Diffusion Equation

The diffusion equation is:

with some initial conditions
and Dirichlet boundary conditions and .

Full discretization

Similar as in Poisson Equation
By Taylor’s expansion:

So for the true solution we obtain

Now use

with and

A method is convergent if for a fixed and for ever we have

Theorem

If then this method converges.

Proof

We will use for .
We have

so we can get

So by we get:

Proof 2

Apply Lax Equivalence Theorem.

Theorem

This method is stable for norm iff .

Proof

Look at eigenvalues of
noting that is a TST matrix with ,
Eigenvalues of are
Hence the eigenvalues of are ,
and if we will have an eigenvalue of modulus
(for small enough ) which is not stable.

Semidiscretization (with Euler)

Semidiscretization
Write for .
Then we can make the equation into:

Now we can use the Euler method method to solve this,
(which yields a full discretization)
but we can also use the Reverse-Euler Method
which yields:

Reverse Euler has enhanced stability.
This means that we can pick larger
i.e. we can have
Note that for Reverse-Euler Method,
we would need to solve the linear system of equations
that arises from the implicit method.

Crank Nicolson scheme

Crank-Nicolson method
Using the Trapezoidal rule (ODEs) after Semidiscretization we get:

Lemma

This method is stable for .

Proof

Define and
where is a TST matrix with ,
Let .
Amazingly, and have the same set of eigenvectors,
hence also does and they are all Normal Matrix
with eigenvalues given by:

So this method is stable for any !!!

Theorem

This method converges for .

Proof

We can verify that .
We also have:

Hence for the error we have:

so:

All eigenvalues of are greater than
(by Gershgorin Theorem)
and we have already seen thus:

for some .
Thus taking will result in error.