Step 4: Diffusion Equation in 1-D

Step 4: Diffusion Equation in 1-D#

The one-dimensional diffusion equation is:

\frac{\partial u}{\partial t} = ν \frac{\partial^{2} u}{\partial x^{2}}

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 $\frac{\partial^{2} u}{\partial x^{2}}$ #

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 $u_{i + 1}$ and $u_{i - 1}$ around $u_{i}$ :

$u_{i + 1} = u_{i} + Δ x \frac{\partial u}{\partial x} |_{i} + \frac{Δ x^{2}}{2} \frac{\partial^{2} u}{\partial x^{2}} |_{i} + \frac{Δ x^{3}}{3!} \frac{\partial^{3} u}{\partial x^{3}} |_{i} + O (Δ x^{4})$

$u_{i - 1} = u_{i} - Δ x \frac{\partial u}{\partial x} |_{i} + \frac{Δ x^{2}}{2} \frac{\partial^{2} u}{\partial x^{2}} |_{i} - \frac{Δ x^{3}}{3!} \frac{\partial^{3} u}{\partial x^{3}} |_{i} + O (Δ x^{4})$

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 $O (Δ x^{4})$ or higher (and really, those are very small), then we can rearrange the sum of these two expansions to solve for our second-derivative.

$u_{i + 1} + u_{i - 1} = 2 u_{i} + Δ x^{2} \frac{\partial^{2} u}{\partial x^{2}} |_{i} + O (Δ x^{4})$

Then rearrange to solve for $\frac{\partial^{2} u}{\partial x^{2}} |_{i}$ and the result is:

\frac{\partial^{2} u}{\partial x^{2}} = \frac{u_{i + 1} - 2 u_{i} + u_{i - 1}}{Δ x^{2}} + O (Δ x^{2})

Back to Step 4#

We can now write the discretized version of the diffusion equation in 1D:

\frac{u_{i}^{n + 1} - u_{i}^{n}}{Δ t} = ν \frac{u_{i + 1}^{n} - 2 u_{i}^{n} + u_{i - 1}^{n}}{Δ x^{2}}

As before, we notice that once we have an initial condition, the only unknown is $u_{i}^{n + 1}$ , so we re-arrange the equation solving for our unknown:

u_{i}^{n + 1} = u_{i}^{n} + \frac{ν Δ t}{Δ x^{2}} (u_{i + 1}^{n} - 2 u_{i}^{n} + u_{i - 1}^{n})

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 $0.5 \leq x \leq 1$ and $u = 1$ everywhere else. We are ready to go!

import numpy as np
from matplotlib import pyplot as plt

nx = 41
dx = 2 / (nx - 1)
nt = 20 # the number of timesteps
nu = 0.3 # diffusion coefficient
sigma = 0.2 # sigma is a numerical parameter, we will learn it later
dt = sigma * dx**2 / nu  # dt is defined using sigma 

u = np.ones(nx) # a numpy array with nx elements all equal to 1
u[int(0.5/dx):int(1/dx+1)] = 2 # setting u = 2 between 0.5 and 1 

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

for n in range(nt): # iterate through 0 to nt-1, in total nt steps
    un = u.copy() # copy the existing values of u into un
    for i in range(1, nx-1): # skip first and last component!
        u[i] = un[i] + nu * dt / dx**2 * (un[i+1] - 2 * un[i] + un[i-1])
        
plt.plot(np.linspace(0,2,nx),u)

[<matplotlib.lines.Line2D at 0x7f27c8150550>]

_images/cd9646958269b2c237c943321ccc83df86cd16c0c3a0f09e8decafad73ad4b59.png

Step 4: Diffusion Equation in 1-D

Contents

Step 4: Diffusion Equation in 1-D#

Discretizing ∂2u∂x2#

Back to Step 4#

Discretizing $\frac{\partial^{2} u}{\partial x^{2}}$ #