How Do You Numerically Solve Coupled PDEs with Continuity Constraints?

[keyns]

I'm new, hi all.

I have two coupled equations, one of which is continuity. Basically, my problem comes down to the following system:

(1) $u=f(v)$ (similarly, $v=g(u)$. Here, u and v are the components of a vector field, ie u=u(x,y) and v=v(x,y).

(2) Continuity: $\nabla \cdot \textbf{u} = 0$ or $u_{x}+v_{y}=0$

From here, I can find the following expressions

$u_{x} = -g_{y} \left( u \right)$
$v_{y} = -f_{x} \left( v \right)$

Which I think leaves an equation of the form $G \left( u,u_{x},u_{y} \right)=0$ and $F \left( v,v_{x},v_{y} \right)=0$

It seems to me my original problem has two variables and I have two equations. I think this should be solvable, but I don't know how. Any help please? Thanks in advance!

--edit-- p.s. I'm looking for a numerical (discrete) solution.

 

[Mute]

I'm new, hi all.

I have two coupled equations, one of which is continuity. Basically, my problem comes down to the following system:

(1) $u=f(v)$ (similarly, $v=g(u)$. Here, u and v are the components of a vector field, ie u=u(x,y) and v=v(x,y).

If u = f(v) and v = g(u), don't you just have a (possibly nonlinear) system of equations to solve for v and u? What is the continuity condition (2) for? Do you get more than one solution by solving (1) and you need (2) to choose a solution of interest?

 

[keyns]

If u = f(v) and v = g(u), don't you just have a (possibly nonlinear) system of equations to solve for v and u? What is the continuity condition (2) for? Do you get more than one solution by solving (1) and you need (2) to choose a solution of interest?

Actually I have only one relation for $u$ and $v$ that I can write it as $u(v)$ or $v(u)$. Sorry for the confusion. Otherwise you would be right. To clarify my equations:

(1) A relation for $u$ and $v$ (if I have $u$, I have $v$ and vice versa)
(2) A relation for $u_{x}$ and $v_{y}$ (if I have $u_{x}$, I have $v_{y}$ and vice versa

--edit-- Which then, after some rewriting, leads to two ODE's $G$ and $F$ as stated before. I just don't know how to solve those.