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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2023, Vol. 40 ›› Issue (5): 779-792.doi: 10.3969/j.issn.1005-3085.2023.05.007

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求解不规则区域上椭圆方程的一种Cartesian网格方法及其在Navier-Stokes方程中的应用

史卫东1,  徐建军2,  岳孝强3   

  1. 1. 山西财经大学应用数学学院，太原 030006;
    2. 中国科学院重庆绿色智能技术研究院，重庆 400714;
    3. 科学工程计算与数值仿真湖南省重点实验室，智能计算与信息处理教育部重点实验室，湘潭 411105
  • 收稿日期:2021-02-03 接受日期:2022-11-18 出版日期:2023-10-15 发布日期:2023-12-15
  • 通讯作者: 徐建军 E-mail: xujianjun@cigit.ac.cn
  • 基金资助:
    国家自然科学基金(11601462; 11971414)；湖南省科技厅科研基金(2018WK4006)；山西财经大学青年科研基金(QN2019023)；科学挑战计划(TZZT2016002).

A Cartesian Grid Method for the Elliptic Equations on Irregular Domains with Application to the Navier-Stokes Equations

SHI Weidong1,  XU Jianjun2,  YUE Xiaoqiang3   

  1. 1. School of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan 030006;
    2. Chongqing Institute of Green and Intelligent Technology, Chinese Academy of Sciences, Chongqing 400714;
    3. Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education, Xiangtan 411105
  • Received:2021-02-03 Accepted:2022-11-18 Online:2023-10-15 Published:2023-12-15
  • Contact: J. Xu. E-mail address: xujianjun@cigit.ac.cn
  • Supported by:
    The National Natural Science Foundation of China (11601462; 11971414); the Project of Scientific Research Fund of Hunan Provincial Science and Technology Department (2018WK4006); the Project of Youth Research Fund of Shanxi University of Finance and Economics (QN2019023); the Science Challenge Project (TZZT2016002).

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摘要：

提出了一种求解不规则边界上有Robin边界条件的椭圆方程的Cartesian网格方法。该椭圆方程经重写后转化为定义在矩形区域上的椭圆界面问题，进而采用水平集浸入界面方法(IIM)对其进行求解。特别地，Robin边界条件采用单边三次插值离散。随后，利用该方法求解定义在不规则区域上的Navier-Stokes程。Navier-Stokes方程的解法器由求解速度方程的虚拟流体方法(GFM)和辅助变量方程的IIM耦合而成。数值测试表明，椭圆方程的解法器能够产生二阶精度的数值解和梯度，而且能够快速收敛，Navier-Stokes方程的解法器产生了二阶精度的速度及一阶精度的压力。圆柱绕流的仿真验证了Navier-Stokes方程解法器的鲁棒性。

关键词: 椭圆方程, Navier-Stokes方程, Cartesian网格方法, 水平集方法, 浸入界面方法

Abstract:

A Cartesian grid method is presented for solving elliptic equation on irregular domains with Robin boundary condition in this paper. The elliptic equation is reformulated into an elliptic interface problem on a larger regular domain, then solved by using the level-set immersed interface method (IIM) recently developed. In particular, the Robin boundary condition is discretized using one-sided cubic interpolation. The method is applied to solving the Navier-Stokes equations on irregular domains. The Navier-Stokes solver couples the ghost fluid method for the velocity equations and the IIM for the auxiliary variable equation. Numerical tests show that second-order accuracy is achieved in both solution and gradient for the elliptic solver, and with fast convergence. The Navier-Stokes solver produces second-order accurate velocity and one-order accurate pressure. The robustness of the Navier-Stokes solver is demonstrated through simulations of flow around a circular cylinder.

Key words: elliptic equation, Navier-Stokes equations, Cartesian grid method, level-set method, immersed interface method

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