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.