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首页> 外文期刊>International journal of numerical methods for heat & fluid flow >Galerkin proper orthogonal decomposition-reduced order method (POD-ROM) for solving generalized Swift-Hohenberg equation
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Galerkin proper orthogonal decomposition-reduced order method (POD-ROM) for solving generalized Swift-Hohenberg equation

机译：解广义Swift-Hohenberg方程的Galerkin固有正交分解降阶方法（POD-ROM）

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Purpose The current paper aims to develop a reduced order discontinuous Galerkin method for solving the generalized Swift-Hohenberg equation with application in biological science and mechanical engineering. The generalized Swift-Hohenberg equation is a fourth-order PDE; thus, this paper uses the local discontinuous Galerkin (LDG) method for it. Design/methodology/approach At first, the spatial direction has been discretized by the LDG technique, as this process results in a nonlinear system of equations based on the time variable. Thus, to achieve more accurate outcomes, this paper uses an exponential time differencing scheme for solving the obtained system of ordinary differential equations. Finally, to decrease the used CPU time, this study combines the proper orthogonal decomposition approach with the LDG method and obtains a reduced order LDG method. The circular and rectangular computational domains have been selected to solve the generalized Swift-Hohenberg equation. Furthermore, the energy stability for the semi-discrete LDG scheme has been discussed. Findings The results show that the new numerical procedure has not only suitable and acceptable accuracy but also less computational cost compared to the local DG without the proper orthogonal decomposition (POD) approach. Originality/value The local DG technique is an efficient numerical procedure for solving models in the fluid flow. The current paper combines the POD approach and the local LDG technique to solve the generalized Swift-Hohenberg equation with application in the fluid mechanics. In the new technique, the computational cost and the used CPU time of the local DG have been reduced.
机译：目的本论文旨在开发一种降阶不连续伽勒金方法，用于求解广义Swift-Hohenberg方程，并将其应用于生物科学和机械工程中。广义的Swift-Hohenberg方程是四阶PDE。因此，本文采用局部不连续伽勒金（LDG）方法。设计/方法/方法首先，通过LDG技术离散化空间方向，因为此过程导致基于时间变量的非线性方程组。因此，为了获得更准确的结果，本文使用指数时间微分方案来求解所获得的常微分方程组。最后，为减少所用的CPU时间，本研究将适当的正交分解方法与LDG方法相结合，并获得了降阶LDG方法。选择了圆形和矩形计算域来求解广义的Swift-Hohenberg方程。此外，还讨论了半离散LDG方案的能量稳定性。结果表明，与没有适当的正交分解（POD）方法的局部DG相比，新的数值程序不仅具有合适的可接受的精度，而且具有较低的计算成本。原创性/价值局部DG技术是解决流体流动模型的有效数值程序。本文将POD方法与局部LDG技术相结合，以解决广义Swift-Hohenberg方程及其在流体力学中的应用。在新技术中，减少了本地DG的计算成本和CPU时间。

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