Unbounded Advection Equation

We consider

for and
with initial condition .
Note that this has exact solution

Downward instability

Consider the Semidiscretization:

using the Euler method:

for
We use Fourier Analysis of Stability:

But then
so the method is unstable for all

Upwind scheme

Consider the Semidiscretization:

using the Euler method:

for
This is veery similar to the previous bit, but now:

So for .
Hence we have stability for , but instability for

Leap-frog method

We Semidiscretization as

But now we solve the ODE using the midpoint rule:

We find the two-step leapfrog method:

The local error is
We use Multi-step:

This is a difference equation
with general solution
where are roots of
In our case:

We have stability iff for all ,
which is true iff