### Non-linear coupled equations are not converging

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Hello everyone

I am trying to simulate non-linear coupled equations and this is my code:

The problem is with changing coefficients or time, either is not converging or it is converging to non-sense(value of u_1 for example becomes zero in whole domain).

What is problem in your opinion?!

Initial condition seems reasonable. Sometimes I think problem is variational formulation, but I am not sure.

These are the equations:

$\frac{\partial u_1}{\partial t}=D\times\left(\Delta u_1+\frac{K}{u_2}\times\Delta u_2\right)\backslash\frac{\partial u_2}{\partial t}-\left(\frac{K}{c}\right)\times\left(\frac{u_2}{u_1}\right)\times\frac{\partial u_1}{\partial t}=X\times\Delta u_2$∂

I am trying to simulate non-linear coupled equations and this is my code:

```
# -*- coding: utf8 -*-
from __future__ import print_function
from fenics import *
import numpy as np
from dolfin import *
from mshr import *
tol = 1E-14
t = 0.0
class InitialConditions(Expression):
def eval(self, values, x):
values[0] = 0.0
values[1] = 315
if (near(x[0], 2.0, 0.2) or near(x[0], 13.0, 0.2)) and \
near(x[1], 1.0, 0.1):
values[0] = 5000.0
values[1] = 315.01
def value_shape(self):
return (2,)
T = 1.0e-1 # final time
num_steps = 100 # number of time steps
dt = T / num_steps # time step size
D = 1.000e-10 # Diffusion Coefficient
K = 8.000e-1 # Thermometric diffusion ratio
c = 4.000e3 # Heat Capacity
X = 2.000e-7 # Thermoemtric diffusion coefficient
# Create mesh and define function space
mesh = RectangleMesh(Point(0., 0.),Point(15.0, 2.0), 30, 30)
P1 = FiniteElement("P", triangle,2)
element = MixedElement([P1, P1])
V = FunctionSpace(mesh, element)
# Define test functions
v_1, v_2 = TestFunctions(V)
# Define functions for concentration and Temperature
u = Function(V)
u_n = Function(V)
# Split system functions to access components
u_1, u_2 = split(u)
# Define boundary condition
z_1 = Expression(('0.0','315.0'), degree = 2)
z_2 = Expression(('0.0','315.0'), degree = 2)
#tol = 1E-14
def boundary_D(x, on_boundary):
return on_boundary and not ((near(x[0], 2.0, 0.5) or near(x[0], 13.0, 0.5)) and\
near(x[1], 2.0, tol)) and not (near(x[0], 0.0, tol) or\
near(x[0], 15.0, tol))
def boundary_D_source(x, on_boundary):
return on_boundary and (near(x[0], 2.0, 0.5) or near(x[0], 13.0, 0.5)) and \
near(x[1], 2.0, tol)
bc_1= DirichletBC(V, z_1, boundary_D_source)
bc_2 = DirichletBC(V, z_2, boundary_D)
bcs = [bc_1, bc_2]
#bcs = bc_2
# Define initial value
f = InitialConditions(degree = 1)
u_n = interpolate(f, V)
S_1 = Source(degree = 2)
u_n1, u_n2 = u_n.split()
# Define variational problem
v_1, v_2= TestFunctions(V)
k = Constant(dt)
K = Constant(K)
D = Constant(D)
c = Constant(c)
X = Constant(X)
g = Constant(-1.0)
r = Constant(-1.00e-6)
F = u_2*((u_1 - u_n1))*v_1*dx + k*D*dot(grad(u_1), grad(v_1*u_2))*dx \
+ k*D*g*v_1*u_2*ds + k*D*K*dot(grad(u_2), grad(v_1))*dx + k*D*K*r*v_1*ds\
+ u_1*((u_2 - u_n2))*v_2*dx - (K/c)*(u_2)*((u_1 - u_n1))*v_2*dx\
+ k*X*dot(grad(u_2), grad(v_2*u_1))*dx + k*X*r*v_2*u_1*ds
vtkfile_1 = File("Project/u1.pvd")
vtkfile_2 = File("Project/u2.pvd")
du = TrialFunction(V)
# Time-stepping
for n in range(num_steps):
_u_1, _u_2 = u_n.split()
vtkfile_2 << (_u_1,t)
# Compute solution
J = derivative(F, u, du)
problem = NonlinearVariationalProblem(F, u, bcs, J)
solver = NonlinearVariationalSolver(problem)
solver.solve()
# Update previous solution
u_n.assign(u)
t += dt
```

The problem is with changing coefficients or time, either is not converging or it is converging to non-sense(value of u_1 for example becomes zero in whole domain).

What is problem in your opinion?!

Initial condition seems reasonable. Sometimes I think problem is variational formulation, but I am not sure.

These are the equations:

$\frac{\partial u_1}{\partial t}=D\times\left(\Delta u_1+\frac{K}{u_2}\times\Delta u_2\right)\backslash\frac{\partial u_2}{\partial t}-\left(\frac{K}{c}\right)\times\left(\frac{u_2}{u_1}\right)\times\frac{\partial u_1}{\partial t}=X\times\Delta u_2$∂

`u`_{1}∂`t`=`D`×(Δ`u`_{1}+`K``u`_{2}×Δ`u`_{2})\∂`u`_{2}∂`t`−(`K``c`)×(`u`_{2}`u`_{1})×∂`u`_{1}∂`t`=`X`×Δ`u`_{2}
Community: FEniCS Project

I cannot reproduce your problem since I get a different error in line 70 NameError: name 'Source' is not defined.

written
7 months ago by
Eleonora Piersanti

Oh I am sorry. Delete that line. I forgot to delete it, it does not affect the code. Thank you for checking my code by the way.

written
7 months ago by
Morteza

### 1 Answer

0

When dealing with such low values for coefficients such as the diffusion and thermometric diffusion maybe it would be a good idea to nondimensionalize your equation. We need to be careful with too little and too big values as we are dealing with float points arithmetic.

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