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

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.