Level 3 · Undergraduate core teaching kit · First- and second-year university
Calculus, vectors, and differential equations
Use the learner record during the live investigation, then use the instructor guide to facilitate comparison, address misconceptions, and assess evidence-bounded reasoning.
Learner lab record
Oscillator convergence and energy-drift study
When does a numerical trajectory represent the differential equation rather than the stepping method's error?
Setup
Use the numerical oscillator bench. Establish a reference run, vary only the time step, then change one physical parameter and repeat the convergence study.
Predict first
- 1. Predict how halving the time step changes trajectory error.
- 2. Predict what happens when the step crosses the displayed stability boundary.
| Variable | Role | Unit |
|---|---|---|
| Mass and spring constant | physical inputs | kg and N/m |
| Time step | numerical independent variable | s |
| Trajectory and phase | dependent | m and rad |
| Relative energy error | numerical diagnostic | % |
Observation columns
Analyze
- 1. Which runs demonstrate convergence?
- 2. Does bounded energy oscillation equal exact conservation?
- 3. How does changing mass alter the physical period?
- 4. State a defensible step-size rule from your evidence.
Conclusion frame
For m=___ and k=___, reducing Δt from ___ to ___ changed energy error from ___ to ___; therefore a justified step is ___.
Instructor guide · 50–65 minutes
Teach the investigation, not the interface
Learning target: Learners separate physical parameters from numerical controls and justify convergence using trajectory and invariant diagnostics.
Prepare
- • Review ω=√(k/m) and oscillator energy.
- • Define convergence before opening the lab.
- • Prepare one stable-looking but inaccurate run.
Facilitation moves
- • Ask which equation stayed fixed across the comparison.
- • Require at least three step sizes.
- • Compare phase error with energy error rather than accepting one plot.
Accessibility and participation
- • Pair plots with numeric error summaries.
- • Describe phase-space motion without requiring color distinction.
- • Provide a three-run table and calculator support.
Evidence of learning
- • A three-step convergence table
- • A physical-versus-numerical variable distinction
- • A justified stability recommendation
Misconception checks
A smooth curve must be accurate.
Interpolation can look smooth while phase and energy drift far from the differential-equation solution.
More steps always repair an unstable model.
Smaller steps address discretization; they do not repair an incorrect equation, boundary condition, or unit conversion.
Extension
Compare symplectic Euler with forward Euler using identical initial conditions and computational cost.