# Hyperbolic PDE¶

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One-way wave equation: $$u_t + c u_x = 0$$ $$u(0,x) = f(x) \quad \text{on} \, [0,1]$$

(Periodic boundary conditions)

## Solution¶

$$u(t,x) = f(x-ct)$$
In [26]:
import numpy as np
import scipy as sp
import matplotlib.pyplot as plt


## Step 2: Set up grid¶

• dx: grid spacing in the $x$-direction
• x: grid coordinates
In [27]:
nx = 100
x = np.linspace(0, 1, nx, endpoint=False)
dx = x[1] - x[0]


## Set time/speed parameters¶

In [33]:
c = 1.0 # speed
T = 1.0 / c # end time
lmbda = 0.93
dt = dx * lmbda / c
nt = int(T//dt)
print('T = %g' % T)
print('tsteps = %d' % nt)
print('    dx = %g' % dx)
print('    dt = %g' % dt)
print('lambda = %g' % lmbda)

T = 1
tsteps = 107
dx = 0.01
dt = 0.0093
lambda = 0.93


## Set Initial Condition¶

(Square wave with amplitude 1)

In [34]:
def f(x):
return ((x>0.4) & (x<0.6)).astype(np.float64)


## Plot solution¶

In [42]:
import time

u = f(x)

fig = plt.figure(figsize=(8,8))
plt.title('u vs x')
line1, = plt.plot(x, u, lw=3, clip_on=False)

def timestepper(n):
uex = f((x - c * (n+1) * dt) % 1.0)
line1.set_data(x, uex)
return [line1]

from matplotlib import animation
from IPython.display import HTML

ani = animation.FuncAnimation(fig, timestepper, frames=nt, interval=20)
html = HTML(ani.to_jshtml())
plt.clf()
html

Out[42]: