I have to resolve a PDE COUPLED system, and I don't find any method for this. Any idea ???

Runge-Kutta?

How do tou want to use RK with it ?

P_ i + P_ j ( dv_ i / dx_ j ) - v_ j ( P_ i / dx_ j ) - L_ {jj} P_ i = 0

Where v_i and dv_i/dx_j are known. L_{jj} is the Laplacian operator. And _i is the Einstein notation. We have to resolve P_i by an implicit method. This is a COUPLED equation due to P_ j ( dv_ i / dx_ j ).

O.Doche

What is dimension of the system? 2D or 3D?

If you know the components of v and its derivatives, they can be substituted into the equation for p. Thus you have one equation to be solved. In this case, there's no equation to couple it to.

However, if components of v and its derivatives are known only through a second equation, then that equation and the equation for p ARE certainly coupled. If the second equation contains p, the two equations need to be solved simultaneously. Otherwise, solve for v and its derivatives first and substitute into the equation for p.

What boundary conditions are in force?

When all this is known, you can think about solving the p equation - or the coupled system.

I have a 3D problem. So I have 3 equations.

P_1 + f(P_1,P_2,P_3 and derivatives)=0 P_2 + f(P_1,P_2,P_3 and derivatives)=0 P_3 + f(P_1,P_2,P_3 and derivatives)=0

P_1, P_2 and P_3 ( and their derivatives ) are unknown. The problem is linear. The problem is that I have P_1, P_2 and P_3 (implicit) in the SAME equation. Thanks O.doche

Sorry, for previous message, it is not readable.

I have a 3D problem. So I have 3 equations.

P_1 + f (P_1,P_2,P_3 and derivatives)=0

P_2 + f (P_1,P_2,P_3 and derivatives)=0

P_3 + f (P_1,P_2,P_3 and derivatives)=0

P_1, P_2 and P_3 ( and their derivatives ) are unknown. The problem is linear and I have P_1, P_2 and P_3 (implicit) in the SAME equation.

Thanks O.doche

And Boundary conditions are known and very simple.

If there are N points then you have a system of 3N coupled equations. You basically have an equation like

AP=0

where A is 3N x 3N and P is 3N x 1. You have to find out efficient way of solving this. Make use of the fact that A will be sparse. Make use of the fact that you have structured grid, some sort ADI ?

There is nothing special to be done because of the coupled system. You have to apply whatever iterative technique you choose to each equation.

It might go like this

1. Initialize P to some value
2. Apply one cycle of iterations to P_1 equation
3. Apply one cycle to P_2 equation
4. Apply one cycle to P_3 equation
5. If not converged go to step 2

In each step you will be using the latest available solution of P_1, P_2 and P_3. So the coupling is taken care of.

Also notice that you have a homogeneous equation so that P is determined only to within a constant (like pressure in incompressible flow). So you may have to anchor the solution in order to prevent drifting. Of course your bc might fix some reference value.

Opps. Sorry.

They sure do look coupled!

when you use iterative method, you can use pointwise exact 3x3 matrix inversion. And you may use multigrid method to solve it efficiently.

1) Have you heard ( or seen ) a book where I can find efficient solution for PDE Coulped Equation resolution ?

2) Praaven, I will try your method soon. What do you think about the computational time ( compare to a similar non-coupled system ). Is there a big difference?

3) Thanks Jim_Park, Junseok Kim, HVN and Praveen C for your help.