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You should have completed Steps 1 and 2 before continuing. This IPython notebook continues the presentation of the 12 steps to Navier-Stokes, the practical module taught in the interactive CFD class of Prof. Lorena Barba.

Step 3: The Diffusion Equation in 1D

The one-dimensional diffusion equation is:
∂u∂t=ν∂2u∂x2
The first thing you should notice is that —unlike the previous two simple equations we have studied— this equation has a second-order derivative. We first need to learn what to do with it!
Discretizing ∂2u∂x2
The second-order derivative can be represented geometrically as the line tangent to the curve given by the first derivative. We will discretize the second-order derivative with a Central Difference scheme: a combination of Forward Difference and Backward Difference of the first derivative. Consider the Taylor expansion of ui+1 and ui−1 around ui:
ui+1=ui+Δx∂u∂x||||i+Δx22∂2u∂x2||||i+Δx33∂3u∂x3||||i+O(Δx4)
ui−1=ui−Δx∂u∂x||||i+Δx22∂2u∂x2||||i−Δx33∂3u∂x3||||i+O(Δx4)
If we add these two expansions, you can see that the odd-numbered derivative terms will cancel each other out. If we neglect any terms of O(Δx4) or higher (and really, those are very small), then we can rearrange the sum of these two expansions to solve for our second-derivative.

ui+1+ui−1=2ui+Δx2∂2u∂x2||||i+O(Δx4)
Then rearrange to solve for ∂2u∂x2||||i and the result is:
∂2u∂x2=ui+1−2ui+ui−1Δx2+O(Δx2)
Back to Step 3
We can now write the discretized version of the diffusion equation in 1D:
un+1i−uniΔt=νuni+1−2uni+uni−1Δx2
As before, we notice that once we have an initial condition, the only unknown is un+1i, so we re-arrange the equation solving for our unknown:
un+1i=uni+νΔtΔx2(uni+1−2uni+uni−1)
The above discrete equation allows us to write a program to advance a solution in time. But we need an initial condition. Let's continue using our favorite: the hat function. So, at t=0, u=2 in the interval 0.5≤x≤1 and u=1 everywhere else. We are ready to number-crunch!

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import numpy                 #loading our favorite library
from matplotlib import pyplot    #and the useful plotting library
%matplotlib inline

nx = 41
dx = 2/(nx-1)
nt = 20    #the number of timesteps we want to calculate
nu = 0.3   #the value of viscosity
sigma = .2 #sigma is a parameter, we'll learn more about it later
dt = sigma*dx**2/nu #dt is defined using sigma ... more later!


u = numpy.ones(nx)      #a numpy array with nx elements all equal to 1.
u[.5/dx : 1/dx+1]=2  #setting u = 2 between 0.5 and 1 as per our I.C.s

un = numpy.ones(nx) #our placeholder array, un, to advance the solution in time

for n in range(nt):  #iterate through time
    un = u.copy() ##copy the existing values of u into un
    for i in range(1,nx-1):
        u[i] = un[i] + nu*dt/dx**2*(un[i+1]-2*un[i]+un[i-1])
        
pyplot.plot(numpy.linspace(0,2,nx), u);
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Learn More
For a careful walk-through of the discretization of the diffusion equation with finite differences (and all steps from 1 to 4), watch Video Lesson 4 by Prof. Barba on YouTube.
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