# Numerical solution of linear PDEs: Computing the Crank-Nicolson matrix automatically

The Crank-Nicolson method rewrites a discrete time linear PDE as a matrix multiplication \(\phi_{n+1}=C \phi_n\). Nonlinear PDE’s pose some additional problems, but are solvable as well this way (by linearizing every timestep). A major advantage here is that going \(k\) steps into the future is just \(\phi_{n+k}=C^{k}\phi_n\), and calculating a matrix power is polynomial time. The method is in general very stable.

For an assignment I had to construct the Crank-Nicolson matrix for a simple linear 1 dimension PDE, which had to be derived by hand. That’s a bit labourus, so I made a method to derive it automatically. As I came across it while sifting through some older files, and could still not find that method elsewhere online, I thought I might as well write it up here.

# Derivation (1D)

This is the standard Crank-Nicolson expression for a linear, 1 dimensional PDE. The time stepping matrix would then be \(\mathbf{C}=\mathbf{A}^{-1}\mathbf{B}\).

\[\begin{align} \phi^{n+1} - \frac{\Delta t}2 f^{n+1} =& \; \phi^n + \frac{\Delta t}2 f^n =\\ \mathbf{A} \phi^{n+1} =& \; \mathbf{B} \phi^n \end{align}\]Since \(f\) is linear, we can expand it into the set of its impulse responses \(\left\{h^k\right\}\). These are the vector outputs \(h^k = f\left(\delta^k\right)\) by applying it to a shifted unit impulse \(\delta^k_i = \delta_{ik}\). We can then rewrite \(f\) in terms of its impulse responses:

\[\begin{equation} f\left(\phi\right)_i = \sum_k h^k_i \phi_k \end{equation}\]The Crank-Nicolson equation can be rewritten by substituting \(f\) for this definition with the goal of reformulating it as a matrix equation. Focusing on a single element \(\phi^{n+1}_i\) and moving the \(\phi^n\) terms not resulting from \(f\left(\phi\right)\) into the sum one obtains

\[\begin{equation} \sum_k \delta_{ik} \phi^{n+1}_i - \frac{\Delta t}2 h^k_i \phi^{n+1}_k = \sum_k \delta_{ik} \phi^n_i + \frac{\Delta t}2 h^k_i \phi^n_k \end{equation}\]Equating this with the definition of the matrix-vector dot product

\((\mathbf{A}\vec x)_i = \sum_k A_{ik} x_k\)
gives the elements of the Crank-Nicolson matrices \(\mathbf{A}\) and \(\mathbf{B}\):

# Implementation

Here is a simple diffusion-advection example in Javascript. Click to reset and apply new values.

Advection(1st derivative):

Diffusion(2nd derivative):

The implementation in Python is a straightforward translation of the
equations above. Note that `int(i == k)`

is the Kronecker delta
\(\delta_{ik}\).

```
import numpy as np
import numpy.linalg as LA
def cn_lin_mat(f, n, dt):
d = np.eye(n) # shifted impulses {delta^k}
h = [f(di) for di in d] # shifted impulse responses {h^k}
A = np.zeros((n, n))
B = np.zeros((n, n))
for i in range(n):
for k in range(1, n+1):
k = k % n
A[i, k] = int(i == k) - dt/2*h[k][i]
B[i, k] = int(i == k) + dt/2*h[k][i]
Ainv = LA.inv(A)
C = Ainv @ B
return C
```

# Higher Dimensions

This generalizes
trivially to higher dimensions.
You just have to rewrite f so that it works on 1 dimensional data,
by reshaping it at appriopiate times.
So, for a 2d linear PDE over a field with width `W`

and height `H`

it would result in the following code. You’d probably want to use
sparse arrays for \(\mathbf{A}\) and \(\mathbf{B}\),
since they have \((WH)^2\) elements, of which most are 0.

```
def f1d(phi):
phi = phi.reshape((H, W))
phi_next = f(phi)
return phi_next.flatten()
C = cn_lin_mat(f1d, W*H, dt)
phi1 = (C @ phi0.flatten()).reshape((H, W))
```