具有双重虚拟边界的基本解方法求解Stokes问题

Method of fundamental solutions with double fictitious boundaries for solving Stokes problems

针对传统的基本解方法求解二维Stokes问题时,基本解矩阵病态程度高的问题,提出一个具有双重虚拟边界的基本解方法(method of fundamental solution,MFS)。基于Laplace分解,Stokes问题被转化成3个Laplace方程,进而使用具有双重虚拟边界的基本解方法求解该Laplace方程组。对应地,这3个Laplace方程的数值解被表示成源点位于2个虚拟边界上的基本解的线性组合。利用在实际边界上的边界配置点处,方程组所要满足的边界条件,得到未知系数。数值实验表明:数值解是高度准确的,与精确解相比,误差约为10-6~10-5,通过在边界上添加噪音,数值解的稳定性得到了验证;与基本解方法相比,具有双重虚拟边界的基本解方法显著地降低了基本解矩阵的条件数。

To avoid the problem that the matrix of the traditional method of fundamental solutions is highly ill-conditioned when solving the Stokes problems with two dimensions,the method of fundamental solutions with double fictitious boundaries is proposed.First,based on Laplace decomposition,Stokes problem is transformed to three Laplace equations.Then these equations are solved by the method of fundamental solutions with double fictitious boundaries.Accordingly,numerical approximate solutions of these three Laplace equations are expressed by the linear combination of fundamental solutions,whose sources distribute on the double fictitious boundaries.At last,numerical experiments show that the numerical solution is highly accurate.Comparing with the exact solution,the error is about 10-6-10-5.Besides,noises are added into the boundary conditions to demonstrate the stability of the method of fundamental solutions with double fictitious boundaries.At the same time,compared with the method of fundamental solutions,the method of fundamental solutions with double fictitious boundaries,apparently reduce the condition number of MFS matrix.