An approach for the efficient solution of the time-dependent linear Boltzmann equation

Kinetic equations have proven to be useful models in different applications as, e.g., neutron transport,
gas dynamics, semiconductors, photon propagation, opinion dynamics or biological network
formation. One of the key challenges in the numerical simulation of kinetic equations is their highdimensionality.
For instance, in neutron transport, the solution depends in general on three spatial,
three angular and one temporal variable. Based on the mixed variational formulation analyzed in
[1] we will present a strategy for the design of efficient numerical approximation schemes. The formulation
of [1] incorporates boundary conditions in a weak sense. Unfortunately, the bilinear form
corresponding to the boundary conditions couples spatial and angular variables in a non-smooth way
rendering a tensor product approximation not straight-forward. As a remedy we introduce an absorbing
layer and consider a perturbed variational problem which leads to matrices with tensor product
structure. Hence, even in the full tensor product approximation, the resulting discrete operators can
be stored efficiently. We provide corresponding error estimates for our approach.