Generalized Curvilinear Advection Formalism for Finite Volume Codes Doing Relativistic Hydrodynamics

While it is possible to numerically evolve the relativistic fluid equations using any chosen coordinate mesh, typically there are distinct computational advantages associated with different types of candidate grids. 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 (AMR) and avoid the appearance of coordinate singularities. Here it is shown that it should be possible to reap the benefits associated with multiple types of 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 particular set of coordinates on a mesh defined by an alternative type of coordinate system. A formalism (a generalization of the much-used Valencia formulation) that will aid in the implementation of such a hybrid scheme is provided. It is further suggested that a preferred approach to modeling astrophysical flows that are dominated by rotation may involve the evolution of inertial-frame cylindrical momenta (i.e., radial momentum, angular momentum, and vertical momentum) and the Jacobi energy—all on a corotating Cartesian coordinate grid.