MATH 4810Computational Modelling

Module 02 · Capstone Project

An ODE Solver Toolkit and Convergence Study

Implement a small library of one-step numerical methods, apply them to a nonlinear model, demonstrate the convergence order of each, and diagnose a stiff case. The project consolidates every skill of Module 2.

Weight Counts toward the module component of the course grade Duration ≈ 14 hours Prerequisite Units 2.1 and 2.2

Module 2 capstone

An ODE Solver Toolkit and Convergence Study

Implement Euler, modified Euler, midpoint, and fourth-order Runge–Kutta. Demonstrate their convergence orders. Apply them to a nonlinear model. Diagnose a stiff case.

§ Project brief

The Module 2 capstone is a small library of ODE solvers accompanied by a full convergence study. You will implement four one-step methods, verify each against an analytical solution, estimate the observed convergence order numerically, apply the methods to a nonlinear model without a closed-form solution, and diagnose a stiff case.

The project is assessed on the correctness of the implementations, the rigour of the convergence study, and the clarity of the interpretation. A well-organised spreadsheet with a clean convergence table and a clear stiffness discussion is the target deliverable.

Implementation plus evidence

A solver that produces numbers is not a deliverable. A solver that produces numbers and demonstrates that its observed order matches the theory is.

§ Learning outcomes addressed

This capstone assesses

  • Implement Euler, modified Euler, midpoint, and fourth-order Runge–Kutta.
  • Verify each method against an analytical solution.
  • Demonstrate the observed convergence order numerically.
  • Apply the methods to a nonlinear initial-value problem.
  • Perform a step-doubling error estimate in the absence of a closed-form solution.
  • Recognise and diagnose stiffness.

§ Suggested models

Model A

Nonlinear pendulum

θ'' + (g/L) sin θ = 0, reduced to a first-order system. No closed-form solution for large amplitude.

Model B

Lotka–Volterra predator–prey

x' = αx − βxy, y' = δxy − γy. Has an invariant that can be used as a verification check.

Model C

Damped oscillator

x'' + 2ζωx' + ω²x = 0. Closed-form solution available; useful as a verification reference.

Model D

SIR epidemic

S' = −βSI/N, I' = βSI/N − γI, R' = γI. Conserves total population.

§ Deliverables

Deliverables and weights
Deliverable Weight What it should contain
Solver implementations 20% Euler, modified Euler, midpoint, and RK4 for a general first-order system. Documented, verified.
Analytical verification 15% Comparison against a closed-form solution for at least one problem, with error table.
Convergence study 25% Observed order for each method at four step sizes. Log-log plot. Comparison to theoretical order.
Application to nonlinear model 15% Simulation of the chosen model with step-doubling error estimate and a claim-support statement.
Stiffness diagnosis 15% A stiff example. Demonstration of the stability limit. Discussion of the practical consequences.
Technical report 10% Three to five pages: model statement, implementations, convergence evidence, stiffness discussion.

§ Milestones

  1. Choose the model and analytical reference Pick one of the suggested models or propose your own. Two hours.
  2. Implement the four solvers Euler, modified Euler, midpoint, RK4. One function per method. Three hours.
  3. Verify against analytical solution Run each method at one step size, compare to the exact solution. Two hours.
  4. Convergence study Run each method at four step sizes, compute observed order. Three hours.
  5. Apply to the nonlinear model Run with step doubling and report estimated error. Two hours.
  6. Diagnose stiffness Construct a stiff example, demonstrate the stability limit. Two hours.

§ Rubric

Assessment criteria
Criterion Excellent Satisfactory Needs work
Solver correctness All four methods correct and verified. Three methods correct; one has a minor error. Two or more methods incorrect.
Convergence study Observed order within 10% of theoretical for all methods. Observed order within 25% for all methods. Observed order does not match theory.
Application Full simulation with error estimate and clear claim-support statement. Simulation with error estimate but limited interpretation. Simulation without error estimate.
Stiffness Clear demonstration of the stability limit; practical implications discussed. Demonstration of the stability limit without discussion. No stiffness example.
Report Well-organised, complete, self-contained. Complete but disorganised. Missing sections or unclear.

§ Common pitfalls

Watch for these

  • Using k₁ in the k₃ stage of RK4. The most common implementation error. k₃ uses k₂.
  • Estimating order from only two step sizes. Two points can always be fitted by a line. Use at least four.
  • Comparing the observed order to a single-point error. The order is the slope of the log-log plot, not a single ratio.
  • Extrapolating a stiff step size from the stable regime. The stability limit is a hard boundary, not a soft one.
  • Claiming the model is correct because the convergence order matches. Convergence order addresses the method, not the model.

§ Submission

What to submit

  • A single ZIP archive named M2-capstone-yourid.zip.
  • The archive contains the solver spreadsheet or code, the verification and convergence tables, the log-log plot, the stiffness demonstration, and the technical report.
  • Submit through the course LMS before the deadline listed on the course schedule.
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