Kinetic problems in plasma physics and radiative transfer (e.g. the Vlasov equation or the Boltzmann equation) are posed in an up to six-dimensional phase space. The unfavorable scaling of the degrees of freedom with dimension, called curse of dimensionality, makes such simulations extremely challenging. Moreover, many of the classic complexity reduction techniques, such as sparse grids, do not cope well with the fine-scale structures inherent in many kinetic problems.
We have developed dynamical low-rank algorithms suited for kinetic problems. My Banff BIRS talk provides an introduction to the general methodology. A major recent advance has been the construction of a mass, momentum, and energy conservative dynamical low-rank algorithm.
We have also shown how to construct dynamical low-rank algorithms that are asymptotic preserving in the corresponding diffusive or fluid limits. Providing a mathematical analysis is challenging as very few rigorous results that show that the solution is low-rank are available and the obtained equations of motion for the low-rank factors are fundamentally nonlinear. In the case of the linear Boltzmann equation close to the diffusive limit such an analysis can be conducted using Chapman-Enskog theory.