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 Fourier Analysis of Stability > 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