Dear community,

I try to solve a transient Poisson equation with Robin BC on an 1x1 UnitSquareMesh domain, which is described by

u′=Δu+f

u=u_0 at t=0

In order to obtain a discretisization in time the Crank-Nicolson method is employed. Due to the implementation of the Crank-Nicolson method, f^{n-1} for the previous time step, called `f_prev`

in the Python code, and f^{n} for the current time step, called `f`

in the Python code, were introduced. Consequently u^{n-1}, called `u_prev`

, and u^{n}, called `u`

, are used.

My source term f, which is just made up for code demonstration, rather than being a reasonable model for any physical system, is described by the following UserExpression:

```
class MySourceConcentration(UserExpression):
def __init__(self, t, u, **kwargs):
super().__init__(**kwargs)
self.t = t
self.u = u
def eval(self, values, x):
# Calculate the source term f for t>0
if self.t >0:
if x[0]>0 and x[0]<=0.5:
# Calculate average of u for the domain x>=0 and x<=0.5
u_avg = assemble(self.u*dx(1)) / assemble(Constant(1.0)*dx(1))
# Calculate the surface area of a particle
area = area_0 - dt * (1 - u_avg)
# Calculate source term f
values[0] = Constant1 * area * (Constant2 - u_avg)
# for the rest of the domain: source term f=0
else:
values[0] = 0
# Set the source term f for t=0
else:
if x[0]>0 and x[0]<=0.5:
values[0] = f_0
else:
values[0] = 0
def value_shape(self):
return ()
```

I was told that updating the UserExpression at the end of the main for-loop by using

```
f_prev = f
```

should be avoided. So I added the following code lines to the existing UserExpression

```
def update(self, t, u):
self.t = t
self.u = u
```

and updated the UserExpression at the end of the main for-loop in the following matter

```
f_prev.update(t, u_prev)
```

My MWCE is running, but I´m unsure, if the implementation is right?

Could you confirm or refute this?

Thanks,

Cou

Here is the full MWCE:

```
from fenics import *
import numpy as np
# Define source term f
class MySourceConcentration(UserExpression):
def __init__(self, t, u, **kwargs):
super().__init__(**kwargs)
self.t = t
self.u = u
def eval(self, values, x):
# Calculate the source term f for t>0
if self.t >0:
if x[0]>0 and x[0]<=0.5:
# Calculate average of u for the domain x>=0 and x<=0.5
u_avg = assemble(self.u*dx(1)) / assemble(Constant(1.0)*dx(1))
# Calculate the surface area of a particle (equation is made up for demonstrationen purposes without physical correctness)
area = area_0 - dt * (1 - u_avg)
# Calculate source term f
values[0] = Constant1 * area * (Constant2 - u_avg)
# for the rest of the domain: source term f=0
else:
values[0] = 0
# Set the source term f for t=0
else:
if x[0]>0 and x[0]<=0.5:
values[0] = f_0
else:
values[0] = 0
def value_shape(self):
return ()
def update(self, t, u):
self.t = t
self.u = u
# Define Constants
Constant1 = 1
Constant2 = 10
D = Constant(0.6) # diffusion coefficient
f_0 = 4 # Source term value for t=0
beta = 1.2
T = 2.0 # final time
num_steps = 10 # number of time steps
dt = T / num_steps # time step size
# Create mesh and define function space
nx = ny = 16
mesh = UnitSquareMesh(nx, ny)
space_dim = mesh.geometry().dim()
V = FunctionSpace(mesh, 'CG', 1)
# Create classes for defining parts of the boundaries and the interior
# of the domain
class Left(SubDomain):
def inside(self, x, on_boundary):
return near(x[0], 0.0)
class Right(SubDomain):
def inside(self, x, on_boundary):
return near(x[0], 1.0)
class Bottom(SubDomain):
def inside(self, x, on_boundary):
return near(x[1], 0.0)
class Top(SubDomain):
def inside(self, x, on_boundary):
return near(x[1], 1.0)
class Obstacle_left(SubDomain):
def inside(self, x, on_boundary):
return (between(x[0], (0.0, 0.5)))
class Obstacle_right(SubDomain):
def inside(self, x, on_boundary):
return (between(x[0], (0.5, 1.0)))
# Initialize sub-domain instances
left = Left()
top = Top()
right = Right()
bottom = Bottom()
obstacle_left = Obstacle_left()
obstacle_right = Obstacle_right()
# Initialize mesh function for interior domains
domains = MeshFunction("size_t", mesh, space_dim-1)
obstacle_left.mark(domains, 1)
obstacle_right.mark(domains, 3)
# Initialize mesh function for boundary domains
boundaries = MeshFunction("size_t", mesh, space_dim-1)
boundaries.set_all(0)
left.mark(boundaries, 1)
top.mark(boundaries, 2)
right.mark(boundaries, 3)
bottom.mark(boundaries, 4)
# Define new measures associated with the interior domains and
# exterior boundaries
dx = Measure('dx', domain=mesh, subdomain_data=domains)
ds = Measure('ds', domain=mesh, subdomain_data=boundaries)
vtkfile = File("UserExpression_update.pvd")
################### Initial values (t=0) ######################################
t=0
u_0 = Constant(1)
u_prev = interpolate(u_0, V)
f_prev = MySourceConcentration(t, u_prev, degree=1)
############ Robin boundary condition j_m * n_m = h*(u-c_oo) ##################
h = 1.0
c_oo = 1.0
#################### Define variational problem ###############################
u = TrialFunction(V)
v = TestFunction(V)
f = MySourceConcentration(t, u_prev, degree=1)
F = (u - u_prev)*v*dx + 0.5*h*beta*dt*(u+u_prev-2*c_oo)*v*ds + 0.5*beta*dt*D*(inner((grad(u)+grad(u_prev)),grad(v)))*dx - 0.5*dt*(f_prev+f)*v*dx(1) - 0.5*dt*(f_prev+f)*v*dx(3)
a, L = lhs(F), rhs(F)
# Time-stepping
u = Function(V)
for n in range(num_steps):
# Update current time
t += dt
# Compute solution
solve(a == L, u)
vtkfile << (u, t)
# Update previous solution
u_prev.assign(u)
f_prev.update(t, u_prev)
```