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We extend the positivity-preserving method of Zhang & Shu (2010, JCP, 229,3091-3120) to simulate the advection of neutral particles in phase space usingcurvilinear coordinates. The ability to utilize these coordinates is importantfor non-equilibrium transport problems in general relativity and also inscience and engineering applications with specific geometries. The methodachieves high-order accuracy using Discontinuous Galerkin (DG) discretizationof phase space and strong stability-preserving, Runge-Kutta (SSP-RK) timeintegration. Special care in taken to ensure that the method preserves strictbounds for the phase space distribution function $f$; i.e., $f\in[0,1]$. Thecombination of suitable CFL conditions and the use of the high-order limiterproposed in Zhang & Shu (2010) is sufficient to ensure positivity of thedistribution function. However, to ensure that the distribution functionsatisfies the upper bound, the discretization must, in addition, preserve thedivergence-free property of the phase space flow. Proofs that highlight thenecessary conditions are presented for general curvilinear coordinates, and thedetails of these conditions are worked out for some commonly used coordinatesystems (i.e., spherical polar spatial coordinates in spherical symmetry andcylindrical spatial coordinates in axial symmetry, both with spherical momentumcoordinates). Results from numerical experiments --- including one example inspherical symmetry adopting the Schwarzschild metric --- demonstrate that themethod achieves high-order accuracy and that the distribution functionsatisfies the maximum principle.

Bound-Preserving Discontinuous Galerkin Methods for Conservative Phase Space Advection in Curvilinear Coordinates