A moment closure based on projections on the boundary of the realizability domain

Teddy Pichard (Ecole Polytechnique)

The method of moments is widely used for the reduction of kinetic models for applications e.g. in radiative transfer, uncertainty quantification, rarefied gases or polydisperse flows. It consists in integrating the kinetic equations against a set of basis functions. However, the resulting system of moment equations is generally under-determined and requires additional closing equations. One possibility to define such a closure consists in reconstructing an underlying kinetic distribution from the moments, then expressing the closure based on this reconstructed function. Such reconstructions are not unique, and we may choose one with particular properties. However, imposing additional constraints to this reconstruction also corresponds to imposing additional constraints to the set of moments or to the closure. One commonly imposes the underlying distribution to be positive or to belong to a certain convex cone, and its moment are called realizable. This work focuses on the geometry of the realizability domain (set of realizable moments), and especially on its boundary. Then, a method to construct realizable closures is proposed using a particular choice of projections onto the boundary of the realizability domain.