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SWE_1D.py
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SWE_1D.py
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import numpy as np
import matplotlib as plt
import math
# Domain parameters
Nx = 512
Lx = 6.
dx = Lx / Nx
# Creating the spatial computational grid:
x = np.arange(dx/2, Lx, dx) - Lx/2
# Physical parameters
H0 = 1.
g0 = 9.81
Eta_B = 0.00005*(np.exp( - (2*x / Lx)**2 )) * np.ones(Nx) # Bottom topography
# Gravity wave speed
c0 = np.sqrt(g0 * H0)
# Spatial Differentiation function (second-order)
def ddx(f):
xp1 = np.roll(f, -1)
xm1 = np.roll(f, 1)
d = (xp1 - xm1) / (2*dx)
return d
# Spatial Differentiation function (forth-order)
def ddx4(f):
xp1 = np.roll(f, -1)
xp2 = np.roll(f, -2)
xm1 = np.roll(f, 1)
xm2 = np.roll(f, 2)
d4 = ( -1/12 * xp2 + 2/3 * xp1 - 2/3 * xm1 + 1/12 * xm2) / (dx)
# These coefficients are based on the output of Coefficients.py
return d4
### Constrain temporal grid:
# Initial conditions
# CASE a
u_init = np.zeros(Nx)
h_init = H0 + 1e-2 * np.exp( - (x / 0.2)**2 ) - Eta_B
#h_init = H0* np.ones(Nx) - Eta_B
# CASE b
# pi=math.pi
# u_init = 3e-2 * np.cos( 4*pi*x/Lx) + 1e-1 * np.exp( - (x / 0.2)**2 )
# h_init = H0 *np.ones(Nx) - Eta_B
# Time-stepping cfl factor
cfl = 0.2
# Target end-time for the simulation
final_time = 3 * (Lx/2) / c0
# Initial time
time0 = 0.
# State how frequently we want to store outputs
out_freq = final_time / 200
Nouts = int(final_time / out_freq) + 1
out_ind = 1
# The times for the stored outputs
times = np.arange(Nouts) * out_freq
# An array to store the spatio-temporal evolution
simulated_h = np.zeros((Nouts, Nx))
simulated_u = np.zeros((Nouts, Nx))
# Store the initial conditions
simulated_u[0,:] = u_init
simulated_h[0,:] = h_init
# Initialize some working variables
time = time0
u = u_init.copy()
h = h_init.copy()
# Loop through time until we get to the target date
cnt = 0
# Discretization Schemes
# Temporal scheme order (Choose between 1, 2 or 3)
t_order = 3
# Spatial scheme order (Choose between 2 or 4)
s_order = 4
# Storing dudt and dhdt in an array
dudt = np.zeros((Nx,t_order))
dhdt = np.zeros((Nx,t_order))
while time < final_time:
# Get dt
dt = cfl * dx / (np.max(np.abs(u)) + c0)
if s_order == 2:
dudt[:,0] = - u * ddx(u) - g0 * ddx(h+Eta_B)
dhdt[:,0] = - ddx(u * h)
elif s_order == 4:
dudt[:,0] = - u * ddx4(u) - g0 * ddx4(h+Eta_B)
dhdt[:,0] = - ddx4(u * h)
# Update fields (1st-order temporal)
if t_order == 1:
# Compute RHS of evolution equations and save
dt_now = dt
u += dudt[:,0] * dt_now
h += dhdt[:,0] * dt_now
if t_order == 2:
#Update fields (2nd-order temporal)
if cnt == 0:
dt_now = dt/10.
u += dudt[:,0] * dt_now
h += dhdt[:,0] * dt_now
else:
dt_now = dt
u += ( 1.5 * dudt[:,0] - 0.5 * dudt[:,-1] ) * dt_now
h += ( 1.5 * dhdt[:,0] - 0.5 * dhdt[:,-1]) * dt_now
if t_order == 3:
#Update fields (3rd-order temporal)
if cnt == 0:
dt_now = dt/10.
u += dudt[:,0] * dt_now
h += dhdt[:,0] * dt_now
if cnt == 1:
dt_now = dt/10.
u += ( 1.5 * dudt[:,0] - 0.5 * dudt[:,-1] ) * dt_now
h += ( 1.5 * dhdt[:,0] - 0.5 * dhdt[:,-1]) * dt_now
else:
dt_now = dt
u += ( 5/3 * dudt[:,0] - 5/6 * dudt[:,-1] + 1/6 * dudt[:,-2] ) * dt_now
h += ( 5/3 * dhdt[:,0] - 5/6 * dhdt[:,-1] + 1/6 * dhdt[:,-2]) * dt_now
# Update the history of dudt and dhdt values
dudt = np.roll(dudt, -1, axis=1)
dhdt = np.roll(dhdt, -1, axis=1)
# Update time
time += dt_now
# u_vec[cnt] = u[cnt]
# h_vec[cnt] = h[cnt]
cnt += 1
# Output, if appropraite
if (time >= out_ind * out_freq):
simulated_h[out_ind,:] = h
simulated_u[out_ind,:] = u
if (out_ind % 10 == 0):
print('Stored output {0:d} of {1:d}.'.format(out_ind, Nouts-1))
out_ind += 1
np.savez('simulation_results.npz',
times = times,
x = x,
simulated_h = simulated_h,
simulated_u = simulated_u,
Eta_B = Eta_B,
H0 = H0,
g0 = g0,
Lx = Lx)