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.