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A moving finite element method for solving two-dimensional coupled Burgers’ equations at high Reynolds numbers
Applied Mathematics Letters ( IF 3.7 ) Pub Date : 2022-06-21 , DOI: 10.1016/j.aml.2022.108271
Xiaohua Zhang , Xinmeng Xu

The coupled Burgers’ equations at high Reynolds numbers usually have sharp gradients or are discontinuous in the solution. Therefore, it is difficult to obtain analytical solutions. This paper aims to use the moving finite element method proposed by Li et al. (2001) to get stable and high-precision numerical solutions for the coupled Burgers’ equations at high Reynolds numbers. The method decouples the mesh equation and partial differential equation (PDE) into two unrelated parts, mesh reconstruction and PDE solver. The mesh reconstruction constructs the harmonic mapping between the physical and logical domains through an iterative method so that the mesh structure maintains harmonics after multiple numerical integrations. We compute three classic numerical examples. Numerical results show that the moving finite element method effectively solve the coupled Burgers’ equations at high Reynolds numbers, obtain stable numerical results, and achieve higher numerical accuracy. During the evolution of the solution, the mesh is always concentrated in the position where the solution has sharp gradients.



中文翻译:

求解高雷诺数下二维耦合 Burgers 方程的移动有限元方法

高雷诺数下的耦合 Burgers 方程通常具有陡峭的梯度或在解中是不连续的。因此,很难获得解析解。本文旨在利用李等人提出的移动有限元法。(2001)获得高雷诺数下耦合伯格斯方程的稳定和高精度数值解。该方法将网格方程和偏微分方程 (PDE) 解耦为两个不相关的部分,即网格重建和 PDE 求解器。网格重构通过迭代方法构建物理域和逻辑域之间的谐波映射,使网格结构在多次数值积分后保持谐波。我们计算了三个经典的数值例子。数值结果表明,移动有限元法有效地求解了高雷诺数下的耦合Burgers方程组,获得了稳定的数值结果,并获得了较高的数值精度。在解的演化过程中,网格总是集中在解具有急剧梯度的位置。

更新日期:2022-06-21
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