Basic Equations¶

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}

where the eigenvector matrix $\mathbf{R}$ and the diagonal matrix of eigenvalues $\mathbf{\Lambda}$ are given

\begin{equation} \mathbf{W} = \mathbf{R}^{-1}\mathbf{Q} \qquad\text{,}\qquad \mathbf{\Lambda}= \begin{pmatrix} -c & 0 \\ 0 & c \end{pmatrix} \qquad\text{,}\qquad \mathbf{R} = \begin{pmatrix} Z & -Z \\ 1 & 1 \end{pmatrix} \qquad\text{and}\qquad \mathbf{R}^{-1} = \frac{1}{2Z} \begin{pmatrix} 1 & Z \\ -1 & Z \end{pmatrix} \end{equation}

Here $Z = \rho c$ with $c = \sqrt{\mu/\rho}$ represents the seismic impedance.

This notebook implements both upwind and Lax-Wendroff schemes for solving the free source version of the elastic wave equation in a homogeneous media. 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}

# 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
# Show Plot in The Notebook
matplotlib.use("nbagg")
import matplotlib.pyplot as plt
import matplotlib.animation as animation

1. Initialization of setup¶

# Initialization of setup
# --------------------------------------------------------------------------
nx    = 800        # number of grid points 
c     = 2500       # acoustic velocity in m/s
ro    = 2500       # density in kg/m^3
Z     = ro*c       # impedance
mu    = ro*c**2    # shear modulus

xmax  = 10000      # Length in m 
eps   = 0.5        # CFL
tmax  = 2.0        # simulation time in s
isnap = 10         # plotting rate
sig   = 200        # argument in the inital condition
x0    = 5000       # position of the initial condition

imethod = 'upwind'  # 'Lax-Wendroff', 'upwind'

# Initialize Space
x, dx = np.linspace(0,xmax,nx,retstep=True)

# use wave based CFL criterion
dt = eps*dx/c    # calculate time 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))
Qa = np.zeros((2,nx))

2. Initial condition¶

Seismic disturbances are introduced through specification of a particular spatial initial condition, in this case we use a Gaussian function.

Exercise 1¶

Implement the spatial initial condition given by:

\begin{equation} Q(x,t=0) = e^{-1/\sigma^2 (x - x_{o})^2} \end{equation}

Then, visualize the initial condition in a given plot.

#################################################################
# INITIALIZE THE SOURCE TIME FUCTION HERE!
#################################################################


#################################################################
# PLOT THE SOURCE TIME FUNCTION HERE!
#################################################################

# 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()

3. Solution for the homogeneous problem¶

Upwind finite volume scheme¶

We decompose the solution into right propagating $\mathbf{\Lambda}^{+}$ and left propagating eigenvalues $\mathbf{\Lambda}^{-}$ where

\begin{equation} \mathbf{\Lambda}^{+}= \begin{pmatrix} -c & 0 \\ 0 & 0 \end{pmatrix} \qquad\text{,}\qquad \mathbf{\Lambda}^{-}= \begin{pmatrix} 0 & 0 \\ 0 & c \end{pmatrix} \qquad\text{and}\qquad \mathbf{A}^{\pm} = \mathbf{R}^{-1}\mathbf{\Lambda}^{\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}^{+}\Delta\mathbf{Q}_{l} - \mathbf{A}^{-}\Delta\mathbf{Q}_{r}) \end{equation}

with corresponding flux term given by

\begin{equation} \mathbf{F}_{l} = \mathbf{A}^{+}\Delta\mathbf{Q}_{l} \qquad\text{,}\qquad \mathbf{F}_{r} = \mathbf{A}^{-}\Delta\mathbf{Q}_{r} \end{equation}

Lax-Wendroff finite volume scheme¶

The upwind solution presents a strong diffusive behaviour. 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 $\mathbf{A}$ 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}(\mathbf{Q}_{i+1}^{n} - \mathbf{Q}_{i-1}^{n}) + \frac{1}{2}\Big(\frac{dt}{dx}\Big)^2\mathbf{A}^2(\mathbf{Q}_{i+1}^{n} - 2\mathbf{Q}_{i}^{n} + \mathbf{Q}_{i-1}^{n}) \end{equation}

Exercise 2¶

Initialize all relevant matrices, i.e $R$, $R^{-1}$, $\mathbf{\Lambda}^{+}$, $\mathbf{\Lambda}^{-}$, $\mathbf{A}^{+}$, $\mathbf{A}^{-}$, $\mathbf{A}$.

#################################################################
# INITIALIZE ALL MATRICES HERE!
#################################################################
# R =        
# Rinv =                
# Lp =        
# Lm =       
# Ap =                         
# Am =                        
# A =

# Initialize all matrices
# --------------------------------------------------------------------------

R = np.array([[Z, -Z],[1, 1]])        # Eq. 8.43
Rinv = np.linalg.inv(R)               # Eq. 8.43
Lp = np.array([[0, 0], [0, c]])       # Eq. 8.49
Lm = np.array([[-c, 0], [0, 0]])      # Eq. 8.49
Ap = R @ Lp @ Rinv                    # Eq. 8.50     
Am = R @ Lm @ Rinv                    # Eq. 8.50    
A = np.array([[0, -mu], [-1/ro, 0]])  # Eq. 8.35

4. Finite Volumes solution¶

# Initialize animated plot
# ---------------------------------------------------------------
fig = plt.figure(figsize=(10,6))
ax1 = fig.add_subplot(2,1,1)
ax2 = fig.add_subplot(2,1,2)
line1 = ax1.plot(x, Q[0,:], 'k', x, Qa[0,:], 'r--')
line2 = ax2.plot(x, Q[1,:], 'k', x, Qa[1,:], 'r--')
ax1.set_ylabel('Stress')
ax2.set_ylabel('Velocity')
ax2.set_xlabel(' x ')
plt.suptitle('Homogeneous F. volume - %s method'%imethod, size=16)

plt.ion()    # set interective mode
plt.show()

# ---------------------------------------------------------------
# Time extrapolation
# --------------------------------------------------------------- 
for i in range(nt): 
    if imethod =='Lax-Wendroff':        
        for j in range(1,nx-1):
            dQ1 = Q[:,j+1] - Q[:,j-1]
            dQ2 = Q[:,j-1] - 2*Q[:,j] + Q[:,j+1]
            Qnew[:,j] = Q[:,j] - 0.5*dt/dx*(A @ dQ1)\
            + 1./2.*(dt/dx)**2 * (A @ A) @ dQ2    # Eq. 8.56
            
        # Absorbing boundary conditions
        Qnew[:,0] = Qnew[:,1]
        Qnew[:,nx-1] = Qnew[:,nx-2]

    elif imethod == 'upwind': 
        for j in range(1,nx-1):
            dQl = Q[:,j] - Q[:,j-1]
            dQr = Q[:,j+1] - Q[:,j]
            Qnew[:,j] = Q[:,j] - dt/dx * (Ap @ dQl + Am @ dQr)  # Eq. 8.54
            
        # Absorbing boundary conditions 
        Qnew[:,0] = Qnew[:,1]
        Qnew[:,nx-1] = Qnew[:,nx-2]
    else:
        raise NotImplementedError

    Q, Qnew = Qnew, Q
    
    # --------------------------------------   
    # Animation plot. Display solution
    if not i % isnap: 
        for l in line1:
            l.remove()
            del l               
        for l in line2:
            l.remove()
            del l 
        # -------------------------------------- 
        # Analytical solution (stress i.c.)
        Qa[0,:] = 1./2.*(np.exp(-1./sig**2 * (x-x0 + c*i*dt)**2)\
        + np.exp(-1./sig**2 * (x-x0-c*i*dt)**2))

        Qa[1,:] = 1/(2*Z)*(np.exp(-1./sig**2 * (x-x0+c*i*dt)**2)\
        - np.exp(-1./sig**2 * (x-x0-c*i*dt)**2))
        
        # -------------------------------------- 
        # Display lines
        line1 = ax1.plot(x, Q[0,:], 'k', x, Qa[0,:], 'r--', lw=1.5)
        line2 = ax2.plot(x, Q[1,:], 'k', x, Qa[1,:], 'r--', lw=1.5)
        plt.legend(iter(line2), ('F. Volume', 'Analytic'))
        plt.gcf().canvas.draw()