Diffusion process

The heat equation describes the distribution of heat (or variation in temperature) in a given region over time. For a function u(x,y,t) of two spatial variables (x,y) and a time variable t, the heat equation is

where k is a constant.

The numerical solution can be approximated using various methods such as the explicit, implicit or Crank-Nicolson methods. In the following, we’ll use the explicit method.

Implementation

We begin by setting simulation parameters

n = 60
k = 1

and we create a group representing the heated surface

G = dana.zeros((n,n))

We connect each group cell to its 4 neighbours and we write the heat equation using the forward difference/ central difference (FCTS) equation.

K = numpy.array([[0, 1, 0],
                 [1, 0, 1],
                 [0, 1, 0]])
G.connect(G.V, K, 'N', shared=True)
G.dV = 'dt*k*(N/4-V)'

We run 2500 iterations, keeping left, right and up border at temperature 1 while bottom border is kept at temperature 0.

for i in range(2500):
    G.compute(dt=1)
    G.V[0,:] = 0
    G.V[:,n-1] = G.V[n-1,:] = G.V[:,0] = 1

And finally, we visualize the result:

fig = plt.figure(figsize=(10,7.5))
plt.imshow(G.V, cmap=plt.cm.hot, origin='lower',
           extent=[0,n,0,n],
           interpolation='bicubic', vmin=0, vmax=1)
plt.colorbar()
CS = plt.contour(G.V, 5, colors='k')
plt.clabel(CS, inline=1, fontsize=16)
plt.grid()
plt.show()