A New Way to Conserve Total Energy for Eulerian Hydrodynamic Simulations with Self-Gravity

We propose a new method to conserve the total energy to round-off error in grid-based codes for hydrodynamic simulations with self-gravity. A formula for the energy flux due to the work done by the the self-gravitational force is given, so the change in total energy can be written in conservative form. Numerical experiments with the code Athena show that the total energy is indeed conserved with our new algorithm and the new algorithm is second order accurate. We have performed a set of tests that show the numerical errors in the traditional, non-conservative algorithm can affect the dynamics of the system. The new algorithm only requires one extra solution of the Poisson equation, as compared to the traditional algorithm which includes self-gravity as a source term. If the Poisson solver takes a negligible fraction of the total simulation time, such as when FFTs are used, the new algorithm is almost as efficient as the original method. This new algorithm is useful in Eulerian hydrodynamic simulations with self-gravity, especially when results are sensitive to small energy errors, as for radiation pressure dominated flow.

💡 Research Summary

The paper addresses a long‑standing problem in grid‑based (Eulerian) hydrodynamic simulations that include self‑gravity: the total energy of the system is not conserved when gravity is treated as a source term. In conventional schemes the Poisson equation ∇²φ = 4πG ρ is solved to obtain the gravitational potential φ, the acceleration g = −∇φ is computed, and the term ρ g·v is added to the energy equation as a non‑conservative source. Because this source is not expressed as a divergence, round‑off errors accumulate, leading to noticeable energy drift over long integrations. This drift can corrupt the dynamics, especially in radiation‑pressure‑dominated flows or in problems that require many dynamical times.

The authors propose a reformulation that casts the work done by gravity into a flux form, allowing the total energy equation to be written in fully conservative form: ∂ₜE + ∇·