Summary of observations

The characteristics of the displayed curves can be summarized as follows:

• The Backward Euler scheme always gives a monotone solution, lying above the exact curve.
• The Crank-Nicolson scheme gives the most accurate results, but for $\Delta t=1.25$ the solution oscillates.
• The Forward Euler scheme gives a growing, oscillating solution for $\Delta t=1.25$; a decaying, oscillating solution for $\Delta t=0.75$; a strange solution $u^n=0$ for $n\geq 1$ when $\Delta t=0.5$; and a solution seemingly as accurate as the one by the Backward Euler scheme for $\Delta t = 0.1$, but the curve lies below the exact solution.