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  

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

作　　者：史卫东 徐建军[2] 岳孝强 SHI Weidong;XU Jianjun;YUE Xiaoqiang(School of Applied Mathematics,Shanxi University of Finance and Economics,Taiyuan 030006;Chongqing Institute of Green and Intelligent Technology,Chinese Academy of Sciences,Chongqing 400714;Hunan Key Laboratory for Computation and Simulation in Science and Engineering,Key Laboratory of Intelligent Computing&Information Processing of Ministry of Education,Xiangtan 411105)

机构地区：[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

出　　处：《工程数学学报》2023年第5期779-792,共14页Chinese Journal of Engineering Mathematics

基　　金：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)。

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。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.

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 

分 类 号：O241.82[理学—计算数学]