Appendix C: Adiabatic check: Evrard's test

Figure C1: The evolution of total (solid line), kinetic (doted), thermal (dash-doted) and gravitational (dashed) energies for an adiabatic collapse of 4096 particles distributed by Monte Carlo according to a 1/r-density profile

Figure C2: A zoom in the total energy displayed in Fig. C1 revealing the conservation error in the critical phase of the adiabatic shock

One possible three-dimensional SPH energy-conservation test is the spherical collapse of an initially-cold adiabatic-gas, with a radial profile $\rho\propto 1/r$([Evrard 1988]; HK89; [Nelson & Papaloizou 1994]; [Steinmetz M. & Müller 1993]), called Evrard's test.

The initial configuration was obtained by Monte Carlo by distributing 4096 particles spherically, according to a 1/r radial-profile. As in HK89, the total mass, M, the cutoff-radius, R, and the gravitational constant G are equal to unity, and the initial specific thermal-energy, u=0.05, was homogeneously distributed in the system. The adopted number of time-bins was 4, starting from a root time-step, $\Delta t=0.001953125$. The softening length, $\epsilon=0.0928$, was estimated from Eq. (26), where both $E_{\rm G}$ and $\epsilon\simeq 0.1$ are iteratively calculated during the initializations. The aperture parameter $\theta$ was set as the two-bit floating $\theta=0.25$.

Thermal energy was explicitly integrated keeping fixed the entropy:

$$T\frac{{\rm d}s}{{\rm d}t}=\frac{{\rm d}u}{{\rm d}t}+\frac{P}{\rho} {\vec\nabla}\cdot{\vec v}=0$$ (C1)

so that both spatial and time inaccuracies promote an artificial heating (cooling).

Some of the results are shown in Fig. C1 in terms of the evolution of the integral energies: total, E, thermal, U, kinetic, K, and potential, $E_{\rm G}$. The graph in Fig. C1 does agree to other works in literature (e.g., HK89; SM93). The total energy was conserved within $\sim 2\%$. The behavior of the total-energy error is illustrated in Fig. C2, showing that a numerical cooling roughly mimics the potential energy rate. Both total linear and angular momenta were better conserved than one part in 10⁵.