Low-rank solution of two-particle kinetic equations

• Applicant: Professorin Dr. Katharina Kormann
• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 530709913

Kinetic equations are often considered the most comprehensive model for the description of plasmas, gases and fluids that is computationally feasible. Nevertheless the kinetic equations are derived from the many-particle distribution function under certain assumptions. For most phenomena these assumptions are satisfactory but in particular on short time-scale and for small mean-free path, where the molecular chaos assumption fails, they come to their limit. The most prominent example of such a state are dense gases. A more comprehensive description of kinetic processes consists of the two-particle method that describes an $N$-body problem kinetically and includes the two-particle interactions exhaustively. The numerical solution of this model has not been addressed to the best of our knowledge due to the high dimensionality of the 12-dimensional phase-space. On the other hand, low-rank approximations have seen a rapid development in recent years and have been successful in mitigating the curse of dimensionality under certain assumptions. In particular, the low-rank solution of the Vlasov and the Boltzmann equations has been shown to yield good compression and accurate results in a number of recent publications. It is the goal of this project to develop, implement, and analyze a first numerical simulation tool for the two-particle kinetic equation based on low-rank compression. We will consider a representation that couples a kinetic description of the one-particle distribution function with a two-particle correction function and a field equation describing the (electric) field induced by the particles. The numerical solution strategy will focus on an efficient low-rank splitting and a structure-preserving spatial discretization. We will therefore derive an adaptive dynamical low-rank approximation that is based on a structure-preserving truncation, a moment-correction by coupling to macroscopic fluid models, as well as a carefully-adapted active-flux discretization of the projected subsystems of the dynamical low-rank approximation. We plan to compare the results of our new solver to the solution of the Boltzmann equation and compare characteristic properties like the relaxation time for distributions, in particular far from Maxwellian equilibrium.

• DFG Programme: Research Grants