Conservation of Gravitational Energy

The total energy of an astronomical system is of great interest since the evolution of the system depends on it. Hence we solve the hydrodynamical equations by taking account of the conservation in numerical simulations of astrophysical objects. However, gravitational energy is often taken into account as a source term, and the total energy including gravity is not guaranteed to be conserved. This is partly because it increases the computational cost to solve hydrodynamical equations in the fully conservative form, i.e., those without any source term. This paper shows that the total energy and momentum of a system are fully conserved if the source terms due to gravity are properly taken into account. The error in the total energy is reduced as small as the round-off error. The method is applicable both when the gravitational force is given by the Poisson equation or explicitly as a function. The former case is confirmed by numerical simulations of 2D fragmentation of a self-gravitating gas. This method is applicable not only to the Cartesian uniform grid but an octree grid often used in adaptive mesh refinement.