F11NC - Numerical ODEs

Course leader(s):

Mariya Ptashnyk (Edinburgh, September)

Aims

This module provides an introduction to the derivation and analysis of techniques for the numerical approximation of ordinary differential equations. The theory is reinforced through a practical computational mathematics assignment.

Syllabus

1. Linear Multistep Methods (1.1 1. Local Truncation Error, 1.2 2. Stability Analysis, 1.3 3. Convergence Analysis, 1.4 4. Absolute Stability and Stiffness, 1.5 5. Implementation)

2. Runge-Kutta Methods (2.1 1. Derivation of Runge-Kutta Methods, 2.2 2. Absolute Stability, 2.3 3. Time-Step Control, 2.4 4. Implementation)

3. Invariants in ODEs (3.1 1. Linear Invariants, 3.2 2. Quadratic Invariants, 3.3 3. Invariants in Linear Systems)

Learning outcomes

By the end of the course, students should be able to do the following:

construct approximate solutions of Ordinary Differential Equations (ODEs) using linear multistep and Runge-Kutta numerical methods
analyse important properties of the approximation methods including convergence and different forms of stability
derive a timestep size and error control mechanism for Runge-Kutta approximations.
compare and contrast the theoretical limitations, efficiency and accuracy of different methods for different types of ODEs.
implement different approximation methods on test problems and experimentally evaluate their accuracy and rate of convergence.

Further details

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SCQF Level: 11

Credits: 15