Numerical Analysis II

Abstract

The central topic of this course is the numerical treatment of ordinary differential equations. It focuses on the derivation, analysis, efficient implementation, and practical application of single step methods and pay particular attention to structure preservation.

Objective

The course aims to impart knowledge about important numerical methods for the solution of ordinary differential equations. This includes familiarity with their main ideas, awareness of their advantages and limitations, and techniques for investigating stability and convergence. Further, students should know about structural properties of ordinary diferential equations and how to use them as guideline for the selection of numerical integration schemes. They should also acquire the skills to implement numerical integrators in Python and test them in numerical experiments.

Content

Chapter 1. Some basics 1.1. What is a differential equation? 1.2. Some methods of resolution 1.3. Important examples of ODEs Chapter 2. Existence, uniqueness, and regularity in the Lipschitz case 2.1. Banach fixed point theorem 2.2. Gronwall’s lemma 2.3. Cauchy-Lipschitz theorem 2.4. Stability 2.5. Regularity Chapter 3. Linear systems 3.1. Exponential of a matrix 3.2. Linear systems with constant coefficients 3.3. Linear system with non-constant real coefficients 3.4. Second order linear equations 3.5. Linearization and stability for autonomous systems 3.6 Periodic Linear Systems Chapter 4. Numerical solution of ordinary differential equations 4.1. Introduction 4.2. The general explicit one-step method 4.3. Example of linear systems 4.4. Runge-Kutta methods 4.5. Multi-step methods 4.6. Stiff equations and systems 4.7. Perturbation theories for differential equations Chapter 5. Geometrical numerical integration methods for differential equation 5.1. Introduction 5.2. Structure preserving methods for Hamiltonian systems 5.3. Runge-Kutta methods 5.4. Long-time behaviour of numerical solutions Chapter 6. Finite difference methods 6.1. Introduction 6.2. Numerical algorithms for the heat equation 6.3. Numerical algorithms for the wave equation 6.4. Numerical algorithms for the Hamilton-Jacobi equation in one dimension Chapter 7. Stochastic differential equations 7.1. Introduction 7.2. Langevin equation 7.3. Ornstein-Uhlenbeck equation 7.4. Existence and uniqueness of solutions in dimension one 7.5. Numerical solution of stochastic differential equations

Resources