The source free elastic wave equation can be written in terms of a coupled first-order system
\begin{align} \partial_t \sigma - \mu \partial_x v & = 0 \\ \partial_t v - \frac{1}{\rho} \partial_x \sigma & = 0 \end{align}with $\rho$ the density and $\mu$ the shear modulus. This equation in matrix-vector notation follows
\begin{equation} \partial_t \mathbf{Q} + \mathbf{A} \partial_x \mathbf{Q} = 0 \end{equation}where $\mathbf{Q} = (\sigma, v)$ is the vector of unknowns and the matrix $\mathbf{A}$ contains the parameters $\rho$ and $\mu$. The above matrix equation is analogous to the advection equation $\partial_t q + a \partial_x q = 0$. Although it is a coupled system, diagonalization of $\mathbf{A} = \mathbf{R}^{-1}\mathbf{\Lambda}\mathbf{R}$ allows us to implement all elements developed for the solution of the advection equation in terms of fluxes. It turns out that the decoupled version is
\begin{equation} \partial_t \mathbf{W} + \mathbf{\Lambda} \partial_x \mathbf{W} = 0 \end{equation}with $\mathbf{W} = \mathbf{R}^{-1}\mathbf{Q}$, where the eigenvector matrix $\mathbf{R}$ and the diagonal matrix of eigenvalues $\mathbf{\Lambda}$ are given for the heterogeneous case
\begin{equation} \mathbf{\Lambda}_i= \begin{pmatrix} -c_i & 0 \\ 0 & c_i \end{pmatrix} \qquad\text{,}\qquad \mathbf{A}_i= \begin{pmatrix} 0 & -\mu_i \\ -1/\rho_i & 0 \end{pmatrix} \qquad\text{,}\qquad \mathbf{R} = \begin{pmatrix} Z_i & -Z_i \\ 1 & 1 \end{pmatrix} \qquad\text{and}\qquad \mathbf{R}^{-1} = \frac{1}{2Z_i} \begin{pmatrix} 1 & Z_i \\ -1 & Z_i \end{pmatrix} \end{equation}Here $Z_i = \rho_i c_i$ represents the seismic impedance. In comparison with the homogeneous case, here we allow coefficients of matrix $\mathbf{A}$ to vary for each element
This notebook implements the Lax-Wendroff scheme for solving the free source version of the elastic wave equation in a heterogeneous media. This solution is compared with the one obtained using the finite difference scheme. To keep the problem simple we use as spatial initial condition a Gauss function with half-width $\sigma$
\begin{equation} Q(x,t=0) = e^{-1/\sigma^2 (x - x_{o})^2} \end{equation}
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# Import all necessary libraries, this is a configuration step for the exercise.
# Please run it before the simulation code!
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.animation as animation
# Show the plots in the Notebook.
plt.switch_backend("nbagg")
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# Initialization of setup
# --------------------------------------------------------------------------
nx = 800 # number of grid points
c0 = 2500 # acoustic velocity in m/s
rho = 2500 # density in kg/m^3
Z0 = rho*c0 # impedance
mu = rho*c0**2 # shear modulus
rho0 = rho # density
mu0 = mu # shear modulus
xmax = 10000 # in m
eps = 0.5 # CFL
tmax = 1.5 # simulation time in s
isnap = 10 # plotting rate
sig = 200 # argument in the inital condition
x0 = 2500 # position of the initial condition
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# Finite Differences setup
# --------------------------------------------------------------------------
dx = xmax/(nx-1) # calculate space increment
xfd = np.arange(0, nx)*dx # initialize space
mufd = np.zeros(xfd.size) + mu0 # initialize shear modulus
rhofd = np.zeros(xfd.size) + rho0 # initialize density
# Introduce inhomogeneity
mufd[int((nx-1)/2) + 1:nx] = mufd[int((nx-1)/2) + 1:nx]*4
# initialize fields
s = np.zeros(xfd.size)
v = np.zeros(xfd.size)
dv = np.zeros(xfd.size)
ds = np.zeros(xfd.size)
s = np.exp(-1./sig**2 * (xfd-x0)**2) # Initial condition
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# Finite Volumes setup
# --------------------------------------------------------------------------
A = np.zeros((2,2,nx))
Z = np.zeros((1,nx))
c = np.zeros((1,nx))
# Initialize velocity
c = c + c0
c[int(nx/2):nx] = c[int(nx/2):nx]*2
Z = rho*c
# Initialize A for each cell
for i in range(1,nx):
A0 = np.array([[0, -mu], [-1/rho, 0]])
if i > nx/2:
A0= np.array([[0, -4*mu], [-1/rho, 0]])
A[:,:,i] = A0
# Initialize Space
x, dx = np.linspace(0,xmax,nx,retstep=True)
# use wave based CFL criterion
dt = eps*dx/np.max(c) # calculate tim step from stability criterion
# Simulation time
nt = int(np.floor(tmax/dt))
# Initialize wave fields
Q = np.zeros((2,nx))
Qnew = np.zeros((2,nx))
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# Initial condition
#----------------------------------------------------------------
sx = np.exp(-1./sig**2 * (x-x0)**2)
Q[0,:] = sx
# ---------------------------------------------------------------
# Plot initial condition
# ---------------------------------------------------------------
plt.plot(x, sx, color='r', lw=2, label='Initial condition')
plt.ylabel('Amplitude', size=16)
plt.xlabel('x', size=16)
plt.legend()
plt.grid(True)
plt.show()
We decompose the solution into right propagating $\mathbf{\Lambda}_i^{+}$ and left propagating eigenvalues $\mathbf{\Lambda}_i^{-}$ where
\begin{equation} \mathbf{\Lambda}_i^{+}= \begin{pmatrix} -c_i & 0 \\ 0 & 0 \end{pmatrix} \qquad\text{,}\qquad \mathbf{\Lambda}_i^{-}= \begin{pmatrix} 0 & 0 \\ 0 & c_i \end{pmatrix} \qquad\text{and}\qquad \mathbf{A}_i^{\pm} = \mathbf{R}^{-1}\mathbf{\Lambda}_i^{\pm}\mathbf{R} \end{equation}This strategy allows us to formulate an upwind finite volume scheme for any hyperbolic system as
\begin{equation} \mathbf{Q}_{i}^{n+1} = \mathbf{Q}_{i}^{n} - \frac{dt}{dx}(\mathbf{A}_i^{+}\Delta\mathbf{Q}_{l} - \mathbf{A}_i^{-}\Delta\mathbf{Q}_{r}) \end{equation}with corresponding flux term given by
\begin{equation} \mathbf{F}_{l} = \mathbf{A}_i^{+}\Delta\mathbf{Q}_{l} \qquad\text{,}\qquad \mathbf{F}_{r} = \mathbf{A}_i^{-}\Delta\mathbf{Q}_{r} \end{equation}The upwind solution presents a strong diffusive behavior. In this sense, the Lax-Wendroff perform better, with the advantage that it is not needed to decompose the eigenvalues into right and left propagations. Here the matrix $\mathbf{A}_i$ can be used in its original form. The Lax-Wendroff follows
\begin{equation} \mathbf{Q}_{i}^{n+1} = \mathbf{Q}_{i}^{n} - \frac{dt}{2dx}\mathbf{A}_i(\Delta\mathbf{Q}_{l} + \Delta\mathbf{Q}_{r}) + \frac{1}{2}\Big(\frac{dt}{dx}\Big)^2\mathbf{A}_i^2(\Delta\mathbf{Q}_{l} - \Delta\mathbf{Q}_{r}) \end{equation}
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# Initialize animated plot
# ---------------------------------------------------------------
fig = plt.figure(figsize=(10,6))
ax1 = fig.add_subplot(2,1,1)
ax2 = fig.add_subplot(2,1,2)
ax1.axvspan(((nx-1)/2+1)*dx, nx*dx, alpha=0.2, facecolor='b')
ax2.axvspan(((nx-1)/2+1)*dx, nx*dx, alpha=0.2, facecolor='b')
ax1.set_xlim([0, xmax])
ax2.set_xlim([0, xmax])
ax1.set_ylabel('Stress')
ax2.set_ylabel('Velocity')
ax2.set_xlabel(' x ')
line1 = ax1.plot(x, Q[0,:], 'k', x, s, 'r--')
line2 = ax2.plot(x, Q[1,:], 'k', x, v, 'r--')
plt.suptitle('Heterogeneous F. volume - Lax-Wendroff method', size=16)
ax1.text(0.1*xmax, 0.8*max(sx), '$\mu$ = $\mu_{o}$')
ax1.text(0.8*xmax, 0.8*max(sx), '$\mu$ = $4\mu_{o}$')
plt.ion() # set interective mode
plt.show()
# ---------------------------------------------------------------
# Time extrapolation
# ---------------------------------------------------------------
for j in range(nt):
# Finite Volume Extrapolation scheme-------------------------
for i in range(1,nx-1):
# Lax-Wendroff method
dQl = Q[:,i] - Q[:,i-1]
dQr = Q[:,i+1] - Q[:,i]
Qnew[:,i] = Q[:,i] - dt/(2*dx)*A[:,:,i] @ (dQl + dQr)\
+ 1/2*(dt/dx)**2 *A[:,:,i] @ A[:,:,i] @ (dQr - dQl)
# Absorbing boundary conditions
Qnew[:,0] = Qnew[:,1]
Qnew[:,nx-1] = Qnew[:,nx-2]
Q, Qnew = Qnew, Q
# Finite Difference Extrapolation scheme---------------------
# Stress derivative
for i in range(1, nx-1):
ds[i] = (s[i+1] - s[i])/dx
# Velocity extrapolation
v = v + dt*ds/rhofd
# Velocity derivative
for i in range(1, nx-1):
dv[i] = (v[i] - v[i-1])/dx
# Stress extrapolation
s = s + dt*mufd*dv
# --------------------------------------
# Animation plot. Display solutions
if not j % isnap:
for l in line1:
l.remove()
del l
for l in line2:
l.remove()
del l
line1 = ax1.plot(x, Q[0,:], 'k', x, s, 'r--')
line2 = ax2.plot(x, Q[1,:], 'k', x, v, 'r--')
plt.legend(iter(line2), ('F. Volume', 'f. Diff'))
plt.gcf().canvas.draw()
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