Numerical Solution of Ordinary Differential Equations (AA5)

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	AA5
Semester	1
Course Title	Numerical Solution of Ordinary Differential Equations
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning outcomes	Numerical methods for initial value problems for ordinary differential equations are studied in this course; an introduction to numerical methods for the two-point boundary value problem is also given. Learning Objectives: Understanding the basic facts for initial value problems and the two-point boundary value problem. Understanding the fundamental qualitative characteristics of numerical methods for initial value problems, like consistency, order of accuracy, stability and convergence. It is expected that after taking the course the student will have: Understand the basic facts for initial value problems as well as for the two-point boundary value problem. Know the basic numerical methods for initial value problems and are familiar with their advantages and drawbacks. Understand the role of consistency, order of accuracy and stability of numerical methods for initial value problems. Know the basic numerical methods for initial value problems. Know the basic properties of finite difference and finite element methods for the two-point boundary value problem.
General Competences	Production of free, creative and inductive thinking. Consolidation, deepening and application of mathematical knowledge. Familiarity with numerical methods for initial as well as for boundary value problems.

Short introduction to the theory of initial value problems.
Analysis of the Euler methods: order of accuracy; stability properties, A-stability and B-stability; error estimates under various Lipschitz conditions (global, local and one-sided); a posteriori error estimates.
Runge-Kutta and collocation methods: stability properties, order of accuracy, embedded pairs of methods and adaptive time step selection.
Multistep methods: elements of the theory of difference equations, the root condition and stability, order of accuracy, one-leg methods, and G-stability.
Introduction to the theory of the two-point boundary value problem: energy method and elliptic regularity.
Finite difference methods for the two-point boundary value problem.
Finite element method: construction of finite element spaces for various boundary conditions, Galerkin and Ritz methods, the Nitsche trick. Error estimates in the case of indefinite operators.

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