For binary-collision Boltzmann equations, a tailored family of approximate collision operators makes phi-divergence moment closures dissipate an approximate entropy, yielding symmetric-dissipative moment hierarchies and an entropy-stable space-time DG method.
Villani, A review of mathematical topics in collisional kinetic theory, Handb
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Analysis of Moment Closures Using $\varphi$-Divergences for Rarefied Dynamics with Binary Collisions and Their Galerkin Discretizations
For binary-collision Boltzmann equations, a tailored family of approximate collision operators makes phi-divergence moment closures dissipate an approximate entropy, yielding symmetric-dissipative moment hierarchies and an entropy-stable space-time DG method.