Crank-Nicolson with Variable Diffusivity

We have implemented the Crank-Nicolson (CN) method for a two-variable reaction-diffusion system with constant grid parameters, $\Delta t$ and $\Delta x$, and system parameters (including diffusion coefficients $D_u$ and $D_v$).

When we keep all of these parameters constant, we can define constants

that define the non-zero entries of the two tridiagonal matrices defined by the CN stencil.

To reuse our previous implementation of the CN method when diffusion coefficients $D_u$ and $D_v$ are functions of time $t$ (the properties of the material that the molecular species $u$ and $v$ diffuse in change) we can use Python lambda functions.

Suppose that

where $D_{u,0}$ and $D_{v,0}$ are the constant diffusion coefficients we used previously.

Then in our Python code, sigma_u and sigma_v become

sigma_u = lambda t: float((1.+0.05*t)*D_u*dt)/float(2.*dx*dx)
sigma_v = lambda t: float((1.+0.05*t)*D_v*dt)/float(2.*dx*dx)

We also need to turn matrices A_u, B_u, A_v, and B_v into lambda functions (see below).

Reusing our previous code with these modifications looks as follows - with a kymograph of our numerical results and the time behaviour of sigma_u shown below.

import numpy
from matplotlib import pyplot
numpy.set_printoptions(precision=3)

L = 1.
J = 100
dx = float(L)/float(J-1)
x_grid = numpy.array([j*dx for j in range(J)])

T = 200
N = 1000
dt = float(T)/float(N-1)
t_grid = numpy.array([n*dt for n in range(N)])

D_v = float(10.)/float(100.)
D_u = 0.01 * D_v

k0 = 0.067
f_vec = lambda U, V: numpy.multiply(dt, numpy.subtract(numpy.multiply(V, 
                     numpy.add(k0, numpy.divide(numpy.multiply(U,U), numpy.add(1., numpy.multiply(U,U))))), U))
g = lambda u, v: -f(u,v)
 
sigma_u = lambda t: float((1.+0.05*t)*D_u*dt)/float(2.*dx*dx)
sigma_v = lambda t: float((1.+0.05*t)*D_v*dt)/float(2.*dx*dx)

total_protein = 2.26

no_high = 10
U = numpy.array([0.1 for i in range(no_high,J)] + [2. for i in range(0,no_high)])
V = numpy.array([float(total_protein-dx*sum(U))/float(J*dx) for i in range(0,J)])

A_u = lambda t: numpy.diagflat([-sigma_u(t) for i in range(J-1)], -1) +\
      numpy.diagflat([1.+sigma_u(t)]+[1.+2.*sigma_u(t) for i in range(J-2)]+[1.+sigma_u(t)]) +\
      numpy.diagflat([-sigma_u(t) for i in range(J-1)], 1)
        
B_u = lambda t: numpy.diagflat([sigma_u(t) for i in range(J-1)], -1) +\
      numpy.diagflat([1.-sigma_u(t)]+[1.-2.*sigma_u(t) for i in range(J-2)]+[1.-sigma_u(t)]) +\
      numpy.diagflat([sigma_u(t) for i in range(J-1)], 1)
        
A_v = lambda t: numpy.diagflat([-sigma_v(t) for i in range(J-1)], -1) +\
      numpy.diagflat([1.+sigma_v(t)]+[1.+2.*sigma_v(t) for i in range(J-2)]+[1.+sigma_v(t)]) +\
      numpy.diagflat([-sigma_v(t) for i in range(J-1)], 1)
        
B_v = lambda t: numpy.diagflat([sigma_v(t) for i in range(J-1)], -1) +\
      numpy.diagflat([1.-sigma_v(t)]+[1.-2.*sigma_v(t) for i in range(J-2)]+[1.-sigma_v(t)]) +\
      numpy.diagflat([sigma_v(t) for i in range(J-1)], 1)
        
U_record = []
V_record = []

U_record.append(U)
V_record.append(V)

for ti in range(1,N):
    t = ti*dt
    
    U_new = numpy.linalg.solve(A_u(t), B_u(t).dot(U) + f_vec(U,V))
    V_new = numpy.linalg.solve(A_v(t), B_v(t).dot(V) - f_vec(U,V))
    
    U = U_new
    V = V_new
    
    U_record.append(U)
    V_record.append(V)


U_record = numpy.array(U_record)
V_record = numpy.array(V_record)

fig = pyplot.figure()
fig.set_size_inches(10.5,6.5)
ax1 = fig.add_subplot(122)
ax1.set_xlabel('x')
ax1.set_ylim(T)
heatmap = ax1.pcolor(x_grid, t_grid, U_record, vmin=0., vmax=1.2)
colorbar = pyplot.colorbar(heatmap)
colorbar.set_label('concentration U')
ax2 = fig.add_subplot(121, sharey=ax1)
ax2.set_xlabel('sigma_u')
ax2.set_ylabel('t')
ax2.set_xlim([sigma_u(0), sigma_u(T)])
line = ax2.plot([sigma_u(t) for t in t_grid], t_grid)

png

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