Ch23 Discussion

Discussion Guide

Opening prompt (5 min). "Your simulation gives smooth, plausible output. How do you know it isn't smoothly, plausibly wrong?" Surface: compare to a known solution, check a conserved quantity, test the convergence order, sanity-check equilibria. This frames the whole chapter's verification discipline.

Prompt 1 — order vs cost. RK4 costs 4 RHS evaluations per step, Euler 1. Walk the room through: for 6-digit accuracy, who does less total work, and by how much? (Answer: RK4, by ~4 orders of magnitude — F1.) Then: when might Euler still be the right call? (Very short runs; a throwaway estimate; a stiff problem where neither explicit method is appropriate anyway.)

Prompt 2 — the method-of-lines threshold. Put the 1D heat MOL system on the board: $du_i/dt = (u_{i-1} - 2u_i + u_{i+1})/\Delta x^2$. Ask: "What is this? Point at it." Lead them to it is a system of ODEs — and u = u + dt*rhs(u) is Euler. Then: "If Euler-in-time is what Ch. 24's explicit scheme uses, what could you swap in to get a better solver?" (RK4 in time.) This is the chapter's payoff; make them say it.

Prompt 3 — stability vs accuracy. Show explicit Euler on $y' = -y$ at $h = 1, 2, 3$ (→ 0, oscillate, blow up) beside backward Euler (→ 0.5, 0.333, 0.25, always sane). Ask: "The true solution just decays to zero — why does the explicit method explode?" Draw out that the failure is stability, not accuracy, and connect it to why stiff chemistry and fine PDE grids need implicit methods.

Mini group activity (20 min). In pairs: hand-integrate the harmonic oscillator $y_1' = y_2$, $y_2' = -y_1$ from $(1, 0)$ one Euler step and one RK4 step, both with $h = 0.5$, and compute the energy $\tfrac12(y_1^2 + y_2^2)$ after each. (Euler gives $(1, -0.5)$, energy $0.625$ — energy grew 25%! RK4 gives $(0.877604, -0.479167)$, energy $0.499895$ — barely moved.) Debrief: Euler pumps energy into a conserved system (orbits spiral out), which is why plain Euler is disqualified for long dynamical runs and why symplectic integrators exist. This single computation makes the §23.4/§23.6 energy-drift point concrete and memorable.

Exit question. "Name the one line that proves the heat solver is an ODE integrator." (Target: u = u + dt*rhs(u) — Euler's method on the method-of-lines system.)