Kinetic equations play an important role in applications ranging from fusion and astrophysical plasmas to radiative heat transfer and models in biology. Moreover, all of these fields rely heavily on computer simulation. The most challenging aspect of numerically solving kinetic equations is, without doubt, the up to six-dimensional phase space of these equations. However, this is by no means the only challenge encountered. In many problems a stringent CFL condition needs to be overcome, small scale structures and large gradients (similar to computational fluid dynamics) have to be resolved, and conservative numerical methods that preserve certain physical invariants are desired.
This summer school will familiarize the participants with many of the numerical methods that are used to treat kinetic equations. This includes both methods that employ some form of complexity reduction and schemes that directly discretize phase space. The former category includes particles methods, such as particle-in-cell (PIC) schemes, and dynamic low-rank approximations. In the latter category we will primarily consider semi-Lagrangian schemes, such as those based on Lagrange and spline interpolation or a discontinuous Galerkin formulation. Conservative numerical methods will be a common theme in many of the lectures. We will also consider some topics related to the implementation of these schemes. Finally, current research topics in the field will be considered.
The lectures will be complemented by hands-on exercise sessions in which the participants will gain experience implementing the numerical methods. The main prerequisites are a basic knowledge of numerical analysis and some familiarity with a programming language, such as Python or Matlab.