MA8404 Numerical solution of time dependent differential equations, autumn 2023

Official Description and Learning Outcomes

• Wednesdays 08:15-10:00, R50
• Fridays 14:15-16:00, R51

• Week 38. Monday 18.09, 10.15-12 in 656, Sentralbygg 2. Tuesday 19.09, 08:15 to 10:00 in 656, Sentralbygg 2. The lectures scheduled for 20.09 and 22.09 are cancelled.

• Brynjulf Owren
• E-mail: Brynjulf [dot] Owren [at] ntnu [dot] no
• Office: 1350

The curriculum will be taken from a few monographs and articles. Some useful resources are listed below, and there might be more. But they will all be available electronically, through a weblink or posted as pdfs.

• Hairer, Lubich, Wanner, Geometric Numerical Integration (GNI), Springer (2006)
• Hairer, Nørsett, Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems (HNW) 2nd ed, Springer (1993)
• Hairer, Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Equations (HW) 2nd ed, Springer (1996)

The aim of this course is to teach students how to derive, analyse and implement numerical methods for solving time dependent differential equations. To achieve these aims, students will numerically solve mathematical problems and mathematically analyse the methods used for numerical solution. This course will cover the basic topics of stability, accuracy, and efficiency for various numerical algorithms. At the end of the course, the student will be able to:

• Select and implement numerical methods that are most suitable for a given application.
• Analyse the methods, using suitable techniques.
• Participate in scientific discussions and simple research related to the course topics.

Topics:

• Classical methods for ordinary differential equations
• Structure preserving integrators
• Integration methods for PDEs, including multisymplectic and splitting methods

Minor adjustments might be possible, depending on the students interests.

Lecture plan

Order theory, week 36. Start with M1s.mp4, then M2s.mp4 etc.

• The part done in the last week on energy preserving and multi-symplectic methods for PDEs

As a part of the course, you will do a project that will be presented in the last week of the course. You can either propose a project by yourself, and that must be approved by the lecturer. Otherwise, you can select a project from the list that you find here. Whether your project is one of the proposed one or suggested by yourself, register it here. If it is not approved, you will be notified by the lecturer. Please do it before Monday evening September 18.

The exam is oral and you can choose between the dates December 15, 2023 and January 08, 2024. Please sign up with the spread sheet link sent by email on November 5.

A time slot will be assigned for you, and you will draw one topic (and possibly a reserve topic). The topics used two years ago are found below, but it is clear that at least some among 8-10 will be skipped this year.

1. Explain the idea behind collocation methods. Show how collocation methods constitute a subclass of Runge-Kutta methods. Give a quick account of the C and D conditions and their relation to collocation. Discuss linear stability, Padé approximants and the Ehle conjecture. Discuss Radau schemes, Lobatto schemes, DIRK and SDIRK methods with respect to linear stability,
2. Order theory. Explain the main steps in order theory and B-series for Runge-Kutta methods, including the use of rooted trees.
3. Explain contractivity for ordinary differential equations. Then define B-stability and show the criterion for a Runge-Kutta method to be B-stable. Give a proof for the B-stability of Gauss-Legendre methods based on collocation.
4. Explain the general format and important subclasses of linear multistep methods such as the Adams methods (explicit and implicit), BDF methods and even Milne-Simpson and Nyström methods. What is Dahlquist´s first barrier for multistep methods. Then discuss absolute stability for linear multistep methods, including Dahlquists second barrier, A(alpha)-stability, A(0)-stability etc.
5. Explain what are first integrals in differential equations, and the ability of Runge-Kutta methods to preserve first integrals of polynomial type. Also discuss the preservation of more general first integrals by means of discrete gradient methods. Discuss reversibility in differential equations, and explain the adjoint of a method and what it means that a method is symmetric. What is a reversible numerical method.
6. Give a brief introduction to Hamiltonian systems and explain the role of symplecticity, and what it means that a numerical method is symplectic. Discuss symplectic Runge-Kutta methods. Explain the notion of backward error analysis, why it is important when solving Hamiltonian systems, the connection to symplectic schemes, and illustrate the power of this by discussing some results for long time integration. What are variational integrators, how are they derived and what are their properties.
7. Explain the form of energy preserving (or even Hamiltonian) PDEs and discuss in particular the variational derivative. Discuss the use of discrete gradient method for conserving energy in partial differential equations.
8. What are IMEX methods? Discuss the subclass based on Diagonal-Implicit-Runge-Kutta methods combined with explicit Runge-Kutta schemes.
9. Discuss the notion of multisymplecticity, the multi-symplectic form, the energy and momentum conservation laws associated to it. Explain what multi-symplectic methods are, and mention a few of their pros and cons.
10. Give a presentation of exponential integrators. You can include the Lawson methods as a general example and also derive the standard Lie-Euler method.