Asymptotic preserving high order generalized upwind SBP schemes with IMEX time integration applied to kinetic transport models

Kinetic models universally describe physical processes relevant to natural and engineering sciences at the level of hyperbolic balance laws. They are characterized by the inclusion of collision operators as source terms, modeling particle interaction. Compared to macroscopic fluid models built from PDEs in space and time for averaged quantities, kinetic fluid models are closer to particle descriptions. Their deeper level offers more insight into less understood phenomena, e.g. rarefied gases or compressible turbulence with extreme demands on direct numerical simulation. Resolving small scales requires further mathematical modeling, whereby kinetic models constitute viable candidates to build upon. They bridge scales by their multiscale nature with respect to the ratio of mean free path and characteristic length.

For vanishing ratios, reasonable kinetic equations converge towards a macroscopic model such as the compressible Euler equations. Close to the limit, the macroscopic model is sufficiently accurate at reduced cost.

However, the multiscale nature of kinetic models may vary locally in space and time in concert with the viability of the corresponding macroscopic model. Requiring the full kinetic model, major numerical challenges are high dimensionality, nonlinearity of physically interesting collision operators, discrete preservation of the asymptotic limit, and stiffness caused by multiple scales. This project intends to advance numerical schemes for kinetic models in order to forward the understanding of multiscale behavior and underresolved fluid flow. The overall goal is to devise novel numerical techniques for kinetic models which stand on firm mathematical ground regarding stability, accuracy and asymptotics. To this end, we follow the path of asymptotic preserving schemes to pass from kinetic to macroscopic models on the discrete level. We will utilize the structure preserving, discretization independent summation-by-parts (SBP) framework in space, owning provable accuracy and stability properties, in particular considering its extension to upwind SBP schemes which include upwind mechanisms for additional artificial dissipation. This will be combined with a new avenue to implicit-explicit (IMEX) time integration based on suitable splittings of the space-discretized equations. Thus, new high order asymptotic preserving IMEX upwind SBP schemes will be designed for kinetic equations related to neutron transport, rarefied gases and turbulence. We strive for predictive stability by a unified analysis of the linear and nonlinear stability properties and asymptotic preservation of the newly developed fully discrete schemes, including the interplay between asymptotic preservation and entropy stability in the nonlinear case.