MATH 4810 Computational Modelling

Module 02 · Numerical ODEs

Numerical Methods for Ordinary Differential Equations

Turn continuous initial-value problems into stepping rules. Compare Euler, modified Euler, midpoint, and fourth-order Runge–Kutta methods, and learn to demonstrate convergence rather than assume it.

2 units · 6 lessons 2 laboratories Assignment 2 · 10% ≈ 20 hours

Module 02

Numerical Methods for Ordinary Differential Equations

Discretization, one-step methods, error analysis, and convergence testing for deterministic dynamical systems.

§ Overview

This module directly supports the modelling of deterministic dynamical systems. You will move from the definition of the initial-value problem through progressively higher-order one-step methods, and you will learn to quantify truncation error, demonstrate convergence order, and recognise when an explicit method is impractical.

Unit 2.1

One-step numerical methods.

Unit 2.2

Runge–Kutta and adaptive reasoning.

Laboratories 3–4

Euler, modified Euler, and Runge–Kutta modelling.

Assignment 2

ODE simulation and error analysis.

§ Learning outcomes

By the end of Module 2 you will be able to

  • Apply Euler, modified Euler, midpoint, and fourth-order Runge–Kutta methods to ordinary differential equations.
  • Analyze sources of truncation, discretization, and roundoff error.
  • Assess consistency, stability, convergence, and step-size sensitivity.
  • Compare analytical and numerical solutions, including observed order of convergence.
  • Recognise stiffness and explain the limitations of explicit methods.

2.1 One-Step Numerical Methods

The core idea: replace the derivative with a discrete approximation and step forward. This unit develops the family of one-step methods and the error analysis that distinguishes them.

Lesson 1

Discretization and Euler's method

Turn a continuous initial-value problem into a stepping rule.

 Lesson 2

Modified Euler and midpoint methods

Second-order corrections and what they buy you in accuracy per step.

 Lesson 3

Local and global error

Truncation error per step versus accumulated error over an interval.

2.2 Runge–Kutta and Adaptive Reasoning

Fourth-order Runge–Kutta is the workhorse of deterministic simulation. This unit covers its derivation, implementation, cost, and the practical questions of step-size selection and method limitations.

Lesson 1

Fourth-order Runge–Kutta method

Derivation, implementation, and the four-stage cost per step.

 Lesson 2

Step-size selection and convergence

Convergence order in practice, and how to demonstrate it numerically.

 Lesson 3

Stiffness awareness and method limitations

When explicit methods force impractically small steps.

§ Laboratories

Lab 3

Euler and modified Euler methods

Implement both methods in a spreadsheet, compare against an analytical solution, and measure observed error.

Lab 4

Runge–Kutta modelling

Implement fourth-order Runge–Kutta, demonstrate fourth-order convergence, and compare cost against accuracy.

§ Assignment 2

ODE simulation and error analysis · 10%

Model a dynamical system with an ODE, implement at least two numerical methods, and present an error analysis that demonstrates convergence order and identifies the dominant error source.

Full assignment brief →

§ Common pitfalls

Watch for these

  • A smaller time step is not automatically sufficient to guarantee a correct result. Step-size reduction controls one error source; it does not fix a wrong equation, an incorrect boundary condition, or a misspecified model.
  • Numerical convergence does not establish model validity. A converged solution can be a converged solution to the wrong problem.
  • A graphical fit does not by itself establish model adequacy. Visual agreement can coexist with systematic error, especially over short observation windows.
👤