{"title":"An Efficient Second-Order Algorithm Upon MAC Scheme for Nonlinear Incompressible Darcy–Brinkman–Forchheimer Model","authors":"Pengshan Wang, Wei Liu, Gexian Fan, Yingxue Song","doi":"10.1007/s00021-024-00851-w","DOIUrl":null,"url":null,"abstract":"<p>In this paper, the Marker and Cell scheme based on a two-grid algorithm is proposed for the two-dimensional incompressible Darcy–Brinkman–Forchheimer equations in porous media. The motivation of the two-grid Marker and Cell algorithm is figuring out a nonlinear equation on a coarse grid with mesh size <i>H</i> and a linear equation on a fine grid with mesh size <i>h</i>. A small positive parameter <span>\\(\\varepsilon \\)</span> is introduced. By using it, the non-differentiable nonlinear term can be transformed into the term which is twice continuously differentiable. The error estimates of the velocity and pressure in the <span>\\(L^2\\)</span> norms are obtained, which show <span>\\(O(\\varepsilon +H^4+h^2)\\)</span>. Second-order accuracy for some terms of velocity in the <span>\\(H^1\\)</span> norms is also obtained. Several numerical experiments are provided to confirm the availability of this efficient second-order algorithm. Behavior of the fluid flow with different Brinkman number is considered.</p>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2024-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Fluid Mechanics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00021-024-00851-w","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, the Marker and Cell scheme based on a two-grid algorithm is proposed for the two-dimensional incompressible Darcy–Brinkman–Forchheimer equations in porous media. The motivation of the two-grid Marker and Cell algorithm is figuring out a nonlinear equation on a coarse grid with mesh size H and a linear equation on a fine grid with mesh size h. A small positive parameter \(\varepsilon \) is introduced. By using it, the non-differentiable nonlinear term can be transformed into the term which is twice continuously differentiable. The error estimates of the velocity and pressure in the \(L^2\) norms are obtained, which show \(O(\varepsilon +H^4+h^2)\). Second-order accuracy for some terms of velocity in the \(H^1\) norms is also obtained. Several numerical experiments are provided to confirm the availability of this efficient second-order algorithm. Behavior of the fluid flow with different Brinkman number is considered.
本文针对多孔介质中的二维不可压缩达西-布林克曼-福克海默方程，提出了基于双网格算法的 Marker and Cell 方案。双网格 Marker and Cell 算法的动机是在网格尺寸为 H 的粗网格上计算非线性方程，在网格尺寸为 h 的细网格上计算线性方程。通过使用它，不可微的非线性项可以转化为两次连续可微项。得到了速度和压力在 \(L^2\) 规范下的误差估计，显示了 \(O(\varepsilon +H^4+h^2)\).在 \(H^1\) 规范下，一些速度项的二阶精度也得到了。提供了几个数值实验来证实这种高效二阶算法的可用性。考虑了不同布林克曼数的流体流动行为。
期刊介绍：
The Journal of Mathematical Fluid Mechanics (JMFM)is a forum for the publication of high-quality peer-reviewed papers on the mathematical theory of fluid mechanics, with special regards to the Navier-Stokes equations. As an important part of that, the journal encourages papers dealing with mathematical aspects of computational theory, as well as with applications in science and engineering. The journal also publishes in related areas of mathematics that have a direct bearing on the mathematical theory of fluid mechanics. All papers will be characterized by originality and mathematical rigor. For a paper to be accepted, it is not enough that it contains original results. In fact, results should be highly relevant to the mathematical theory of fluid mechanics, and meet a wide readership.
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