Same time integration code in Matlab and python, matlab is stable, python blows up

Question

Here is an algorithm of Exponential time differencing method, using the original Matlab code from Oxford

clc
clear

% viscosity
nu = 0.5;

% Spatial grid and initial condition:
N = 128;
x = 2*pi*(0:N-1)'/N;
u = cos(x).*(1+sin(x));
v = fft(u);

% Precompute various ETDRK4 scalar quantities:
h = 0.01;
tot = 5;

% time step
k = [0:N/2-1 0 -N/2+1:-1]';

% wave numbers
L = 1/16*k.^2 - 1/16^3*nu*k.^4;

% Fourier multipliers
E = exp(h*L);
E2 = exp(h*L/2);
M =16;

% no. of points for complex means
r = exp(1i*pi*((1:M)-.5)/M); % roots of unity

% construct things on the
LR = h*L(:,ones(M,1)) + r(ones(N,1),:);

Q = h*real(mean( (exp(LR/2)-1)./LR ,2) );
f1 = h*real(mean( (-4-LR+exp(LR).*(4-3*LR+LR.^2))./LR.^3 ,2));
f2 = h*real(mean( (2+LR+exp(LR).*(-2+LR))./LR.^3 ,2));
f3 = h*real(mean( (-4-3*LR-LR.^2+exp(LR).*(4-LR))./LR.^3 ,2));

% Main time-stepping loop:
uu=u;tt=0;
vv=v;
NvNV = [];
aa = [];
bb = [];
cc = [];
NvNv = [];
NaNa = [];
NbNb = [];
NcNc = [];

nmax = floor(tot/h);
g = -0.5i*k;
%
for n = 1:nmax

    t = n*h;

    Nv = g.*fft(real(ifft(v)).^2);

    a = E2.*v + Q.*Nv;

    Na = g.*fft(real(ifft(a)).^2);

    b = E2.*v + Q.*Na;

    Nb = g.*fft(real(ifft(b)).^2);

    c = E2.*a + Q.*(2*Nb-Nv);

    Nc = g.*fft(real(ifft(c)).^2);

    v = E.*v + Nv.*f1 + 2*(Na+Nb).*f2 + Nc.*f3;
    u = real(ifft(v));
    uu = [uu,u];
    vv = [vv,v];
    tt = [tt,t];
    aa = [aa,a];
    bb = [bb,b];
    cc = [cc,c];
    NvNv = [NvNv,Nv];
    NaNa = [NaNa,Na];
    NbNb = [NbNb,Nb];
    NcNc = [NcNc,Nc];
end

Python code

    # spatial domain: 0,2pi
# time domain: 0,2

# init
import numpy as np
from matplotlib import pyplot as plt

np.seterr(all='raise')

plt.style.use('siads')

np.random.seed(0)

pi = np.pi

# viscosity
nu = 0.5

# mesh
mesh = 128

# time restriction
tot = 5


# distributed x point
x = np.linspace(0, 2.0 * pi, mesh, endpoint=False)

# force IC
u0 = np.cos(x)*(1.0 + np.sin(x))

N = mesh
k_array_noshift = np.fft.fftfreq(mesh)*mesh

u = u0
v = np.fft.fft(u)

h = 0.01

## follow them, set -N/2 to zer
k_array_noshift[mesh / 2] = 0

## Linear part in fourier space
L = 1.0/16.0*k_array_noshift**2 - nu*1.0/16.0**3 * k_array_noshift**4

## Fourier mulitplier
E = np.exp(h * L)
E2 = np.exp(h * L / 2.0)

## select number of points on the circle
M = 16

## choose radius 1, since k^2-k^4 ranges a lot... so 1 is enough to make sure only one singular point
r = np.exp(1j * np.pi * (np.arange(1, M + 1) - 0.5) / M)
r = r.reshape(1, -1)
r_on_circle = np.repeat(r, mesh, axis=0)

## define hL on M copies
LR = h * L
LR = LR.reshape(-1, 1)
LR = np.repeat(LR, M, axis=1)

## obtain hL on circle
LR = LR + r_on_circle

## important quantites used in ETDRK4

# g = -0.5*i*k
g = -0.5j * k_array_noshift

# averaged Q, f1,f2,f3
Q = h*np.real(np.mean( (np.exp(LR/2.0)-1)/LR, axis=1 ))

f1 = h*np.real(np.mean( (-4.0-LR + np.exp(LR)*(4.0-3.0*LR+LR**2))/LR**3, axis=1 ))

f2 = h*np.real(np.mean( (2.0+LR + np.exp(LR)*(-2.0 + LR))/LR**3, axis=1 ))

f3 = h*np.real(np.mean( (-4.0-3.0*LR - LR**2 + np.exp(LR)*(4.0 - LR))/LR**3, axis=1 ))


def compute_u2k_zeropad_dealiased(uk_):

    u2k = np.fft.fft(np.real(np.fft.ifft(uk_))**2)
    # three over two law
    # N = uk.size
    #
    # # map uk to uk_fine
    #
    # uk_fine = np.hstack((uk[0:N / 2], np.zeros((N / 2)), uk[-N / 2:])) * 3.0 / 2.0
    #
    # # convert uk_fine to physical space
    # u_fine = np.real(np.fft.ifft(uk_fine))
    #
    # # compute square
    # u2_fine = np.square(u_fine)
    #
    # # compute fft on u2_fine
    # u2k_fine = np.fft.fft(u2_fine)
    #
    # # convert u2k_fine to u2k
    # u2k = np.hstack((u2k_fine[0:N / 2], u2k_fine[-N / 2:])) / 3.0 * 2.0

    return u2k


print 'dt =', h

# np.linalg.norm(np.fft.ifft(uk_0)-u0) # very good

ntsnap = int(tot/h)
isnap = 0
tsnap = np.linspace(0, tot, ntsnap)
usnap = np.zeros((mesh, ntsnap))
uksnap = np.zeros((mesh, ntsnap),dtype=complex)

aasnap = np.zeros((mesh, ntsnap),dtype=complex)
bbsnap = np.zeros((mesh, ntsnap),dtype=complex)
ccsnap = np.zeros((mesh, ntsnap),dtype=complex)

Nvsnap = np.zeros((mesh, ntsnap),dtype=complex)
Nasnap = np.zeros((mesh, ntsnap),dtype=complex)
Nbsnap = np.zeros((mesh, ntsnap),dtype=complex)
Ncsnap = np.zeros((mesh, ntsnap),dtype=complex)


tcur = 0.0

## main loop time stepping
while tcur <= tot and isnap < ntsnap:

    # print tcur

    # record snap
    # if abs(tcur - tsnap[isnap]) < 1e-2:
    print ' current progress =', tcur / tsnap[-1] * 100, ' % '
    u = np.real(np.fft.ifft(v))
    usnap[:, isnap] = u
    uksnap[:, isnap] = v



    Nv = g * np.fft.fft(np.real(np.fft.ifft(v))**2)

    a = E2 * v + Q * Nv

    Na = g * np.fft.fft(np.real(np.fft.ifft(a))**2)

    b = E2 * v + Q * Na

    Nb = g * np.fft.fft(np.real(np.fft.ifft(b))**2)

    c = E2 * a + Q * (2.0*Nb - Nv)

    Nc = g * np.fft.fft(np.real(np.fft.ifft(c))**2)

    v = E*v + Nv*f1 + 2.0*(Na + Nb)*f2 + Nc*f3

    aasnap[:, isnap] = a
    bbsnap[:, isnap] = b
    ccsnap[:, isnap] = c

    Nvsnap[:, isnap] = Nv
    Nasnap[:, isnap] = Na
    Nbsnap[:, isnap] = Nb
    Ncsnap[:, isnap] = Nc


    # algo: ETDRK4
    # # 1. compute nonlinear part in first fractional step
    # u2k = compute_u2k_zeropad_dealiased(uk)
    # Nuk = g * u2k
    #
    # # 2. update fractional step
    # uk_a = E2 * uk + Q * Nuk
    #
    # # 3. compute nonlinear part in second fractional step
    # Nuk_a = g * compute_u2k_zeropad_dealiased(uk_a)
    #
    # # 4. update fractional step
    # uk_b = E2 * uk + Q * Nuk_a
    #
    # # 5. compute nonlinear part in third fractional step
    # Nuk_b = g * compute_u2k_zeropad_dealiased(uk_b)
    #
    # # 6. update fractional step
    # uk_c = E2 * uk_a + Q * (2.0 * Nuk_b - Nuk)
    #
    # # 7 compute nonlinear part in the fourth fractional step
    # Nuk_c = g * compute_u2k_zeropad_dealiased(uk_c)
    #
    # # final, update uk
    # uk = E * uk + Nuk * f1 + 2.0 * (Nuk_a + Nuk_b) * f2 + Nuk_c * f3

    # record time
    tcur = tcur + h
    isnap = isnap + 1

# save uksnap
np.savez('output_uk',uksnap=uksnap,f1=f1,f2=f2,f3=f3,Q=Q,LR=LR,L=L,g=g, Nasnap=Nasnap, Nvsnap=Nvsnap, Nbsnap=Nbsnap, Ncsnap=Ncsnap,
         aasnap=aasnap, bbsnap=bbsnap, ccsnap=ccsnap, usnap=usnap)


# plot snapshots
plt.figure(figsize=(16,16))
for isnap in xrange(ntsnap):
    plt.plot(x, usnap[:,isnap])
plt.xlabel('x')
plt.ylabel('u')
plt.savefig('snapshots.png')

# plot contours
from matplotlib import cm

fig = plt.figure(figsize=(8, 8))
X, Y = np.meshgrid(x, tsnap)
Z = usnap.transpose()
V = np.linspace(-10, 10, 500)
surf = plt.contourf(X, Y, Z, V, cmap=cm.seismic)
plt.savefig('contour.png')

Python code runs to some extent says some value goes to NaN, but not for matlab. Same code, exactly

So I wish to know why and what's going on, you can run the code yourself.

What makes me feel more weird is that the first iteration is extremely good match between the two!

Answer 1

For more information, please check out this linkedIn link.

My LinkedIn Post that explains this problem

Finally!!!!!!!! Everything works!!!!!!!!!!!!!!!!!

I found I can make python works the same well as matlab(note that python always blows up to inf at certain physical time), by simply

changing numpy.fft to scipy.fft

Note that I tested, numpy.fft and scipy.fft evaluated in my code is almost identical but different around 1e-16 ..

But this matters for such a chaotic system simulation.

If I want my code to blow up, simply change scipy.fft to numpy.fft.

From my personal viewpoint as an user, numpy.fft must have some mysterious issue inside! Since matlab fft and scipy fft have no trouble in my code at all.