Efficient Stabilization of Advection Terms Involved in Separated Representations of Boltzmann and Fokker-Planck Equations (Communications in Computational Physics, 2018)
The fine description of complex fluids can be carried out by describing theevolution of each individual constituent (e.g. each particle, each macromolecule, etc.).This procedure, despite its conceptual simplicity, involves many numerical issues, themost challenging one being that related to the computing time required to update thesystem configuration by describing all the interactions between the different individuals.Coarse grained approaches allow alleviating the just referred issue: the systemis described by a distribution function providing the fraction of entities that at certaintime and position have a particular conformation. Thus, mesoscale models involvemany different coordinates, standard space and time, and different conformationalcoordinates whose number and nature depend on the particular system considered.Balance equation describing the evolution of such distribution function consists of anadvection-diffusion partial differential equation defined in a high dimensional space.Standard mesh-based discretization techniques fail at solving high-dimensional modelsbecause of the curse of dimensionality. Recently the authors proposed an alternativeroute based on the use of separated representations. However, until now theseapproaches were unable to address the case of advection dominated models due to stabilizationissues. In this paper this issue is revisited and efficient procedures for stabilizingthe advection operators involved in the Boltzmann and Fokker-Planck equationwithin the PGD framework are proposed.