Finite difference method for 3D diffusion/heat equation

Question

I'm trying to use finite differences to solve the diffusion equation in 3D. I think I'm having problems with the main loop. In particular the discrete equation is:

With Neumann boundary conditions (in just one face as an example):

Now the code:

import numpy as np
from matplotlib import pyplot, cm
from mpl_toolkits.mplot3d import Axes3D ##library for 3d projection plots
%matplotlib inline

kx = 15     #Number of points
ky = 15
kz = 15
largx = 90  #Domain length.
largy = 90
largz = 90   

dt4 = 1/2 #Time delta (arbitrary for the time).
dx4 = largx/(kx-1)    #Position deltas.
dy4 = largy/(ky-1)
dz4 = largz/(kz-1) 

Tin = 25    #Initial temperature
kapp = 0.23

Tamb3d = 150 #Ambient temperature

#Heat per unit of area. One for each face.
qq1=0 / (largx*largz)
qq2=0 / (largx*largz)
qq3=0 / (largz*largy)
qq4=0 / (largz*largy)
qq5=0 / (largx*largy)
qq6=1000 / (largx*largy)


x4 = np.linspace(0,largx,kx)   
y4 = np.linspace(0,largy,ky)
z4 = np.linspace(0,largz,kz)


#Defining the function.
def diff3d(tt):
    
    w2 = np.ones((kx,ky,kz))*Tin         #Temperature array
    wn2 = np.ones((kx,ky,kz))*Tin
    
    for k in range(tt+2):
        wn2 = w2.copy()
        w2[1:-1,1:-1,1:-1] = (wn2[1:-1,1:-1,1:-1] + 
                        kapp*dt4 / dy4**2 *                                       
                        (wn2[1:-1, 2:,1:-1] - 2 * wn2[1:-1, 1:-1,1:-1] + wn2[1:-1, 0:-2,1:-1]) +  
                        kapp*dt4 / dz4**2 *                                       
                        (wn2[1:-1,1:-1,2:] - 2 * wn2[1:-1, 1:-1,1:-1] + wn2[1:-1, 1:-1,0:-2]) +
                        kapp*dt4 / dx4**2 *
                        (wn2[2:,1:-1,1:-1] - 2 * wn2[1:-1, 1:-1,1:-1] + wn2[0:-2, 1:-1,1:-1]))

        #Neumann boundary (dx=dy=dz for the time)
        w2[0,:,:] =   w2[0,:,:] + 2*kapp* (dt4/(dx4**2)) * (w2[1,:,:] - w2[0,:,:] - qq1 * dx4/kapp)
        w2[-1,:,:] =  w2[-1,:,:] + 2* kapp*(dt4/(dx4**2)) * (w2[-2,:,:] - w2[-1,:,:] + qq2 * dx4/kapp)
        w2[:,0,:] =   w2[:,0,:] + 2*kapp* (dt4/(dx4**2)) * (w2[:,1,:] - w2[:,0,:] - qq3 * dx4/kapp)
        w2[:,-1,:] =  w2[:,-1,:] + 2*kapp* (dt4/(dx4**2)) * (w2[:,-2,:] - w2[:,-1,:] + qq4 * dx4/kapp)
        w2[:,:,0] =   w2[:,:,0] + 2 *kapp* (dt4/(dx4**2)) * (w2[:,:,-1] - w2[:,:,0] - qq5 * dx4/kapp)
        w2[:,:,-1] =   w2[:,:,-1] + 2 *kapp* (dt4/(dx4**2)) * (w2[:,:,-2] - w2[:,:,-1] + qq6 * dx4/kapp)
       
    w2[1:,:-1,:-1] = np.nan    #We'll only plot the "outside" points.
    w2_uno = np.reshape(w2,-1)    

    #Plotting
    fig = pyplot.figure()
    X4, Y4, Z4 = np.meshgrid(x4, y4,z4)
    ax = fig.add_subplot(111, projection='3d')
    img = ax.scatter(X4, Y4, Z4, c=w2_uno, cmap=pyplot.jet())
    fig.colorbar(img)
    pyplot.show()

For 5000 iterations (qq6 = 1000/Area) we get:

I'm only applying heat on the top surface. Somehow I end up with the bottom surface heating up.

Adding to that I'm trying to confine the region to which I apply heat (just a small part of one face). When I try that it seems like heat is only transferred in one direction, ignoring all others. Applying (qq1 = 1000/Area) for half of the front face (the other part is adiabatic, q = 0) we end up with:

This is pretty odd. I suspect I'm having some trouble in the main loop (or maybe in the boundary conditions) that I'm not finding.

Answer 1

there are something wrong with this code: w2[:,:,0] = w2[:,:,0] + 2 kapp (dt4/(dx4**2)) * (w2[:,:,-1] - w2[:,:,0] - qq5 * dx4/kapp) please check it again. and I am working on a similar project recently. Do you have a moment to share some experience with me.