Solving PDE with implicit euler in python - incorrect output

I will try and explain exactly what's going on and my issue.

This is a bit mathy and SO doesn't support latex, so sadly I had to resort to images. I hope that's okay.

I don't know why it's inverted, sorry about that. At any rate, this is a linear system Ax = b where we know A and b, so we can find x, which is our approximation at the next time step. We continue doing this until time t_final.

This is the code

import numpy as np

tau = 2 * np.pi
tau2 = tau * tau
i = complex(0,1)

def solution_f(t, x):
    return 0.5 * (np.exp(-tau * i * x) * np.exp((2 - tau2) * i * t) + np.exp(tau * i * x) * np.exp((tau2 + 4) * i * t))

def solution_g(t, x):
    return 0.5 * (np.exp(-tau * i * x) * np.exp((2 - tau2) * i * t) - np.exp(tau * i * x) * np.exp((tau2 + 4) * i * t))

for l in range(2, 12):
    N = 2 ** l #number of grid points
    dx = 1.0 / N #space between grid points
    dx2 = dx * dx
    dt = dx #time step
    t_final = 1
    approximate_f = np.zeros((N, 1), dtype = np.complex)
    approximate_g = np.zeros((N, 1), dtype = np.complex)

    #Insert initial conditions
    for k in range(N):
        approximate_f[k, 0] = np.cos(tau * k * dx)
        approximate_g[k, 0] = -i * np.sin(tau * k * dx)

    #Create coefficient matrix
    A = np.zeros((2 * N, 2 * N), dtype = np.complex)

    #First row is special
    A[0, 0] = 1 -3*i*dt
    A[0, N] = ((2 * dt / dx2) + dt) * i
    A[0, N + 1] = (-dt / dx2) * i
    A[0, -1] = (-dt / dx2) * i

    #Last row is special
    A[N - 1, N - 1] = 1 - (3 * dt) * i
    A[N - 1, N] = (-dt / dx2) * i
    A[N - 1, -2] = (-dt / dx2) * i
    A[N - 1, -1] = ((2 * dt / dx2) + dt) * i

    #middle
    for k in range(1, N - 1):
        A[k, k] = 1 - (3 * dt) * i
        A[k, k + N - 1] = (-dt / dx2) * i
        A[k, k + N] = ((2 * dt / dx2) + dt) * i
        A[k, k + N + 1] = (-dt / dx2) * i

    #Bottom half
    A[N :, :N] = A[:N, N:]
    A[N:, N:] = A[:N, :N]

    Ainv = np.linalg.inv(A)

    #Advance through time
    time = 0
    while time < t_final:
        b = np.concatenate((approximate_f, approximate_g), axis = 0)
        x = np.dot(Ainv, b) #Solve Ax = b
        approximate_f = x[:N]
        approximate_g = x[N:]
        time += dt
    approximate_solution = np.concatenate((approximate_f, approximate_g), axis=0)

    #Calculate the actual solution
    actual_f = np.zeros((N, 1), dtype = np.complex)
    actual_g = np.zeros((N, 1), dtype = np.complex)
    for k in range(N):
        actual_f[k, 0] = solution_f(t_final, k * dx)
        actual_g[k, 0] = solution_g(t_final, k * dx)
    actual_solution = np.concatenate((actual_f, actual_g), axis = 0)

    print(np.sqrt(dx) * np.linalg.norm(actual_solution - approximate_solution))

It doesn't work. At least not in the beginning, it shouldn't start this slow. I should be unconditionally stable and converge to the right answer.

What's going wrong here?

from Solving PDE with implicit euler in python - incorrect output