# A numerical solution of 1D Gaussian advection equation by Upwind Scheme.  import matplotlib.pyplot as pl import numpy as np  t0 = 0.0 dt = 0.1 t_final = 2 x0 = -2.0 dx = 0.1 x_max = 4.0 T = np.arange(t0, t_final+dt, dt) X = np.arange(x0, x_max+dx, dx) U = [0.0 for _ in X] fig, ax = pl.subplots() nframe = 0  for t in T:     if t == t0:         for i in range(len(X)):             U[i] = np.exp(-X[i]**2)     else:         U_prime = [0.0 for _ in X]         for i in range(len(X)):             if i == 0:                 U_prime[i] = np.exp(-(x0-(1-np.cos(t)))**2)             elif i == len(X)-1:                 U_prime[i] = np.exp(-(x_max-(1-np.cos(t)))**2)             else:                 a = np.sin(t)                 if a > 0:                     U_prime[i] = U[i] - a*dt/dx*(U[i] - U[i-1])                 else:                     U_prime[i] = U[i] - a*dt/dx*(U[i+1] - U[i])         U = U_prime     ax.clear()     line, = ax.plot(X, U, 'b-', linewidth=3)     ax.set_xlabel('x')     ax.set_ylabel('u')     ax.set_xlim([x0, x_max])     fig.savefig('upwind1d/frame_'+"%03d" % nframe+'.png')     nframe += 1  # To generate GIF: convert frame_* GaussianUpwind1D.gif