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 (\( \theta=1/2 \))