# Learner lab record: Oscillator convergence and energy-drift study

Course: Calculus, vectors, and differential equations

Name: ____________________  Date: ____________________  Group: ____________________

## Investigation question

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.

## Variables

| 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 | % |

## Predict before changing controls

1. Predict how halving the time step changes trajectory error.

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2. Predict what happens when the step crosses the displayed stability boundary.

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## Observation table

| mass | spring k | step | steps/period | max energy error | stable? |
| --- | --- | --- | --- | --- | --- |
|   |   |   |   |   |   |
|   |   |   |   |   |   |
|   |   |   |   |   |   |
|   |   |   |   |   |   |
|   |   |   |   |   |   |
|   |   |   |   |   |   |

## Analyze

1. Which runs demonstrate convergence?

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2. Does bounded energy oscillation equal exact conservation?

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3. How does changing mass alter the physical period?

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4. State a defensible step-size rule from your evidence.

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## Evidence-bounded conclusion

For m=___ and k=___, reducing Δt from ___ to ___ changed energy error from ___ to ___; therefore a justified step is ___.

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