Finite differences wave equation

Question

I am trying to solve the wave equation (and later animate the solution) using this finite difference method. However, it seems to only yield the same solution for every time step, such that the animation remains constant. What is wrong?

Tension=100 #N
mu = .1 #

v= (Tension/mu)**0.5
L=1 #string length
Nx=200
SimulationTime=.2


x = linspace(0, L, Nx+1) # Mesh points in space
 
dx = x[1] - x[0]
Nt = 1000
dt = SimulationTime/Nt
t = linspace(0, Nt*dt, Nt+1)
V = v*dt/dx
V2 = V**2

Lambda=.1

u = zeros(Nx+1) # t
u_1 = zeros(Nx+1) # t-1dt
u_2 = zeros(Nx+1) # t-2dt

# function that can be used to describe initial conditions
#Note - I should be zero at the boundary and dI/dx should not have a very large value at the boundary.  
I = lambda x: 1e-3*x*(1-x)  
#I = lambda x: 1.e-3*np.sin(2*np.pi*x/Lambda) 
#I = lambda x: 1.e-3*signal.gausspulse(x-Nx*dx/2, fc=10)

for i in range(0,int(Nx)):
     u_1[i] = I(x[i]) #u_1 - solution at time n-1 - Here we put initial conditions for t=0; 


n=0  #First timepoint
for i in range(1, Nx):
     u[i] = u_1[i] -0.5*V2*(u_1[i-1] - 2*u_1[i] + u_1[i+1])  


# Use variable user_data to store u  for each iteration over time.          
user_data = {}
user_data['x'] = x
user_data['u'] = []
#(n, u)


u_2[:] = u_1
u_1[:] = u
for n in range(1, Nt):
     #Compute the values of the whole string at a given timepoint
     for i in range(1, Nx):
          u[i] = -u_2[i]+2*u_1[i]+V2*(u_1[i+1]-2*u_1[i]+u_1[i-1])
     u[0]=0
     u[Nx]=0
     u_2[:] = u_1
     u_1[:] = u

     user_data['u'].append(u)

I tried a lot of different tweaks, but the solution seems to remain constant.

Here is the animation part:

from matplotlib.animation import FuncAnimation


# Example data (list of lists)
data = user_data['u']
data.pop(0)


# Create figure and axis
fig, ax = plt.subplots()
x_data = np.arange(len(data[0]))

# Plot the initial data
line, = ax.plot(x_data, data[0], label='Series 1')

# Customize plot
ax.set_xlim(0, len(data[0]) - 1)
#ax.set_ylim(min(min(data)), max(max(data)))
ax.set_xlabel('X-axis')
ax.set_ylabel('Y-axis')
ax.legend()

# Animation update function
def update(frame):
    line.set_ydata(data[frame])
    return line,

# Create animation
animation = FuncAnimation(fig, update, frames=len(data), interval=1000, blit=True)

# Show the plot
plt.show()

from IPython.display import HTML
HTML(animation.to_jshtml())

Answer 1

You are appending (a reference to) u at every timestep. So each one will be ... the latest value of u[]!

You need to append a COPY:

user_data['u'].append(u.copy())