Advection Equation

We consider the equation:

for and
with initial conditions for
and Dirichlet boundary conditions at
and at

Euler instability

Suppose we try

We have where

But is not Normal Matrix.
The eigenvalues of are bounded by for
but the is which is only for .

Crank-Nicolson method

Crank-Nicolson method
We use Semidiscretization:

and Trapezoidal rule (ODEs)

Lemma

This is stable for all .

Proof

We find
where and are TAntiST Matrix and hence Normal Matrix

Then shares its eigenvectors with and so:

is also Normal Matrix so , thus the method is stable.