def neglap2d(n):
"""
Returns the Chebyshev collocation approximation to -D_x^2-D_y^2 on
the square [-1,1]^2 on the interior points only
"""
D, x = cheb(n + 1)
D2 = np.dot(D, D)
D2 = D2[1:-1, 1:-1]
D2 = csr_matrix(D2)
I = identity(n)
L = -(kron(D2, I) + kron(I, D2)) # FD negative Laplacian
return L
# p14 - solve nonlinear BVP U_xx = exp(u), u(-1)=u(1)=0 by iteration.
import numpy as np
from chebPy import cheb
from scipy.linalg import solve
from matplotlib import pyplot as plt
N=16
D,x=cheb(N)
D2=np.dot(D,D)
D2=D2[1:N,1:N]
u=np.zeros(N-1)
err=np.zeros(N-1)
change = 0.1 # arbitrary small number
it = 0
while True:
unew=solve(D2,np.exp(u))
err = np.abs(unew-u)
change = err.max()
u = unew
it += 1
if(change < 1e-15):
break
s=np.zeros(N+1)
s[0]=0.0
s[N]=0.0
s[1:N]=u
xx=np.linspace(-1.,1.,100)
# p35.py - Allen-Cahn eq. # same problem as in p34.py, except BCs that are imposed explicitly from chebPy import cheb import numpy as np from scipy.linalg import solve import pylab as p import mpl_toolkits.mplot3d.axes3d as p3 from matplotlib import cm # Differentiation matrix and initial data: N = 20 D, x = cheb(N) D2 = np.dot(D, D) # Use full size matrix D2[0, :] = 0. # for convenience D2[N, :] = 0. eps = 0.01 dt = min(.01, 50. * N**(-4.) / eps) t = 0. v = .53 * x + .47 * np.sin(-1.5 * np.pi * x) # Solve PDE by Euler formula and plot results: tmax = 100. tplot = 2 nplots = int(round(tmax / tplot)) plotgap = int(round(tplot / dt)) dt = float(tplot) / float(plotgap) xx = np.linspace(-1., 1., 40) vv = np.polyval(np.polyfit(x, v, N), xx) plotdata = np.zeros((nplots + 1, xx.size))
# x variable in [-A,A], Fourier - trigonometric series:
A = 3.
Nx = 50
dx = 2 * A / float(Nx)
x = -A + dx * np.arange(1, Nx + 1)
# Create entries for Derivative matrix - a Toeplitz matrix with an entry from 'entries' on each diagonal.
entries = np.zeros(Nx)
entries[0] = -1 / (3 * (dx / A)**2) - 1 / 6.
entries[1:Nx] = .5 * (-1)**np.arange(2, Nx + 1) / np.sin(
(np.pi * dx / A) * np.arange(1, Nx) / 2)**2
# Form derivative matrix as a Toeplitz matrix using 'entries'
D2x = (np.pi / A)**2 * toeplitz(entries)
# y variable in [-1,1], Chebyshev:
Ny = 15
Dy, y = cheb(Ny)
D2y = np.dot(Dy, Dy)
# Prepare boundary conditions
Abc = -np.array([[Dy[0, 0], Dy[0, Ny]], [Dy[Ny, 0], Dy[Ny, Ny]]])
BC = np.zeros((2, Ny - 1))
Dybc = np.zeros((2, Ny - 1))
Dybc[0, 0:Ny - 1] = Dy[0, 1:Ny]
Dybc[1, 0:Ny - 1] = Dy[Ny, 1:Ny]
BC[:, 0:Ny - 1] = solve(Abc, Dybc[:, 0:Ny - 1])
# Grid and initial data:
xx, yy = np.meshgrid(x, y)
vv = np.exp(-8. * ((xx + 1.5)**2 + yy**2))
dt = 5. / float(Nx + Ny**2)
vvold = np.exp(-8. * ((xx + dt + 1.5)**2 + yy**2))
# x variable in [-A,A], Fourier - trigonometric series:
A = 3.
Nx = 50
dx = 2*A/float(Nx)
x = -A + dx*np.arange(1,Nx+1)
# Create entries for Derivative matrix - a Toeplitz matrix with an entry from 'entries' on each diagonal.
entries = np.zeros(Nx)
entries[0] = -1/(3*(dx/A)**2)-1/6.
entries[1:Nx] = .5*(-1)**np.arange(2,Nx+1)/np.sin((np.pi*dx/A)*np.arange(1,Nx)/2)**2
# Form derivative matrix as a Toeplitz matrix using 'entries'
D2x = (np.pi/A)**2*toeplitz(entries)
# y variable in [-1,1], Chebyshev:
Ny =15
Dy,y = cheb(Ny)
D2y = np.dot(Dy,Dy)
# Prepare boundary conditions
Abc = - np.array([ [Dy[0,0],Dy[0,Ny]],
[Dy[Ny,0],Dy[Ny,Ny]] ])
BC = np.zeros((2,Ny-1))
Dybc = np.zeros((2,Ny-1))
Dybc[0,0:Ny-1] = Dy[0,1:Ny]
Dybc[1,0:Ny-1] = Dy[Ny,1:Ny]
BC[:,0:Ny-1] = solve(Abc,Dybc[:,0:Ny-1])
# Grid and initial data:
xx,yy = np.meshgrid(x,y)
vv = np.exp(-8.*((xx+1.5)**2+yy**2))
dt = 5./float(Nx+Ny**2)
