Finite Difference Methods: Theory, Application, and Python Programming

A: The hard number is the safety margin to the junction max, often 10–15 °C. If boundary heat flux is optimistic, the FTCS solution may look stable yet never cross the diagnostic threshold, so the safety mechanism isn’t triggered in simulation. That’s a systematic fault exposure issue, not a numerical preference.

A: The boundary is Fo = αΔt/Δx² ≈ 0.5. With Δx² = 1e‑6 m², Δt lands near 0.5×1e‑6/9e‑5 ≈ 5e‑4 s. Miss that order and the solution can look fine for thousands of steps before it blows up.

A: The threshold is the thermal gradient, not the mean temperature. Numerical diffusion flattens peaks, so the model misses cyclic strain range. That lines up with low-cycle thermal fatigue appearing well before creep or corrosion would be active.

A: The limit here is θ = 0.5. Crank–Nicolson is unconditionally stable but not monotone, so sharp transients plus large Δt cause ringing. Reducing Δt attacks the phase error without corrupting the boundary physics.