Lecturer: Christos Pervolianakis
Office: Ernst-Abbe-Platz 2, Room 3533
Email: christos.pervolianakis (AT) uni-jena.de
Dates: Every Monday, 2pm--4pm at August-Bebel-Straße 4 - SR 114
Description
Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations. The aim of this seminar is to present the basic results of one-step methods and multi-methods such as consistency, stability and convergence. Moreover, the implementation of these numerical methods for the approximation of the solution of initial value problems are also crucial. More detailed syllabus, you will find here.
Implicit Euler with Newton method: pdf and here its implemetantion in Python.
Lotka-Volterra Equations: pdf and here its implemetantion in Python with Explicit Euler and Implicit Midpoint Method.
16-10-2023 Both the theoretical and programming exercises are not mandatory but it is highly recommended to try it. If you have any question, please send me an email.
18-10-2023 The Exercises 1 is uploaded.
13-11-2023 The Exercises 2 is uploaded.
07-01-2024 The Exercises 3 is uploaded.
16-10-2023 Review of basic result about existence and uniqueness of an initial value problem.
23-10-2023 Review of basic result about stability an initial value problem. Here is the first two lectures in pdf. More about existence and uniqueness of an ODE system can be found in [2, Chapter 1], [4, Section 5.1, 5.2, 5.3] and [4, Chapter 0, 1].
30-10-2023 One-step methods, Local description of one-step methods, order of convergence of two-stage second order Runge-Kutta. Consistency, see [4, Section 5.5].
06-11-2023 Examples of one-step methods. Global description of one-step methods, stability, see [4, Section 5.6, 5.7].
13-11-2023 Stability, convergence, see [3, Section 5.7]. Solvability of Runge-Kutta methods, see [2, II.7, Theorem 7.2].
20-11-2023 Review of numerical integration (Newton-Cotes), see see [4, Section 3.2].
27-11-2023 Solution of exercises (Exercise 1, 5, Set 2). Prove that the matrix and vector of the Butcher tableu of RKM are obtained from numerical integration, page [4, pages 373-374].
04-12-2023 Proof of sufficiently conditions for the order of convergence of Runge-Kutta methods, see [1, Chapter 3].
11-12-2023 Proof of sufficiently conditions for the order of convergence of Runge-Kutta methods, see [1, Chapter 3].
18-12-2023 Finish the proof of sufficiently conditions for the order of convergence of Runge-Kutta methods, see [1, Chapter 3]. Stiff Problems. A-stability, see [4, Section 5.9].
08-01-2024 A-stability, see [4, Section 5.9]. From [4, Subsection 5.9.2], Definition 5.9.2, Theorem 5.9.1 (without proof), Theorem 5.9.2 (without proof). Introdution to B-stability, see [3, IV.12, pages 180-181].
15-01-2024 B-stability, see [3, IV.12, pages 180-181]. Algebraic stability, see [3, V.9, pages 357-358 Lemma 9.2].
22-01-2024 Exercises.
29-01-2024 Exercises. Here are the solutions.
[1] J. Butcher, Numerical Methods for Ordinary Differential Equations, Wiley, 2003
[4] W. Gautschi, Numerical analysis, Boston, MA: Birkhäuser, Second edition.
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