Copyright (C) 2010-2020 Luke Olson
Copyright (C) 2020 Andreas Kloeckner
The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
import numpy as np
import scipy as sp
import matplotlib.pyplot as plt
Consider $$u_t+au_x=0$$ with periodic boundary conditions.
Set up parameters:
a
for the advection speedlmbda
for the CFL numberdx
for the grid spacing in $x$dt
for the time stepks
for the range of wave numbers to considera = 1
lmbda = 0.6/a
dx = .1
dt = dx*lmbda
ks = np.arange(1,16)
Find $\omega(\kappa)$. Recall $\lambda = ah_t / h_x$.
ETBS: $$ u_{k, \ell + 1} = \lambda u_{k - 1 , \ell} + (1 - \lambda) u_{k, \ell} $$
Recall:
#clear
kappa = ks*dx
p_ETBS = 1
q_ETBS = lmbda*np.exp(-1j*kappa) + (1-lmbda)
s_ETBS = q_ETBS/p_ETBS
omega_ETBS = 1j*np.log(s_ETBS)/dt
Again recall $\lambda = ah_t / h_x$.
Lax-Wendroff: $$ u_{k, \ell + 1} - u_{k, \ell} = -\frac{\lambda}2 (u_{k + 1, \ell} - u_{k - 1, \ell}) + \frac{\lambda^2}{2} ( u_{k + 1, \ell} - 2 u_{k, \ell} + u_{k - 1, \ell}) $$
#clear
p_LW = 1
q_LW = (
# u_{k,l}
1 - 2*lmbda**2/2
# u_{k+1,l}
+ np.exp(1j*kappa) * (-lmbda/2 + lmbda**2/2)
# u_{k-1,l}
+ np.exp(-1j*kappa) * (lmbda/2 + lmbda**2/2)
)
s_LW = q_LW/p_LW
omega_LW = 1j*np.log(s_LW)/dt
plt.plot(ks, omega_ETBS.real, label="ETBS")
plt.plot(ks, omega_LW.real, label="Lax-Wendroff")
plt.plot(ks, a*ks, color='black', label='exact')
plt.legend(loc="best")
<matplotlib.legend.Legend at 0x7f3c400c0850>
plt.plot( ks, omega_ETBS.imag, label="ETBS")
plt.plot( ks, omega_LW.imag, label="Lax-Wendroff")
plt.legend(loc="best")
<matplotlib.legend.Legend at 0x7f3c40009410>