptive snapshot selection algorithm exploits the results of this analysis to measure the control error. A snapshot choosing rule aiming at keeping the error estimate close to a target selection error tolerance is proposed, which is similar to the standard rules found in adaptive time-stepping ordinary differential equations (ODEs) solvers. An incremental singular value decomposition (ISVD) algorithm is used to update the SVD on-the-fly when a new snapshot is available. The purpose of this adaptive selection strategy is to save memory without storing snapshots, while producing a smaller error. Numerical experiments for the 2-D time-domain Maxwell's equations nicely illustrate the performance of the resulting POD-based ROM with adaptive snapshot selection. (C) 2019 Elsevier Inc. All rights reserved.
机译：在这项工作中，我们报告了用间断伽辽金（DG）方法离散的时域麦克斯韦方程组的降阶模型（ROM）。我们利用之前关于适当正交分解（POD）[1,2]的结果，特别是波动方程[3]，提出了一种基于POD的ROM，具有自适应快照选择策略，其中快照由高阶间断伽辽金时域（DGTD）解算器生成。后者建立在非结构单纯形网格上，并将用于定义单元界面电场和磁场数值通量的中心格式与用于相关半离散方程时间积分的二阶蛙跳（LF2）时间格式相结合。基于POD的ROM是通过将全局半离散DG格式投影（Galerkin投影）到POD基向量生成的低维空间来建立的。受[2,3]中所述方法的启发，我们推导了适用于特定建模和离散化设置的基于POD的ROM的误差范围。自适应快照选择算法利用该分析的结果来测量控制误差。提出了一种与自适应时间步进常微分方程（ODE）求解器中的标准规则相似的快照选择规则，旨在使误差估计接近目标选择误差容限。当新快照可用时，使用增量奇异值分解（ISVD）算法动态更新SVD。这种自适应选择策略的目的是在不存储快照的情况下节省内存，同时产生较小的错误。二维时域麦克斯韦方程组的数值实验很好地说明了基于POD的ROM的自适应快照选择的性能。（C） 2019爱思唯尔公司版权所有。

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