![\[ \begin{array}{l} \vec{u}_{t}+(\vec{u}\cdot\nabla)\vec{u} +\nabla p =\frac{1}{Re} \Delta \vec{u}\\ \nabla \cdot \vec{u} = 0 \end{array} \]](form_311.png)
最近我们在移动网格方面的研究在以下几个方面，
For imcompressible, the singularity is comparatively weak bacause the velocity is divergence free that the mechanism to construct control function is much more difficult than those numerical examples we mentioned before. And to update the velocity to the new mesh efficiently but still divergence free is still a problem.
我们研究了对不可压流构造控制函数的方案，并且作为我们提出来的一般的网格 间插值的公式，开发了一个新的旧网格到新网格插值的方案，使得不可压条件能 够保持。下面是两个算例：
We studied the stragety to construct monitor function for imcompressible flow and give a general frame to update the solution to the new mesh, while at the same time keep certain constraint of the solution. The following are two numerical examples:
我们使用原始变量进行求解，计算的区域是标准的正方形，我们取双周期边界条件
![\[ \begin{array}{rcl} \vec{u}(-1,y;t) &=& \vec{u}(1,y;t) \\ \vec{u}(x,-1;t) &=& \vec{u}(x,1;t) \\ p(-1,y;t) &=& p(1,y;t) \\ p(-1,y;t) &=& p(1,y;t) \end{array} \]](form_312.png)
初值为
![\[ \begin{array}{rcl} u &=& \left\{\begin{array}{ll} \tanh(\rho(y - 0.25)) & \mathrm{for\ } y < 0.5;\\ \tanh(\rho(0.75 - y)) & \mathrm{for\ } y > 0.5. \end{array}\right.\\ v &=& \delta \sin(2\pi x) \end{array} \]](form_313.png)
得到的一个计算结果的图形为：
![\[ \begin{array}{l} \rho_t + \vec{u}\cdot\nabla\rho = 0\\ \vec{u}_{t}+(\vec{u}\cdot\nabla)\vec{u} +\nabla p = \left(\begin{array}{c} 0 \\ \rho \end{array}\right) \\ \nabla \cdot \vec{u} = 0 \end{array} \]](form_314.png)
![\[ \begin{array}{l} \displaystyle\frac{\partial \rho }{\partial t}+\displaystyle\frac{\partial \rho u}{\partial x}+\displaystyle\frac{\partial \rho v}{\partial y} = 0 \\ \displaystyle\frac{\partial \rho u}{\partial t}+\displaystyle\frac{\partial \left( p+\rho u^{2}\right) }{\partial x}+\displaystyle\frac{\partial \rho uv}{\partial y} = 0 \\ \displaystyle\frac{\partial \rho v}{\partial t}+\displaystyle\frac{\partial \rho uv}{\partial x}+\displaystyle\frac{\partial \left( p+\rho v^{2}\right) }{\partial y} = 0 \\ \displaystyle\frac{\partial e}{\partial t}+\displaystyle\frac{\partial \left( u\left( p+e\right) \right) }{\partial x}+\displaystyle\frac{\partial \left( v\left( p+e\right) \right) }{\partial y} = 0 \end{array} \]](form_315.png)
状态方程为
![\[ e=\frac{p}{\gamma -1}+\frac{1}{2}\rho \left( u^{2} + v^{2} \right) \]](form_316.png)
![\[ \left(\rho, u, v, p\right) = \left\{ \begin{array}{ll} (1.1, 0.0, 0.0, 1.1), & \rm{if\ } x>0.5,y>0.5, \\ (0.5065, 0.8939, 0.0, 0.35), & \rm{if\ } x<0.5, y>0.5, \\ (1.1, 0.8939, 0.8939, 1.1), & \rm{if\ } x<0.5, y<0.5, \\ (0.5065, 0.0, 0.8939, 0.35), & \rm{if\ } x>0.5, y<0.5. \end{array}\right. \]](form_317.png)
![$ [0, 4]\times[0, 1] $](form_318.png)
. A right moving shock is initially positioned at 
and makes a 
degree angle with the 
-axis. The inflow is with Mach number 10. The boundary condition at bottom is the exact post shock condition from $0$ to 
and is reflective for the rest. At the top boundary, the flow values are set to describe the exact motion of the Mach 10 shock. On the left and right boundaries, the inflow and outflow boundary conditions are used respectively.
1.4.7