Experimenting with Dispersion and Dissipation

Copyright (C) 2010-2020 Luke Olson
Copyright (C) 2020 Andreas Kloeckner

MIT License

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import numpy as np
import scipy as sp
import matplotlib.pyplot as plt

Problem Description

Consider $$ u_t + a u_x = 0$$ with periodic BC on the interval $[0,1]$.

a = 1.0
T = 4.0 / a # end time for four cycles

Set up the Grid

• dx will be the grid spacing in the $x$-direction
• x will be the grid coordinates
• xx will be really fine grid coordinates

nx = 90
k = 1
x = np.linspace(0, 2*np.pi, nx, endpoint=False)
dx = x[1] - x[0]
xx = np.linspace(0, 2*np.pi, 1000, endpoint=False)

Now define an initial condition:

def f(x):
    u = np.sin(k * x)
    return u

plt.plot(xx, f(xx), lw=3, clip_on=False)

[<matplotlib.lines.Line2D at 0x7f5845513130>]

Setting the Time Step

Now we need a time step. Let $$ \Delta t = \Delta x \frac{\lambda}{a}$$ with CFL number $\lambda$.

lmbda = 0.9
dt = dx * lmbda / a
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 = 4
tsteps = 63
    dx = 0.0698132
    dt = 0.0628319
lambda = 0.9

Make an index list, called $J$, so that we can access $J+1$ and $J-1$ easily.

J = np.arange(0, nx)  # all vertices
Jm1 = np.roll(J, 1)
Jp1 = np.roll(J, -1)

Run and Plot Final Solution

plotit = True

uETBS = f(x)
uLW = f(x)
if plotit:
    fig = plt.figure(figsize=(10,10))
    ax = fig.add_subplot(111)
    ax.set_title('u vs x')

for n in range(0, nt):
    uETBS[J] = uETBS[J] - lmbda * (uETBS[J] - uETBS[Jm1])
    
    uLW[J] = uLW[J] - lmbda * (1.0 / 2.0) * (uLW[Jp1] - uLW[Jm1]) \
           + (lmbda**2 / 2.0) * (uLW[Jp1] - 2 * uLW[J] + uLW[Jm1])
    
    uex = f((xx - a * (n+1) * dt) % (2*np.pi))

if plotit:
    ax.plot(x, uETBS, '-', clip_on=False, label="ETBS")
    #ax.plot(x, uLW, '-', clip_on=False, label="Lax-Wendroff")
    ax.plot(xx, uex, 'k-', clip_on=False, label='exact')
    ax.legend(frameon=False)
    plt.show()