A generalized advection formalism for relativistic fluid simulations

While it is possible to numerically evolve the relativistic fluid equations using any chosen coordinate mesh, typically there will be computational advantages associated with certain choices. For example, astrophysical flows that are governed by rotation tend to give rise to advection variables that are naturally conserved when a cylindrical mesh is used. On the other hand, Cartesian-like coordinates afford a more straightforward implementation of adaptive mesh refinement and avoid the appearance of coordinate singularities. Here we show how it may be possible to reap the benefits associated with multiple coordinate systems simultaneously in numerical simulations. This could be accomplished by implementing a hybrid numerical scheme: one that evolves a set of state variables adapted to one set of coordinates on a mesh defined by an alternative set of coordinates. We provide a formalism (a generalization of the much-used Valencia formulation) by which this can be done. We suggest that a preferred approach to modeling astrophysical flows that are dominated by rotation involves the evolution of inertial-frame cylindrical momenta (i.e. radial momentum, angular momentum and vertical momentum) and the Jacobi energy on a corotating Cartesian coordinate grid.