CFD - Flat Plate Boundary Layer

I want to solve Continuity and x-momentum Equations for a boundary layer over a flat plate. It is a 0 pressure gradient flow.

My equations are diverging, and I can't seem to find why they are. I am using an under-relaxation technique to obtain convergence. Could anyone help to spot my mistake?

Using a central/forward/backward difference scheme based on where I am at on my grid.

MATLAB Code is below:

clc, clear
close all
%% Parameters
rho = 1.225; % kg/m^3 - Air sea level standard day
U = 10; % m/s - freestream
mu = 1.789e-5; % kg/(m*s) - Air sea level standard day
nu = mu/rho; % m^2/s
L = 0.25; % m
H = 0.05; % m
 
% nx and ny defines matrix grid, dx and dy for step size
nx = 51;
ny = 11;
dx = .005; % 50 grid points in x-dir - .005m spacing
dy = .005; % 10 grid points in y-dir - .005m spacing
 
% Re based on length
Re_L = U*L/nu;
 
% Filling mesh with 0 to begin
v = zeros(ny,nx); % All initial v are 0
u = ones(ny,nx)*U; % Filling the entire mesh with all u = 10
 
% Boundary Conditions
% Goes (j,i) each segment, fills mesh with known values
u(:,1) = U; % Inlet
v(:,1) = 0; % Inlet
u(1,:) = 0; % No Slip wall
v(1,:) = 0; % No Slip wall
u(ny,:) = U; % Free stream (top of mesh)

%% Solving
% Guess values of alpha
alphau = .1; % use larger than .001 values per McQ %guess, will need to figure out best value for this 
alphav = .01;

% Somehow -.01 au and -.005 av is working better
 
% Differences
diff_u=[];
diff_v=[];
diff_u_max=[];
diff_v_max=[];
diff_both=[];
 
% Tolerance and convergence for main while loop
tol = .001;
z = 0; % Starting the index for iterations

while z < 100 % This is number of iterations it performs
    for i = 2:1:nx-1
        for j = 2:1:ny-1
            if i == nx
                % use backwards
                v_old = v(j,i);
                %v(j,i) = ((u(j,i-1) - u(j,i))/dx)*dy + (v(j-1,i));
                v(j,i) = ((-u(j,i-1) - u(j,i))/dx)*dy + (v(j-1,i));
                change_v = v_old - v(j,i);
                v(j,i)= v_old + (alphav * change_v);
                u_old = u(j,i);
                u(j,i) = u(j,i-1) + (((-v(j,i) + v(j-1,i)) / dy) * dx);
                change_u = u_old - u(j,i+1);
                u(j,i+1) = u_old + (alphau * change_u);
                diff_u = [diff_u change_u];
                diff_v = [diff_v change_v];
            elseif j == ny
                v_old = v(j,i);
                v(j,i) = v(j-1,i);
                u_old = u(j,i+1);
                u(j,i+1) = U;
            else
                % This is central differences
            u_old = u(j,i+1);
            % Using u from Peter_Puttkammer
            % u(j,i+1) = u(j,i) + (v(j,i) * ((u(j+1,i) - (2*u(j,i)) + u(j-1,i)) / u(j,i)) * dx/(dy^2)) - (((u(j+1,i) - u(j-1,i)) / 2)*v(j,i)/u(j,i)*dx/dy);
            u(j,i+1) = (2*dx/u(j,i)) * (((mu/rho)*((u(j-1,i) + 2*u(j,i) + u(j+1,i))/(dy^2))) - (v(j,i)*((u(j-1,i) - u(j+1,i))/(2*dy)))) + u(j,i-1);
            change_u = u_old - u(j,i+1);
            u(j,i+1) = u_old + (alphau * change_u);
            v_old = v(j+1,i);    
            %v(j+1,i)=(2*dy*(-(-u(j,i-1)+u(i+1))/(2*dx)));
            v(j+1,i) = v(j-1,i+1) - dy/2/dx*(u(j,i+1)-u(j,i)+u(j-1,i+1)-u(j-1,i));
            change_v = v_old - v(j+1,i);
            v(j+1,i)= v_old + (alphav * change_v);
            diff_u = [diff_u change_u];
            diff_v = [diff_v change_v];
            end
        end
    end
    diff_U = max(abs(diff_u));
    diff_V = max(abs(diff_v));
    diff_u_max = [diff_u_max diff_U];
    diff_v_max = [diff_v_max diff_V];
    diff = [diff_U diff_V];
    max_diff = max(diff);
    
    if abs(max_diff) < tol
        break
    end
    z = z + 1;     
end

% Plotting to show convergence throughout the iterations
figure
hold on
plot(diff_u_max)
plot(diff_v_max)
hold off
xlabel('Iterations');
ylabel('Chnage in u and v');
title('Convergence of u and v Throughout Iterations');
legend('u difference', 'v difference');

[Abdallah Daher]

I am having the same issue. Were you able to figure it out?