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

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