English: Prothero-Robinson test problem using ${\displaystyle \lambda =-20}$ with ${\displaystyle N=5}$ (top) and ${\displaystyle N=10}$ steps (bottom), comparing explicit Euler (left panels) to implicit Euler (right panels). Blue curve is exact solution, green curves are local transients passing through computed points (red circles). For ${\displaystyle N=5}$ (${\displaystyle h=0.2}$) the explicit method is unstable (top left) as indicated by oscillations of exponentially growing amplitude. Reducing the step size (${\displaystyle N=10}$, lower left), the method regains stability. In the implicit method there is no instability at all, admitting the use of large steps.