# How to solve for pressure using the velocity field?

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Orlando Rojas 2015-1-25

Hi,
I'm trying to solve for the pressure field on a 120x120 grid. I have the velocity field, and I have to use the Pressure Poisson Equation to solve for the pressure. The known boundary condition is P=Patm along the right edge of the grid. Using this I can get all the boundary conditions. I've created my coefficient matrix and I've tried so many different ways to solve the problems but I keep getting and error saying "Out of memory". I've tried solving by the easiest way which is Ax=b, x=A\b. Also by using reduced row echelon form. I'm now trying to solve using lower triangular matrix and the other an upper triangular matrix, but when I try using the function lu on MATLAB, again I get the same "Out of memory" message. Any comments or ideas are appreciated.
Thank you,
Orlando R.
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Star Strider 2015-1-25
OK. Thanks. I didn’t look at the .mat files yet.

### 回答（1 个）

Youssef Khmou 2015-1-26
This problem is for non compressible fluid, first you have to explain the equation you want use, for this two dimensional case, the Poission Pressure equation is D(P)=d((1/Re).D(u)-u.d(u)), D is the Laplacian, d is divergence and Re denotes the Reynold number. What is not clear is that you need only one matrix for velocity u as field of x and y components, anyway i suggest that you use Jacobi method; for (i,j) component of P at iteration m+1 u have :
P(i,j,m+1) = ( P(i-1,j,m) + P(i+1,j,m) + P(i,j-1,m) + P(i,j+1,m) +C(i,j) )/4
C(i,j) is result of right hand side of equation, parallel to this approach try to use gradient function of velocity U to obtain C.
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Youssef Khmou 2015-1-26

For discrete solution, you initialize the matrix P with zeros, you define the boundaries conditions numerically and the size of P(M,N), then you implement the loop:
C1=0.25*(Ux(x-1,y)+Ux(x+1,y)+Ux(x,y-1)+Ux(x,y+1));
C3=0.25*(Uy(x-1,y)+Uy(x+1,y)+Uy(x,y-1)+Uy(x,y+1));
C2=2*(0.5^2)*(Ux(x-1,y)+Ux(x+1,y))*(Uy(x,y-1)*Uy(x,y+1));
P(x,y)=0.25*(P(x-1,y)+P(x+1,y)+P(x,y-1)+P(x,y+1))+rho*(C1+C2+C3);

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