Building a Cosmological Hydrodynamic Code: Consistency Condition, Moving Mesh Gravity and SLH-PM

Building a self-gravitating hydrodynamic code as a combination of a hydrodynamic solver and a gravity solver is discussed. We show that straightforward combining those two solvers generally leads to a code that does not conserve energy locally, and instead a special Consistency Condition ought to be satisfied. A particular example of combining Softened Lagrangian Hydrodynamics (SLH) with a PM gravity solver is used to demonstrate the effect of the Consistency Condition for a self-gravitating hydrodynamic code. The need to supplement the SLH method with the PM gravity solver arose because the Moving Mesh Gravity solver, used in conjunction with the SLH method previously, was found to produce inaccurated results. We also show that most existing cosmological hydrodynamic codes implicitly satisfy the Consistency Condition. Subject headings: cosmology: large-scale structure cosmology: theory dark matter hydrodynamics numerical methods