Building a cosmological hydrodynamic code: Consistency condition, moving mesh gravity, and SLH-P(3)M

Building a self-gravitating hydrodynamic code as a combination of a hydrodynamic solver and a gravity solver is discussed. We show that straightforward combining of those two solvers generally leads to a code that does not conserve energy locally, and instead a special gravitational consistency condition ought to be satisfied. A particular example of combining softened Lagrangian hydrodynamics (SLH) with a P(3)M gravity solver is used to demonstrate the effect of the gravitational consistency condition for a self-gravitating hydrodynamic code. The need to supplement the SLH method with the P(3)M gravity solver arose because the moving mesh gravity solver, used in conjunction with the SLH method previously, was found to produce inaccurate results. We also show that most existing cosmological hydrodynamic codes implicitly satisfy the gravitational consistency condition.