# Unit 1: Numerical Integration of Ordinary Differential Equations

## Contents of a Unit

Each unit is composed of:

- a set of readings that include some short videos to emphasize key ideas. Throughout the readings are embedded questions that are intended to help check your understanding of the material in the readings and videos. Each embedded question also has a corresponding solution, which you may use to check you answer.
- sample problems that are similar to the homework problems. A solution is provided for each sample problem. The sample problems do not have answers to be entered, but we suggest you attempt to solve the sample problems before continuing.

1.1 Overview

- 1.1.1 Measurable Outcomes
- 1.1.2 Pre-requisite Material

1.2 Discretizing ODEs

- 1.2.1 First-Order ODEs
- 1.2.2 An Example of First Order ODE
- 1.2.3 Discretization
- 1.2.4 The Forward Euler Method
- 1.2.5 The Midpoint Method

1.3 Order of Accuracy

- 1.3.1 Errors
- 1.3.2 Local Truncation Error
- 1.3.3 Local Order of Accuracy
- 1.3.4 Definition of Multi-Step Methods
- 1.3.5 Example of Most Accurate Multi-Step Method

1.4 Convergence

- 1.4.1 Types of Errors
- 1.4.2 Convergence of Numerical Methods
- 1.4.3 Rate of Convergence Global Order of Accuracy

1.5 Zero Stability and the Dahlquist Equivalence Theorem

- 1.5.1 Consistency
- 1.5.2 Stability
- 1.5.3 Dahlquist Equivalence Theorem

1.6 Systems of ODE's and Eigenvalue Stability

- 1.6.1 Nonlinear Systems
- 1.6.2 Linear Constant Coefficient Systems
- 1.6.3 Eigenvalue Stability for a Linear ODE
- 1.6.4 Imaginary Eigenvalues

1.7 Stiffness and Implicit Methods

- 1.7.1 Stiffness
- 1.7.2 Spectral Condition Number
- 1.7.3 Implicit Methods for Linear Systems of ODEs
- 1.7.4 Newton-Raphson Implement Implicit Methods on Nonlinear Problems
- 1.7.5 Apply Newton-Rhapson

1.8 Multi-Step Methods

- 1.8.1 Adams-Bashforth Methods
- 1.8.2 Adams-Moulton Methods
- 1.8.3 Backwards Differentiation Methods
- 1.8.4 Backwards Differentiation Excercise

1.9 Runge-Kutta Methods

- 1.9.1 Two-Stage Runge-Kutta Methods
- 1.9.2 Four-Stage Runge-Kutta Method
- 1.9.3 Stability Regions