Advances in Numerical Methods for Hyperbolic and Kinetic Equations


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Summary: Hyperbolic and kinetic partial differential equations play a pivotal role in many applications where the dynamics is governed by transport processes combined with the presence of source terms, like external forces, diffusion or microscopic interactions. In addition to classical  fields such as  fluid dynamics, recently, these equations have gained relevance in modeling complex systems of interacting agents, such as traffc  flow, crowd and swarms dynamics, opinion propagation, wealth distributions, and the spread of diseases.

In this course, we will introduce students to the subject with an emphasis on the design and the analysis of computational methods. In the first part we will focus on hyperbolic conservation laws by introducing the notion of shock, entropy solution and the construction of  finite difference and finite volumes schemes. We will then analyze how to deal with the presence of sources and multiple scales through the concept of asymptotic-preserving (AP) method.

The second part of the course focuses on specific aspects related to kinetic equations. Due to the high number of dimensions and the intrinsic structural properties, the construction of numerical methods requires a careful balance between accuracy and computational complexity. We will discuss some of the main numerical techniques, both from a deterministic point of view and from a probabilistic perspective. We will then show how to extend the construction of AP schemes to this challenging setting.

Finally, the last part of the course is devoted to the case where equations may depend on uncertain parameters. Here we will illustrate some of the basic approaches, such as stochastic Galerkin methods and multifidelity methods.

Assessment: At the end of the course, students can choose between either a written assessment accompanied by a small computational project or a short presentation on topics that expand upon those covered in the module.