david/stationary-rk

Consider the 1st-order RK scheme

\[\begin{align*} y_1 & = u_n - h f(y_1) \\ u_{n+1} & = u_n + h f(y_1).\end{align*}\]

When applied to a first-order upwind differencing semi-discretization of the advection equation with CFL number \(\nu=1\), this scheme reduces to \(u_{n+1}=u_n\).

In fact, the scheme is unstable for CFL numbers \(\nu>2/3\), so this is an example of what can happen when you use an unstable method (the solution doesn’t necessarily blow up!)