2D Laplace Equation

Laplace equation is a 2D second order differential and appears in the Navier Stokes in the diffusion term; it also appears in the third equation of the 2D incompressible NS which is used to couple between pressure and velocity.
Actually in this post we will solve only the right hand side the equation:

∂2p∂x2+∂2p∂y2=0

We already seen how we discretize this - the central difference scheme:

pni+1,j−2pni,j+pni−1,jΔx2+pni,j+1−2pni,j+pni,j−1Δy2=0

This equation is a little different from diffusion only in that there is no time derivative; therefore this equation will give an equilibrium for given boundary conditions. Of course equilibrium will be reached after many iterations but a convergence criterion can be used to stop the solver.
After some arrangements we can take out the equation for the scheme as follows:

pni,j=Δy2(pni+1,j+pni−1,j)+Δx2(pni,j+1+pni,j−1)2(Δx2+Δy2)

For this equation to be solved, we need to provide initial and boundary conditions. We will consider a uniform p=0 everywhere; using the same domain as in the other posts, we put the following BCs:

p = 0 at x = 0

p = y at x = 2

∂p∂y=0 at y = 0, 1
Let's see a representation of these boundaries and explain them:

The domain is initialized with zeros everywhere; the x axis is the right hand side axis; for x = 0 the pressure will always be zero; for x = 2, the pressure will be given by the value of y (hence the diagonal distribution); the last BC actually says that the values of p must be equal in the direction of y axis at y = 0 and y = 1, not to each other, but with the neighboring cells.
This was solved using two approaches: the loop method and function call method; here I will show only the loop method (which is actually not show in Prof. Barba's blog); the second approach can be seen on the GitHub page.
The code for looping method is the following:

A norm of 0.01 is applied as convergence criterion; after the solving is finished the following solution is plotted:

As it was expected, the boundary conditions are kept as defined.
If you want to access the codes for this problem check out my GitHub page; if you have any problems setting this up leave a comment below.