Ch24 Discussion
Discussion Guide
Opening prompt (5 min). "Your simulation ran fine, then you made the grid finer for a better picture, and now it's all NaN. What's your first hypothesis?" Surface intuitions before revealing the CFL answer. Most students blame a coding bug; the point is that nothing in the code is wrong — it's a numerical bug.
Discussion questions: 1. Why is instability a cliff rather than a slope? What is fundamentally different about an error that grows each step versus one that is merely large? (Draw out: amplification factor |G|>1 vs |G|<1 — geometric vs bounded.) 2. The heat limit is r ≤ 1/4 (dt ~ h²); the wave limit is C ≤ 1 (dt ~ h). Why does diffusion punish grid refinement so much harder than wave propagation does? What does this predict about which physics is cheaper to simulate explicitly? 3. In-place update silently switches FTCS to Gauss–Seidel and still produces plausible-looking output. Why is a bug that gives a reasonable wrong answer more dangerous than one that crashes? How would you catch it? (Segue: validation, Case Study 2.) 4. "The purpose of computing is insight, not numbers" (the epigraph). The unstable run produces numbers (Inf, NaN) but zero insight. When has a plausible-but-wrong simulation given false insight? How does validation against a known solution protect you? 5. When is the explicit scheme's simplicity worth its dt limit, and when should you pay for an implicit solve? (Tie r ≤ 1/4 tax to problem size and timescale; connect to Ch. 21 dgesv.)
Mini group activity (20 min): "Find the cliff." In pairs, hand-trace the 2D checkerboard blow-up (example-03) for r = 0.20, 0.25, and 0.30 — three steps each — and report the max interior magnitude at each step. Groups fill a shared table on the board:
| r | step 0 | step 1 | step 2 | step 3 | verdict |
|---|---|---|---|---|---|
| 0.20 | 1 | ? | ? | ? | decays? |
| 0.25 | 1 | ? | ? | ? | holds? |
| 0.30 | 1 | ? | ? | ? | grows? |
They should discover the transition sits right at r = 1/4 (the finite Dirichlet grid decays at 0.25 and grows above; on an infinite grid 0.25 is exactly marginal). Debrief on what "marginally stable" means and why you never run at the limit. This activity makes the abstract stability limit visceral and hand-owned — the whole chapter in twenty minutes of arithmetic.