IFS Seminar: Structure preserving low-rank methods for the Vlasov equation

Abstract: Performing computer simulations of kinetic equations is

extremely expensive due to the unfavorable scaling of the number of

grid points with dimension (called the curse of dimensionality). To

alleviate this often particle in cell schemes are used. However, it is

well known that such methods can struggle with phenomena (such as

Landau damping) that are relatively easy to resolve using grid-based

methods. Moreover, for turbulent problems often a very large number

of particles is required for particle methods negating much of their

advantage.

Recently, dynamical low-rank methods have shown promise for per-

forming such simulations. These methods use an expansion in lower-

dimensional basis functions to break the curse of dimensionality, while

still maintaining a grid. They can be shown (both theoretical and nu-

merical) to be very effective for a range of problems and we will report

on some 6D simulations that have been performed on a single desktop

computer.

A disadvantage of these methods, however, is that in their original for-

mulation, all the underlying physical structure (e.g. conservation of

20mass, momentum, and energy, Hamiltonian structure, etc.) are de-

stroyed. This is in stark contrast to more traditional methods (both

grid and particle based) for which methods have been designed that

conserve at least some of the physical structure. In this talk, we will

report on our recent progress on developing such structure-preserving

methods in the context of dynamical low-rank approximation.