Abstract

The motivation in this article is to foster and conduct an in-depth investigation on reduced-order modeling (ROM) techniques via proper orthogonal decomposition (POD) and discrete empirical interpolation method (DEIM) in conjunction with high-order Legendre spectral element method (SEM) applied for time-dependent linear and nonlinear heat conduction problems. This approach significantly relieves the CPU burden, yet preserves the numerical accuracy of high-order schemes; this is especially pronounced for linear transient problems. For nonlinear cases, we use DEIM to save CPU time for nonlinear counterparts. For this purpose, we first obtain training data by solving the system of full-order models (FOM) using snapshot techniques to construct the POD basis. The well-known generalized single step single solve (GSSSS-1) framework is next employed for the first-order system for the time discretization. Numerical results for both linear and nonlinear heat conduction problems such as in a nuclear reactor, etc., show excellent agreement between FOM solutions and ROM solutions, and the CPU advantages via POD ROM techniques are also prominent for high-order Legendre SEM.

摘要

本文的动机是通过适当的正交分解 (POD) 和离散经验插值法 (DEIM) 结合高阶勒让德谱元法，对降阶建模 (ROM) 技术进行深入研究。 SEM）应用于瞬态线性和非线性热传导问题。这种方法显着减轻了 CPU 负担，同时保留了高阶方案的数值精度；这对于线性瞬态问题尤其明显。对于非线性情况，我们使用 DEIM 为非线性对应项节省 CPU 时间。为此，我们首先通过使用快照技术求解全阶模型 (FOM) 系统来构建 POD 基础来获得训练数据。接下来将著名的广义单步单步求解 (GSSSS-1) 框架用于时间离散化的一阶系统。线性和非线性热传导问题（如核反应堆等）的数值结果表明 FOM 解决方案和 ROM 解决方案之间具有极好的一致性，并且通过 POD ROM 技术的 CPU 优势对于高阶 Legendre SEM 也很突出。