## John von Neumann Lecture: Numerical methods for hyperbolic and kinetic equations (Summer Semester 2019)

### Description of achievement and assessment methods

Written final exam in form of a project report. Students will write a report in which they describe how they apply some of the methods and concepts taught in this course in an application. The report may focus on either a theoretical analysis, on numerical aspects, and/or on practical elements of using numerical methods for hyperbolic and kinetic equations.

### Possibility of re-taking

In the next semester: No

At the end of the semester: Yes

### (Recommended) requirements:

Students are supposed to have a Bachelor degree in Mathematics, Physics or Computer Science. Some basic notions of partial differential equations and numerical analysis are needed. Some knowledge of probability and experience in programming (e.g. MATLAB) may be helpful, even though not strictly necessary.

### Contents

Hyperbolic and kinetic partial differential equations arise in a large number of models in physics and engineering. Prominent examples include the compressible Euler and Navier-Stokes equations, the shallow water equations, the Boltzmann equation and the Vlasov-Fokker-Planck equation. Example of applications area range from classical gasdynamics and plasma physics to semiconductor design and granular gases. Recent studies employ the aforementioned theoretical background in order to describe the collective motion of a large number of particles such as: pedestrian and traffic flows, swarming dynamics and other dynamics driven by social forces. These PDEs have been subjected to extensive analytical and numerical studies over the last decades. It is widely known that their solutions can exhibit very complex behavior including the presence of singularities such as shock waves, clustering and aggregation phenomena, sensitive dependence to initial conditions and presence of multiple spatio-temporal scales.

This course will cover the mathematical foundations behind some of the most important numerical methods for these types of problems. To this goal the first part of the course will be devoted to hyperbolic balance laws where we will introduce the notions of finite-difference, finite volume and semi-Lagrangian schemes. In the second part we will focus on kinetic equations where, due to the high number of dimensions and their intrinsic physical properties, the construction of numerical methods requires a careful balance between accuracy and computational complexity. Finally, we will consider some recent developments related to the construction of asymptotic preserving methods, and to the development of efficient methods for optimal control and uncertainty quantification.

### Study goals

Upon completion of this module, students will have a basic understanding of the mathematical concepts behind some of the key numerical methods for Hyperbolic and Kinetic equations. They will understand the connection between the mathematical structure and its practical implementation, and apply the studied methods in selected applications.

Teaching and learning methods:
Presentation of the material on the blackboard and via slides. Exercises to apply and to deepen the learned knowledge.

Media formats:
List of exercise problems, Lecture notes, MATLAB-Code, blackboard- and slide presentations

A detailed list of references will be provided. Some original works will be provided via download.