The artifacts can be explained by some theory
Exact solution of the scheme:
$$ u^n = A^n,\quad A = \frac{1 - (1-\theta) a\Delta t}{1 + \theta a\Delta t}\thinspace .$$
Stability: \( |A| < 1 \)
No oscillations: \( A>0 \)
Always for Backward Euler (\( \theta=1 \))
\( \Delta t < 1/a \) for Forward Euler (\( \theta=0 \))
\( \Delta t < 2/a \) for Crank-Nicolson (\( \theta=1/2 \))
Concluding remarks:
Only the Backward Euler scheme is guaranteed to always give qualitatively correct results.
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