How can I modify my graph so it displays the proper information for the axes?

Question

I have written a program to solve the Heat Equation ( u_t = k * u_xx ) numerically by method of Finite Differences.

For my problem, u is function of x and t , where 0 < x < L and t > 0 . I have specified L = 1 (the length of the rod) and the terminal time T = 10 seconds for my problem, so I would like for the graph to be displayed on the domain (x,t) \\in {(0,1) x (0, 10)} . However, my axes just don't make sense. It is plotting the x-axis from values of 0 - 40 and the t-axis is showing -0.25 - 0.00 .

How can I edit my code so that when I plot u which depends on x, t the graph will display for values of x ranging from 0 - 1 and t ranging from 0 - 10 seconds??

Thanks in advance for any and all help. it is very greatly appreciated. Here is the code I am working with:

    ## This program is to implement a Finite Difference method approximation
## to solve the Heat Equation, u_t = k * u_xx,
## in 1D w/out sources & on a finite interval 0 < x < L. The PDE
## is subject to B.C: u(0,t) = u(L,t) = 0,
## and the I.C: u(x,0) = f(x).
import numpy as np
import matplotlib.pyplot as plt

# Parameters    
L = 1 # length of the rod
T = 10 # terminal time
N = 40 # spatial values
M = 1600 # time values/hops; (M ~ N^2)
s = 0.25 # s := k * ( (dt) / (dx)^2 )

# uniform mesh
x_init = 0
x_end = L
dx = float(x_end - x_init) / N

x = np.arange(x_init, x_end, dx)
x[0] = x_init

# time discretization
t_init = 0
t_end = T
dt = float(t_end - t_init) / M

t = np.arange(t_init, t_end, dt)
t[0] = t_init

# time-vector
for m in xrange(0, M):
    t[m] = m * dt

# spatial-vector
for j in xrange(0, N):
    x[j] = j * dx

# definition of the solution u(x,t) to u_t = k * u_xx
u = np.zeros((N, M+1)) # array to store values of the solution

# Finite Difference Scheme:

u[:,0] = x * (x - 1) #initial condition

for m in xrange(0, M):
    for j in xrange(1, N-1):
        if j == 1:
            u[j-1,m] = 0 # Boundary condition
        elif j == N-1:
            u[j+1,m] = 0 # Boundary Condition
        else:
            u[j,m+1] = u[j,m] + s * ( u[j+1,m] - 
            2 * u[j,m] + u[j-1,m] )

# for graph    
print u, x, t
plt.plot(u)
plt.title('Finite Difference Approx. to Heat Equation')
plt.xlabel('x-axis')
plt.ylabel('time (seconds)')
plt.axis()
plt.show()

It appears that whatever displays for the x-axis reflects the number of step sizes in space that I take ( N = 40 ) for my code. I thought np.arange(x_init, x_end, dx) would return evenly spaced values within the interval (x_init, x_end) with step size dx ? So what am I doing wrong? Thanks again.

Answer 1

You have some issues with your code as your u turns out to be 40x1601 and not 40x1600. However, I think the plot you may be after (after correcting u) is

corrected_u = u[:,:-1:]
plt.pcolor(t, x, corrected_u)