Geometric Numerical Integration, 7.5 Credits

Contents

In the course numerical integrators for stiff differential equations and for dynamical systems with special geometric structures (symplecticity, reversibility, first integrals, etc.) are introduced and studied. The course starts with an overview of classical numerical methods for ordinary differential equations. This is then followed by a more in-depth analysis of various geometric numerical methods that are of importance in many practical applications. The course is concluded with the study of highly oscillatory problems. Exercises will illustrate and complement the theory presented during the lectures. Computer labs illustrate the implementation and use of the presented numerical methods on academic problems.

Expected learning outcomes

For a passing grade, students must be able to

Knowledge and understanding

• account for the concepts trees and B-series
• account for Hamiltonian systems and symplectic discretization
• account for conservation of invariants
• account for properties of highly oscillatory systems

Skills

• apply and implement classical numerical methods to solve ordinary differential equations, such as Euler's method, Runge-Kutta methods and collocation methods
• apply and implement numerical methods treated in the course that exploit geometric properties of dynamical systems
• communicate the solution of problems both in written and oral form

Judgment and approach

• analyze and assess the geometric properties of dynamical systems and select suitable numerical methods
• analyze and assess numerical solutions of problems on ordinary differential equations.

Required Knowledge

The course requires courses in Calculus in One Variable minimum 15 ECTS and a course in Differential Equations for Engineers, 7,5 ECTS. Proficiency in English equivalent to Swedish upper secondary course English A (IELTS (Academic) with a minimum overall score of 5.5 and no individual score below 5.0. TOEFL PBT (Paper-based Test) with a minimum total score of 530 and a minimum TWE score of 4. TOEFL iBT (Internet-based Test) with a minimum total score of 72 and a minimum score of 17 on the Writing Section). Where the language of instruction is Swedish, applicants must prove proficiency in Swedish to the level required for basic eligibility for higher studies.

Form of instruction

Instruction on the course is mainly conducted in the form of lectures, problem solving sessions and seminars.

Examination modes

Examination on the course is in the form of oral or written presentation of hand-in exercises and seminar tasks, and an oral examination of the theory.

Other regulations

The course can also be included in the subject area of computational science and engineering.