![\[ \frac{\partial U}{\partial t} - y\frac{\partial U}{\partial x} + x\frac {\partial U}{\partial y} = 0 \]](form_270.png)
with initial value
![\[ U(x,y;t) = \left\{\begin{array}{ll} e^{32 ((x-1/2)^2 + y^2)},&\mathrm{if\ } (x-1/2)^2 + y^2 < 1/4; \\ e^{32 ((x+1/2)^2 + y^2)},&\mathrm{if\ } (x+1/2)^2 + y^2 < 1/4; \\ 0, & \mathrm{otherwise}. \end{array}\right. \]](form_271.png)
Mesh and surface of numerical result obtained as following
![\[ u_t + u u_x + u u_y = a \Delta u \]](form_272.png)
![\[ u(x,y;t) = \left( 1 + exp((x+y-t)/(2a)) \right)^{-1} \]](form_273.png)
where 
. There will be a sharp layer located along the line 
. And the steepness of the layer is depended on the parameter 
that the smaller the , the steeper the layer. The mesh and the surface of numerical result as following
in the physical domain is a 
-shaped domain, and the governing equation is given below:
![\[ \begin{array} {rcl} -\Delta \psi &=& Ra \left( {T_ x} + N {C_ x} \right), \\ {T_ t} + {\displaystyle\frac{\partial(T, \psi)} {\partial(x, y)}} &=& \Delta T, \\ {\frac \phi \sigma}{C_t} + {\displaystyle\frac{\partial(C, \psi)} {\partial(x, y)}} &=& {\frac {1} {Le}} \Delta C \end{array} \]](form_277.png)
where 
is the stream function of the flow, 
the temperature, 
the concentration of the constituent, 
the Darcy-modified Rayleigh number, 
the buoyancy ratio, 
the Lewis number, 
the porosity ratio, 
the heat capacity ratio, and
![\[ {\frac {\partial(f,g)} {\partial(x,y)}} := {\frac{\partial f}{\partial x}}{\frac{\partial g}{\partial y}} - {\frac{\partial f}{\partial y}}{\frac{\partial g}{\partial x}}. \]](form_285.png)
with initial value as
![\[ T|_{t=0}=C|_{t=0}=\left\{ \begin{array}{c} 1, x \leq \frac{1}{2},\\ 0, x> \frac{1}{2},\end{array} \right. \]](form_286.png)
and boundary value as
![\[ \psi|_{\partial \Omega} = 0, \left.{\frac{\partial T}{\partial n}}\right|_{\partial \Omega} = 0, \left.{\frac{\partial C}{\partial n}}\right|_{\partial \Omega} = 0. \]](form_287.png)
In this problem, the fluid is initially of different degrees of temperature and concentration of a certain constituent. At the beginning, the warm fluid on the left side of the domain has a less pronounced vertical gradient of hydrostatic pressure than the cold fluid on the right side. This horizontal difference of pressure will start to push the cold fluid to the left side at the bottom and warm fluid to the right side at the top. This keeps the fluid convecting until the cold fluid rests under the warm one. Meanwhile, the diffusion effect will gradually smooth out the temperature and concentration differences between the initially cold and warm fluids. We will stimulate this phenomenon for the case of a large Rayleigh number, 
. Other parameters in the governing equations are 
and 
. In the logical domain, a quasi-uniform triangulation with 1784 elements is shown in Figure fig7}. Physically, if the Rayleigh number is large enough a thin layer of large variation of temperature and concentration will keep existing until the warm fluid settles completely on top of the cold one and eventually the temperature and concentration become uniform in the whole fluid. These phenomena are clearly observed in Figure fig8}. It is seen that the mesh adapts well to the temperature and follows successfully the motion of the thin layer of large temperature and concentration variation.
The mathematical model is a system of coupled nonlinear reaction-diffusion equations:
![\[ \begin{array}{rcl} {u_t} - \Delta u &=& - {\displaystyle\frac R {\alpha \delta}} u e^{\delta (1 - 1/T)}, \\ {T_t} - {\displaystyle\frac 1 {Le}} \Delta T &=& {\displaystyle\frac R {\delta Le}} u e^{\delta (1 - 1/T)} \end{array} \]](form_293.png)
with boundary value as
![\[ \begin{array}{rcccl} u|_{t=0} &=& T|_{t=0} &=& 1, \\ u|_{\partial \Omega} &=& T|_{\partial \Omega} &=& 1 \end{array} \]](form_294.png)
where 
is the ratio of the actor, 
is the tempearture and the other parameters are as 
, 
, 
, 
. The following figures are the numerical result obtained in a 
-shaped domain.
1.4.7