In cosmological simulations, astrophysical gases are typically modeled as inviscid ideal gases that follow the Euler equations, which can be expressed mainly in three different ways: Lagrangian, Eulerian, or arbitrary Lagrange-Eulerian methods. Different methods give specific forms of hydrodynamical equations. When using the Lagrangian approach to specify the field, it is assumed that the observer tracks a specific fluid parcel with its unique characteristics during its movement through space and time. In contrast, the Eulerian approach emphasizes particular locations in space that the fluid passes through as time progresses.