Plotting the transient temperature profile of a 2D grid using finite-difference solver

I understand that for heat conduction through a plane 2D wall with constant thermal conductivity and no energy generated I can use this equasion for an interior node.

I can solve this by hand for small node problems but I really don't know how to set up the for loop to run for a large grid size. Anything would be helpful thank you.

Also if you wanted some variables to use it's for a square grid of various nodes.

T_Left side = 100

T_Top side = 200

T_Right = adiabatic (IDK how you would set this up)

T_Bottom = adiabaic

Answer 1

[Torsten]

Order the temperatures in the grid points

(e.g. T11, T12,T13,...,T1_nx,T21,T22,...,T2_nx,...,T_ny,1,T_ny,2,...T_ny,nx).

Write down the equation for temperature in the (nx*ny) grid points.

Put the coefficients in a matrix A of size (nx*ny) x (nx*ny) on the left-hand side and the constant terms in a vector b of size (nx*ny,1) on the right-hand side.

You'll get a matrix equation

A*T = b

for the temperatures T where A is a matrix with 5 bands.

Now you can solve for T using backslash:

T = A\b.