[mathgra]

Im master student and as a final project my teacher give a steady flow in a lid-driven cavity by using Naiver-Stokes equations in stream function vorticity form with deltax=deltay=0.1 Re=50,100 stopping criteria for both streamfunctions and vorticity is 0.0001.
initial vorticity is 0 everywhere , stream function zero at boundaries and partial derivative of streamfunction with respect to y is 1. So the only top lid is moving.
We use finite difference for the scheme.
I have to drive streamlines and vorticity equations. I try to write matlab code myself program runs but I think I missing some point and I am not sure how to put stopping criteria in loop.


Matlab code that I write
clear all
close all
n=11;
x=linspace(0,1,n);
y=linspace(0,1,n);
h=x(2)-x(1);
criteria=0.0001;
Re=50;
[xx,yy]=meshgrid(x,y);
vortnew=zeros(n,n);
psi=zeros(n,n);
U=1;
run=1;
iter=0;


for i=1:n
for j=1:n
psi(i,j,1)=0;
vortnew(i,j,1)=0;
if j==n
psi(i,j,1)=0;
vortnew(i,j,1)=-2/h;
end
end
end

% vorticity equation
if run==1;

while ( 1>criteria )

vortold=vortnew;
psinew=psi;
iter=iter+1;
for i=1:n
for j=1:n
if j~=1 && j~=n && i~=1 && i~=n % Eger boundaryde degilse

f(i,j)=-((psi(i,j+1)-psi(i,j-1))*(vortnew(i+1,j)-vortnew(i-1,j)))/4*(h^2)
+((psi(i+1,j)-psi(i-1,j))*(vortnew(i,j+1)-vortnew(i,j-1)))/4*(h^2)
+(vortnew(i+1,j)-4*vortnew(i,j)+vortnew(i-1,j)+vortnew(i,j+1))/Re*(h^2);

elseif j==1 && i~=1 && i~=n
vortnew(i,j)=2*(psi(i,j)-psi(i,j+1))/h^2; %Bottom
elseif j==n && i~=1 && i~=n
vortnew(i,j)=2*(psi(i,j)-psi(i,j-1))/h^2-2*U/h; % Top
elseif i==1 && j~=1 && j~=n
vortnew(i,j)=2*(psi(i,j)-psi(i+1,j))/h^2; % Left
elseif i==n && j~=1 && j~=n
vortnew(i,j)=2*(psi(i,j)-psi(i-1,j))/h^2; % Right
elseif (i==1 && j==1) %corners equal to neighbors or zero
vortnew(i,j)=0;
elseif (i==n && j==1)
vortnew(i,j)=0;
elseif (i==1 && j==n)
vortnew(i,j)=(vortnew(i+1,j));
elseif (i==n && j==n)
vortnew(i,j)=(vortnew(i-1,j));
end
end
end
end
for i=1:n
for j=1:m
vortold(i,j)=vortnew(i,j)+(1-0.05)*f(i,j);
% To guarantee convergence she give above statement
end
end
%streamfunction equation non-boundary points
for i=1:n
for j=1:n
if j~=1 && j~=n && i~=1 && i~=n
psi(i,j)=0.25*(h^2*vortnew(i,j)+psi(i+1,j)+psi(i-1,j)+psi(i,j+1)+psi(i,j-1));
else
psi(i,j)=0;
end
end
end
end

[FMDenaro]

Stopping criteria are based on the evaluation of some norm of the (normalized) residuals. You have two equations so you need to apply the norm on two vectors of the residuals.
See https://www.sciencedirect.com/scienc...21999117306939