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

Consider $$u_t+au_x=0$$ with periodic boundary conditions.

Set up parameters:

• a for the advection speed
• lmbda for the CFL number
• dx for the grid spacing in $x$
• dt for the time step
• ks for the range of wave numbers to consider

a = 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:

• $r_k=\delta_{k,j}\Leftrightarrow\hat{\boldsymbol{r}} (\varphi) = e^{- i \theta j}$.
• Index sign flip between matrix and Toeplitz vector.
• $e^{- i \omega (\kappa) h_t} = s (\kappa)$.

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: $$ u_{k, \ell + 1} - u_{k, \ell} = -\frac{\lambda}2 (u_{k + 1, \ell} - u_{k - 1, \ell}) + \frac{\lambda^2}{2} ( u_{k + 1, \ell} - 2 u_{k, \ell} + u_{k - 1, \ell}) $$

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>