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SWE_2D.py
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SWE_2D.py
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import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits import mplot3d
import math
# Domain parameters
Nx = 100
Ny = Nx # Assume unifrom spacing
Lx = 6.
Ly = 6.
dx = Lx / Nx
dy = Ly / Ny
# Creating the spatial computational grid:
x = np.arange(dx/2, Lx, dx) - Lx/2
y = np.arange(dy/2, Ly, dy) - Ly/2
# Physical parameters
f_c = 1 # coriolis force
H0 = 1.
g0 = 9.81
Eta_B = 0.2 * (np.exp( - (2*x / Lx)**2 ) * np.exp( - (2*y /Ly )**2 )) * np.ones((Nx,Ny)) # Bottom topography
# Gravity wave speed
c0 = np.sqrt(g0 * H0)
# Spatial Differentiation function (second-order)
def ddx(f):
xp1 = np.roll(f, -1,axis=1)
xm1 = np.roll(f, 1,axis=1)
d = (xp1 - xm1) / (2*dx)
return d
def ddy(f):
yp1 = np.roll(f, -1, axis=0)
ym1 = np.roll(f, 1, axis=0)
g = (yp1 - ym1) / (2*dy)
return g
# Spatial Differentiation function (forth-order)
def ddx4(f):
xp1 = np.roll(f, -1,axis=1)
xp2 = np.roll(f, -2,axis=1)
xm1 = np.roll(f, 1,axis=1)
xm2 = np.roll(f, 2,axis=1)
dx4 = ( -1/12 * xp2 + 2/3 * xp1 - 2/3 * xm1 + 1/12 * xm2) / (dx)
# These coefficients are based on the output of Coefficients.py
return dx4
def ddy4(f):
yp1 = np.roll(f, -1,axis=0)
yp2 = np.roll(f, -2,axis=0)
ym1 = np.roll(f, 1,axis=0)
ym2 = np.roll(f, 2,axis=0)
dy4 = ( -1/12 * yp2 + 2/3 * yp1 - 2/3 * ym1 + 1/12 * ym2) / (dy)
# These coefficients are based on the output of Coefficients.py
return dy4
### Constrain temporal grid:
# Initial conditions
# CASE a
u_init = np.zeros((Nx,Ny))
v_init = np.zeros((Nx,Ny))
X, Y = np.meshgrid(x, y)
def f(x,y):
return ( (1e-2* np.exp( - (x/ 0.2)**2 - (y / 0.2)**2)))
h_init = H0 + f(X,Y) - Eta_B
ax = plt.axes(projection='3d')
ax.contour3D(X, Y, h_init - H0 + Eta_B, 100, cmap='viridis')
plt.savefig('h_initial.png', dpi=500)
# 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)
# Time-stepping cfl factor
cfl = 0.1
# Target end-time for the simulation
final_time = np.maximum(5 * (Lx/2) / c0 , 5 * (Ly/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,Ny))
simulated_u = np.zeros((Nouts, Nx,Ny))
simulated_v = np.ones((Nouts, Nx,Ny))
# Store the initial conditions
simulated_u[0,:,:] = u_init
simulated_v[0,:,:] = v_init
simulated_h[0,:,:] = h_init
# Initialize some working variables
time = time0
u = u_init.copy()
v = v_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,Ny,t_order))
dvdt = np.zeros((Nx,Ny,t_order))
dhdt = np.zeros((Nx,Ny,t_order))
while time < final_time:
# Get dt
dt = np.min([cfl * dx / (np.max(np.abs(u)) + c0),cfl * dy / (np.max(np.abs(v)) + c0)])
if s_order == 2:
dudt[:,:,0] = f_c*v - u * ddx(u) -v *ddy(u) - g0 * ddx(h+Eta_B)
dvdt[:,:,0] = -f_c*u - u * ddx(v) -v *ddy(v) - g0 * ddy(h+Eta_B)
dhdt[:,:,0] = - ddx(u * h) - ddy(v * h)
elif s_order == 4:
dudt[:,:,0] = f_c*v - u * ddx4(u) -v *ddy4(u) - g0 * ddx4(h+Eta_B)
dvdt[:,:,0] = -f_c*u - u * ddx4(v) -v *ddy4(v) - g0 * ddy4(h+Eta_B)
dhdt[:,:,0] = - ddx4(u * h) - ddy4(v * 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
v += dvdt[:,:,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
v += dvdt[:,:,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
v += ( 1.5 * dvdt[:,:,0] - 0.5 * dvdt[:,:,-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
v += dvdt[:,:,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
v += ( 1.5 * dvdt[:,:,0] - 0.5 * dvdt[:,:,-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
v += ( 5/3 * dvdt[:,:,0] - 5/6 * dvdt[:,:,-1] + 1/6 * dvdt[:,:,-2]) * dt_now
# Update the history of dudt and dhdt values
dudt = np.roll(dudt, -1, axis=2)
dvdt = np.roll(dvdt, -1, axis=2)
dhdt = np.roll(dhdt, -1, axis=2)
# 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
simulated_v[out_ind,:,:] = v
if (out_ind % 10 == 0):
print('Stored output {0:d} of {1:d}.'.format(out_ind, Nouts-1))
out_ind += 1
## Conservation of energy
dx = x[1] - x[0]
KE = np.sum(0.5 * simulated_h * (simulated_u**2 + simulated_v**2), axis=-1) * dx
PE = np.sum(0.5 * g0 * (simulated_h + Eta_B)**2, axis=-1) * dx
np.savez('simulation_results.npz',
times = times,
x = x,
y = y,
simulated_h = simulated_h,
simulated_u = simulated_u,
simulated_v = simulated_v,
Eta_B = Eta_B,
kinetic = KE,
potential = PE,
total = KE + PE,
H0 = H0,
g0 = g0,
Lx = Lx)
#h_anim = plt3D.h_anim(X, Y, simulated_h, Nouts*dt, "h")