IA idea · Numerical methods & error analysis

How small must the step be? Euler's method against an exact solution

AA HLAI HL Solid Also in: Differential equations

Research question

For a differential equation you can solve exactly (cooling, or a population model), how does the error in Euler's method depend on the step size, and how much better is the improved Euler (Heun) method?

Adapt it: change the place, the data or the comparison until the question is yours.

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Why it makes a good exploration

Euler's method is in both HL courses, but the question of how accurate it is usually isn't asked. Comparing with an exact solution makes the error measurable and the pattern provable.

The mathematics you'll need

Euler's method
Exact solution by separation of variables
Global error against h: expect error ∝ h
Improved Euler (new) and error ∝ h²
Log-log plots to find the order

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

Optionally record your own cooling data to set realistic parameters; otherwise no data needed.

Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

Set up and solve the DE exactly.
Apply Euler's method with several step sizes.
Tabulate and plot the error against h.
Implement improved Euler and compare.
Reflect on cost against accuracy.

Pitfalls that cost marks

Measuring the error at only one time.
Not explaining why the error is proportional to h.
Rounding hiding small errors.

Showing personal engagement

Use parameters from your own cooling cup.
Predict the step needed for 0.1 °C accuracy.
Find a DE where Euler's method behaves badly.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Not a natural fit	The core technique sits in the AI course or at HL; an AA SL student could use it only as clearly explained new mathematics.
AA HL	Good fit	Euler's method; Exact solution by separation of variables
AI SL	Not a natural fit	The mathematics is mainly AA or HL (calculus or proof beyond AI SL); an AI SL version would need a data-driven, technology-based approach.
AI HL	Good fit	Euler's method; Exact solution by separation of variables

Level: Solid. Needs some independent work beyond class examples. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Personal engagement (C)

Choose the function or equation yourself and predict how each method will behave before you run it. Hunting for the cases that break a method is engagement an examiner can see.

Reflection (D)

Reflect on error: how it changes with step size or iterations, why a method converges slowly or not at all, and how you know your 'exact' comparison value is correct. For this idea, start with: measuring the error at only one time — say how it affects your answer.

Use of mathematics (E)

SL: The trapezoidal rule or a simple iteration applied correctly, errors tabulated against step size and explained, with any method outside the syllabus explained step by step.

HL: Convergence analysed rather than observed: an error bound derived with calculus or a series, an order of convergence measured and justified, or Euler's method studied against an exact solution.

Criteria A and B (presentation and communication) work the same way for every idea: see the guides to Criterion A and Criterion B.

Taking it further

Apply both methods to a DE with no exact solution and use the error pattern to estimate the true answer.

Extending it for HL

Derive an error bound with a Maclaurin series or calculus, then show your numerical results follow it.

See a complete IA, marked

Our annotated exemplar Is it cheaper to leave the heating on low overnight? A two-temperature model of my bedroom (AI HL) asks a different question, but shows how a complete numerical methods exploration is structured and marked, with an examiner's comment on every criterion. Free excerpts and the full marking table are on its page.

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Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Criteria checklistCheck your plan or draft against the A–E descriptors and self-grade before you hand it in. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, an IB examiner, reads your draft and talks it through with you (€100).