# Speeding up time-dependent diffusion equation

Hi all,

I am working on solving a diffusion equation with a time-dependent diffusion tensor. For that, I am following the Diffusion of a Gaussian function demo, as well as the discussion in the similar thread Time dependent diffusion constant in dolfinx. However, because my diffusion coefficient is a tensor, I cannot use a simple trick from the previous post (`D_.value[0]= 0.1*D`) to reassign the value of the diffusion constituent and have to rewrite the tensor completely. As a result, the following re-assembly of the `A` matrix is taking a very long time on each timestep and I was wondering if there is a way to speed things up.

The example code is as follows (very minor edits to the original Diffusion of a Gaussian function demo). While this example still works quite fast although noticeably slower than the original, this approach results in an extremely large delay for the more complex problem that I am trying to solve.

``````import numpy as np

from mpi4py import MPI
from petsc4py import PETSc

from dolfinx import fem, mesh, io, plot

# Define temporal parameters
t = 0 # Start time
T = 1.0 # Final time
num_steps = 50
dt = T / num_steps # time step size

# Define mesh
nx, ny = 50, 50
domain = mesh.create_rectangle(MPI.COMM_WORLD, [np.array([-2, -2]), np.array([2, 2])],
[nx, ny], mesh.CellType.triangle)
V = fem.FunctionSpace(domain, ("CG", 1))

# Create initial condition
def initial_condition(x, a=5):
return np.exp(-a*(x[0]**2+x[1]**2))
u_n = fem.Function(V)
u_n.name = "u_n"
u_n.interpolate(initial_condition)

# Create boundary condition
fdim = domain.topology.dim - 1
boundary_facets = mesh.locate_entities_boundary(
domain, fdim, lambda x: np.full(x.shape[1], True, dtype=bool))
bc = fem.dirichletbc(PETSc.ScalarType(0), fem.locate_dofs_topological(V, fdim, boundary_facets), V)

uh = fem.Function(V)
uh.name = "uh"
uh.interpolate(initial_condition)

import ufl
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
f = fem.Constant(domain, PETSc.ScalarType(0))

D = ufl.as_tensor(((1, 0), (0, 2)))
a = u * v * ufl.dx + dt*ufl.inner(D * ufl.grad(u), ufl.grad(v)) * ufl.dx
L = (u_n + dt * f) * v * ufl.dx

bilinear_form = fem.form(a)
linear_form = fem.form(L)

A = fem.petsc.assemble_matrix(bilinear_form, bcs=[bc])
A.assemble()
b = fem.petsc.create_vector(linear_form)

solver = PETSc.KSP().create(domain.comm)
solver.setOperators(A)
solver.setType(PETSc.KSP.Type.PREONLY)
solver.getPC().setType(PETSc.PC.Type.LU)

for i in range(num_steps):
t += dt

print(t)
D = ufl.as_tensor(((1 - t, 0), (0, 2 - t)))
a = u * v * ufl.dx + dt*ufl.inner(D * ufl.grad(u), ufl.grad(v)) * ufl.dx
print('----> Assembling')
A = fem.petsc.assemble_matrix(fem.form(a), bcs=[bc])
A.assemble()
solver.setOperators(A)

# Update the right hand side reusing the initial vector
with b.localForm() as loc_b:
loc_b.set(0)
fem.petsc.assemble_vector(b, linear_form)

# Apply Dirichlet boundary condition to the vector
fem.petsc.apply_lifting(b, [bilinear_form], [[bc]])
fem.petsc.set_bc(b, [bc])

# Solve linear problem
solver.solve(b, uh.vector)
uh.x.scatter_forward()

# Update solution at previous time step (u_n)
u_n.x.array[:] = uh.x.array

``````

Constants can have the shape of a tensor.
See for instance https://github.com/FEniCS/dolfinx/blob/deac4f1cba124f09302f8bb501777f3e25a827c4/python/test/unit/fem/test_constant.py#L54

@dokken thanks.
But what to do in the more general case if elements of the `D` tensor are expression (e.g., depend on the spatial coordinates)?

You can make a tensor with `ufl.as_tensor` that is a mixture of `ufl.SpatialCoordinate` and constants (say for time)`
For instance you could do something like

``````x = ufl.SpatialCoordinate(domain)
a = dolfinx.fem.Constant(domain, 0.)
t = dolfinx.fem.Constant(domain, 0.1)
D = ufl.as_tensor(((a*x[0]-t, 0),(x[1], t)))
``````

and update a and t in your time loop.

1 Like

Yup, that’s what I am doing in the code above and it works perfectly fine, however, I was wondering if there is a way to speed up the assembly of the matrix on each timestep after updating `D`. The command

``````A = fem.petsc.assemble_matrix(fem.form(a), bcs=[bc])
A.assemble()
``````

takes vary long, thus slowing down the computations significantly. Any ideas?

One immediate optimisation is to take `fem.form(a)` outside of the loop. This will be looking up the cache on your file system and potentially compiling a form each call. This is very expensive and not its intended use.

This also means you should refactor constant values (e.g. `t`) into a `dolfinx.fem.Constant` and update them. Then you don’t have to recompile the form every time step.

Trying to find more optimisations will likely involve splitting up your stiffness matrix into linear combinations of those with multiples of updating constants. You then won’t have to assemble the FE matrix each time step, just a linear combination of matrices which I believe is scalable.

You can then try to accommodate for these trivial matrix updates in the LU factorisation. If you can reuse the LU factorisation or, like with the matrix operators compose it of a linear combination, you should see a big performance leap.

4 Likes