gyplus = 0; gyminus = 0;
# Time-stepping by leap frog formula:
plotgap = int(round(2./dt)); dt = 2./plotgap;
for n in range(2*plotgap+1):
t = n*dt
if (n+.5)%plotgap < 1.:
i = n/plotgap
ax = subplot(2,2,i+1,projection='3d')
ax.plot_surface(xx, yy, vv, cstride=1,rstride=1,cmap='summer')
title('t = '+str('%.5f' % t)); ax.azim = -138; ax.elev = 38; ax.set_zlim3d(-0.2,1)
# Fourier transform in x-direction
uxx = zeros((Ny+1,Nx)); uyy = zeros((Ny+1,Nx));
for i in range(Ny): # 2nd derivs wrt x in each row
v = vv[i,:];
v_hat = fft(v);
w_hat = (pi/A)**2*k2*v_hat
w = real(ifft(w_hat));
uxx[i,:]= w;
for j in range(Nx): # 2nd derivs wrt y in each column
v = vv[:,j];
# Impose Neumann BC:
uyy[:,j] = chebfft(v);
uyy[0,j] = gyplus; uyy[-1,j] = gyminus;
uyy[:,j] = chebfft(uyy[:,j]);
vvnew = 2*vv - vvold + dt**2*(uxx+uyy);
vvold = vv; vv = vvnew;
show()
"""
Spectral methods in MATLAB. Lloyd
Program 18a
"""
# Chebyshev differentiation via FFT (compare program 11)
from numpy import *
from matplotlib import pyplot as plt
from chebfft import chebfft
xx = arange(-1,1,0.01)
ff = exp(xx)*sin(5*xx)
for N in [10, 20]:
x = cos(pi*arange(0,N+1)/N)
f = exp(x)*sin(5*x)
plt.subplot(220+int(N/10.))
plt.plot(x, f, 'o', xx,ff,'-')
plt.title('f(x), N=' + str(N))
error = chebfft(f)-exp(x)*(sin(5*x)+5*cos(5*x))
plt.subplot(222+int(N/10.))
plt.plot(x,error, marker='o', linestyle='-')
plt.title('error in df(x)/dx, N=' + str(N))
plt.show()
vvold = exp(-8*((xx+dt+1.5)**2+yy**2));
gplus = 0; gminus = 0;
# Time-stepping by leap frog formula:
plotgap = int(round(2./dt)); dt = 2./plotgap;
for n in range(2*plotgap+1):
t = n*dt
if (n+.5)%plotgap < 1.:
i = n/plotgap
ax = subplot(2,2,i+1,projection='3d')
ax.plot_surface(xx, yy, vv, cstride=1,rstride=1,cmap='summer')
title('t = '+str('%.5f' % t)); ax.azim = -138; ax.elev = 38; ax.set_zlim3d(-0.2,1)
# Fourier transform in x-direction
uxx = zeros((Ny+1,Nx)); uyy = zeros((Ny+1,Nx));
for i in range(Ny): # 2nd derivs wrt x in each row
v = vv[i,:];
v_hat = fft(v);
w_hat = (pi/A)**2*k2*v_hat
w = real(ifft(w_hat));
uxx[i,:]= w;
for j in range(Nx): # 2nd derivs wrt y in each column
v = vv[:,j];
uyy[:,j] = chebfft(chebfft(v));
# Impose Dirichlet BC:
uyy[0,j]=gplus; uyy[-1,j]=gminus;
vvnew = 2*vv - vvold + dt**2*(uxx+uyy);
vvold = vv; vv = vvnew;
show()
N = 80
x = cos(pi * arange(0, N + 1) / N)
dt = 8.0 / N**2
v = exp(-200 * x**2)
vold = exp(-200 * (x - dt)**2)
tmax = 4
tplot = 0.075
plotgap = int(round(tplot / dt))
dt = tplot / plotgap
nplots = int(round(tmax / tplot))
plotdata = vstack((v, zeros((nplots, N + 1))))
tdata = 0
for i in range(0, nplots):
for n in range(0, plotgap):
w = chebfft(chebfft(v)).T
w[0] = 0
w[N] = 0
vnew = 2 * v - vold + dt**2 * w
vold = v
v = vnew
plotdata[i + 1, :] = v
tdata = vstack((tdata, dt * i * plotgap))
# Plot results
fig = plt.figure()
ax = axes3d.Axes3D(fig)
X, Y = meshgrid(x, tdata)
ax.plot_wireframe(X, Y, plotdata)
ax.set_xlim(-1, 1)
#Uold = initcond(x-cc*dt) # x is descending, so this is inward-propagating in r
V = 0.0 * x
Vx0 = np.zeros_like(x)
tmax = 4
tplot = 0.1
plotgap = int(np.round(tplot / dt))
dt = tplot / plotgap
nplots = int(np.round(tmax / tplot))
plotdata = np.vstack((U, np.zeros((nplots, N + 1))))
tdata = 0
for i in range(0, nplots):
for n in range(0, plotgap):
# Heun's method
Ux0 = cc * chebfft(U).T # Ux from current U
Vx0 = cc * chebfft(V).T # Vx from current V
Vx0[0] = 0.0 # u_t = 0 at edges
Vx0[N] = 0.0
Ustar = U + dt * Vx0 # full step of U
Uxstar = cc * chebfft(Ustar).T # Vx from current V
Vnew = V + dt * 0.5 * (Ux0 + Uxstar) # V at i+1
Vx1 = cc * chebfft(Vnew).T
Vx1[0] = 0.0 # u_t = 0 at edges
Vx1[N] = 0.0
Unew = U + dt * 0.5 * (Vx0 + Vx1) # U at i+1
