tokens: this may be the largest hurdle in adapting numerical magnetohydrodynamics software to use the proposed scheme. 6. Conclusion We have shown that self-gravity source terms, when properly evaluated, are capable of being fully conservative (e.g., Mikami et al. 2008 ; Springel 2010 ; Katz et al. 2016 ; Hanawa 2019 ). The source terms are derived by guaranteeing their equivalence to a corresponding flux (i.e., for momentum source terms, the divergence of the gravitational stress tensor, and for the energy source term, the divergence of a gravitational energy flux). As presented in this work, the fully conservative source-term scheme is formally second-order accurate in space and time, and does not increase the total number of Poisson solves needed per numerical time step compared to Athena++ v19.0 . The scheme can be implemented alongside a broad class of temporal integrators (e.g., VL2, RK2, RK3, etc.), Riemann solvers, or reconstruction methods (e.g., PLM, PPM, etc.). The three test problems presented in Section 4 exemplify the key advantages of the fully conservative source-term scheme. For the 1D Spitzer sheet advection test problems, we see that the the L 1 errors are nearly identical when the momentum equation is integrated via the divergence of the gravitational stress tensor T g and the conservative source terms. The similarities in the errors and error convergence between the two Spitzer sheet runs are a reflection of the two schemes being equivalent via Equations ( 23 ) and ( 24 ). The Jeans linear stable wave tests demonstrate that the scheme can be used in conjunction with a variety of temporal integrators (e.g., VL2, RK2, and RK3) and reconstruction methods (e.g., PLM and PPM). Although we use the HLLC Riemann solver for all test problems, any solver can be employed; the scheme only mandates that the energy source term uses the same mass fluxes applied in evolving the continuity equation (i.e., the energy source term must be consistent with mass conservation). The source terms in Section 2.3 only guarantee spatial second-order accuracy; however, the Jeans linear wave analysis shows that higher-order temporal integrators and reconstruction methods are still capable of significantly lowering errors at a given resolution. The Jeans instability test demonstrates the algorithm’s ability to conserve total energy. Despite significant changes in each component of the volume-integrated energy, total energy is conserved to round-off error. Total energy conservation (to round-off error) requires that the solution to the discretized Poisson equation (Equation ( 26 )) is of round-off error accuracy at the beginning and final stages of the numerical time step. As noted in Section 3.2 , the algorithm does not require machine-accurate evaluations of the potential at intermediate stages for total energy conservation. This could yield interesting strategies when solving the Poisson equation with iterative methods, e.g., perhaps the convergence
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