The artifacts can be explained by some theory

Exact solution of the scheme: $$u^n = A^n,\quad A = \frac{1 - (1-\theta) a\Delta t}{1 + \theta a\Delta t}\thinspace .$$

Key results:

• Stability: $|A| < 1$
• No oscillations: $A>0$
• $\Delta t < 1/a$ for Forward Euler ($\theta=0$)
• $\Delta t < 2/a$ for Crank-Nicolson ($\theta=1/2$)