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Thanks to Gromov's link condition, it is easy to construct many CAT(0) cube complexes. On the contrary, constructing hyperbolic cube complexes is often a delicate matter. In this talk I will briefly explain the standard technique that is used to show that some Right Angled Coxeter Groups are hyperbolic and I will then introduce a new technique which applies to a larger class of cube complexes (cube complexes with coupled links).
In many applications, the dynamics of gas and plasma can be accurately modeled using kinetic Boltzmann equations. These equations are integro-differential systems posed in a high-dimensional phase space, which is typically comprised of the spatial coordinates and the velocity coordinates. If the system is sufficiently collisional the kinetic equations may be replaced by a fluid approximation that is posed in physical space (i.e., a lower dimensional space than the full phase space). The precise form of the fluid approximation depends on the choice of the moment-closure. In general, finding a suitable robust moment-closure is still an open scientific problem.
In this work we consider two specific closure methods: (1) a regularized quadrature-based closure (QMOM) and (2) a nonextensible entropy-based closure (QEXP).
In QMOM, the distribution function is approximated by Dirac deltas with variable weights and abscissas. The resulting fluid approximations have differing properties depending on the detailed construction of the Dirac deltas. We develop a high-order discontinuous Galerkin scheme to numerically solve resulting fluid equations. We also develop limiters that guarantee that the inversion problem between moments of the distribution function and the weights and abscissas of the Dirac deltas is well-posed.
In QEXP, the true distribution is replaced by a Maxwellian distribution multiplied by a quasi-exponential function. We develop a high-order discontinuous Galerkin scheme to numerically solve resulting fluid equations. We break the numerical update into two parts: (1) an update for the background Maxwellian distribution, and (2) an update for the non-Maxwellian corrections. We again develop limiters to keep the moment-inversion problem well-posed.