a continuous momentum injection $\Delta P=P_{\star}\Delta M$ where $P_{\star}$
is a proportionality constant which depends on the stellar mass $\sim
M_{\star}^{1/2}$). If the momentum injection occurs in a cell with a too low
density, they take 10% of $\Delta M$ out of the sink particle and redistribute
it evenly between the injection cells in order to avoid too large outflow
speeds (and too small integration timestep). Along the same line, Cunningham
et al. (2011); Hansen et al. (2012) extended the work of Krumholz et al.
(2005, 2007) and presented for the first time a combined sub-grid model which
includes protostellar outflows (Matzner and McKee, 1999) and protostellar
radiative heating (Offner et al., 2009), ignoring magnetic fields though.
Their sub-grid model accounts for pre-main sequence evolution as in Offner et
al. (2009) and the calculated protostellar radius is used in the outflow and
radiative heating models. They parameterized their outflow model with
dimensionless parameters which, for instance, sets the fraction of mass
accreted by the protostar or launched in the outflow, and the ejection
velocity as a function of the Kepler speed at the surface. Contrary to the
case of radiative heating where energy is put within the accretion volume,
they inject outflows at a distance comprised between $4\Delta x$ and $8\Delta
x$ from the sink particle, and used angular dependency from Matzner and McKee
(1999). We note that they not only inject momentum, but also mass and thermal
energy. They set the wind temperature to $10^{4}$ K, which is appropriate for
an ionised wind. Using the same tool, Myers et al. (2014) added magnetic field
in their simulations with ORION, but because of the low resolution they used
($>20$ AU), they where not able to launch self-consistent outflows and used
the sub-grid model we just mentioned. The outflows they produce may form
strong shocks at very high temperature, higher than the dust sublimation
temperature so that the dust opacity drops. Their radiative transfer scheme,
primarily designed for dust thermal emission does not allow the gas to cool
efficiently. They thus needed to add a line cooling function for the thermal
budget of the gas.

Federrath et al. (2014) implemented a sub-grid-scale outflow model upon the
sink particles algorithm they implemented in FLASH (Federrath et al., 2010)
for MHD collapse. Their model combines different features of the Li and
Nakamura (2006); Nakamura and Li (2007) and Cunningham et al. (2011)
implementations. They used a normalised velocity profile for momentum
injection, which reproduces the two components outflow/jet with opening angles
of $30^{\circ}$ and $5^{\circ}$ respectively. They also improved the algorithm
for the outflow orientation by recording the angular momentum transfer from
the accreted gas to the sink particle. Their algorithm also conserves mass and
momentum exactly in the sink accretion step. Murray et al. (2018) proposed a
similar implementation in RAMSES. They follow previous works (Cunningham et
al., 2011; Federrath et al., 2014; Offner et al., 2009; Myers et al., 2014)
for the outflow injection as well as the protostellar evolution. Outflows are
there injected within a conical volume about the spin axis of the sink
particle in a radial extent comprised between 4 and 8 cells (at the highest
refinement level) away from the sink particle. Federrath et al. (2014)
performed a parameter study on the number of cells for the radial outflow
direction and found that convergence on the maximum outflow velocity requires
16 cells per outflow radius. Convergence on the mass, linear and angular
momentum of the outflow is achieved though with 8 cells.

All these protostellar outflow sub-grid models are currently widely used in
the community for astrophysical applications on the star formation
rate/efficiency (e.g. Federrath, 2015; Kuiper et al., 2016; Offner and Chaban,
2017; Murray et al., 2018), stellar initial mass function (Krumholz et al.,
2012), origin of turbulence driving (Nakamura and Li, 2011; Offner and Liu,
2018), as well as cluster and massive star formation (Wang et al., 2010;
Cunningham et al., 2011; Kuiper et al., 2015; Li et al., 2018).

In the context of massive star formation, another class of outflows models are
based on the luminosity and ionisation. Sub-grids models of outflows generated
by massive star luminosity have been developed (e.g. Krumholz et al., 2005;
Peters et al., 2014), but the physical mechanisms driving this type of
outflows are not yet well understood. No work in the context of massive star
formation has accounted for the combined effects of magnetic fields and
radiative force with a resolution sufficient to resolve the magnetic outflow
launch. There is certainly a combination of magneto-centrifugal process and
from radiative pressure, depending on the mass of the forming protostar.

###  8.3 Second collapse and pre-main sequences

Ideally, the sub-grid models should be designed to reproduce the results of
high resolution studies, which resolve the protostar scales. The short time
integration after stellar core formation, dominated by the adiabatic
evolution, does not allow to design sub-grid models for protostellar feedback
from full 3D RMHD calculations with a high level of confidence. For these
reason, all recent works mostly rely on the results of 1D spherical symmetry
results to set their protostellar core evolution models. For instance, Kuiper
and Yorke (2013) use the pre-main sequence evolution code STELLAR (Bodenheimer
et al., 2007; Yorke and Bodenheimer, 2008) to determine the protostellar
luminosity of the non-resolved protostars. Pelupessy et al. (2013) developed
the open source Astrophysical Multi-purpose Software Environment (AMUSE), a
component library of simulations involving different physical domains and
scales. For instance, it enables one to couple hydrodynamical codes (GADGET,
ATHENA, AMRVAC) with pre-main sequence and binary evolution codes such as MESA
(Paxton et al., 2011). Recently, Wall et al. (2019) used AMUSE to perform a
star cluster formation model that includes individual star formation from
self-gravitating, magnetized gas, coupled to collisional stellar dynamics. It
couples different tools: the AMR code FLASH (Fryxell et al., 2000), the N-body
code ph4 (McMillan et al., 2012), and the stellar evolution code SeBa
(Portegies Zwart and Verbunt, 1996). We anticipate to see more works using
this strategy of coupling tools and scales in the near future.

Figure 3: Volume rendering from collapsing 1 ${\rm M}_{\odot}$ dense cores at
the protostellar core scale. The left panel shows the results of an ideal MHD
3D simulations, and the right panel the comparison with non-ideal MHD
including ambipolar diffusion and Ohmic diffusion. The color lines indicate
the 3D magnetic field lines, and the color coding the magnetic field
amplitude. Figure reproduced from Vaytet et al. (2018) with permission of A&A.

Recent work from Vaytet et al. (2018) have managed to resolve the second core
formation with non-ideal MHD (Ohm diffusion and ambiploar diffusion) from
collapse 1 ${\rm M}_{\odot}$ dense core (see Figure 3). Interestingly, they
show that the magnetic field lines topology is very different at the first and
second core scales. Ambipolar diffusion is dominating at the first core scale.
The magnetic fields direction is essentially vertical and the magnetic fields
and the gas evolution are decoupled. At the second core scale, the ionisation
rises due to high temperature and the system is back to ideal MHD, but with a
high plasma beta $\beta=P_{\mathrm{therm}}/P_{\mathrm{mag}}$. The weak
magnetic field is efficiently wrapped up by the gas rotation and the resulting
fields direction is essentially toroidal. On the opposite, when ideal MHD is
preserved all the way from the dense core to the stellar core, the magnetic
fields lines are strongly pinched and the field remains poloidal at the second
core scale. The energy balance at the first and second core accretion shocks
is also different. While the accretion shock at the first core has been
classified as a radiative supercritical shock, with all the incident kinetic
energy radiated away (or often referred to as cold accretion), in 1D spherical
core collapse studies (Commerçon et al., 2011a; Vaytet et al., 2012)), recent
3D works have shown that the first core accretion experiences both cold and
hot accretion at the same time at its surface, depending on the accretion
flows morphology. On the surface of the stellar core, the accretion shock has
been found to be subcritical (Vaytet et al., 2012; Tomida et al., 2013; Vaytet
et al., 2018) with a significant part of the incident kinetic energy
transferred to the protostars. Although these results have been investigated
only for the very first stages of the protostar evolution (only one month in
Vaytet et al. (2018)), they indicate that it is not easy to derive general
properties of small scales phenomena to design robust sub-grid models. In
particular, Wurster et al. (2018) show that the magnetic fields evolution of
the protostars they formed in their SPH models is largely affected by
numerical diffusion. Last, the radiative efficiency of the stellar accretion
shock has been shown to be a key process for the pre-main sequence evolution
and sets the radius of the young protostars (Baraffe et al., 2012). We recall
that the sub-grids models designed for the radiative and dynamical feedback
take the protostellar radius as an input. The launching speed of the
protostellar jets as well as the accretion luminosity depends on
$R_{\star}^{-1}$.

##  9 Discussion

We have described almost all the numerical methods that are required to model
the star formation process within the turbulent interstellar medium. These
discretised equations must now be solved for a given set of initial and
boundary conditions, over some prescribed time. This defines the numerical set
up and the overall simulation strategy. Obviously, given the wide range of
spatial and temporal scales involved, one cannot realistically model an entire
galaxy all the way down to the formation of brown dwarves and proto-planetary
disks. Unfortunately, given the complexity of the star formation process, and
the fact that large and small scales are strongly coupled, this is required by
the physics of the problem. Indeed, supersonic turbulence is seeded on large
scales, where kinetic energy is injected through shearing or colliding flows
induced by galactic rotation, spiral waves and extra-galactic accretion flows.
Moreover, turbulence and gas ejection can be triggered by stellar feedback
processes occurring on very small scales, such as radiation driven flows, jets
and ultimately supernovae explosions.

Figure 4: Simulation of the Milky Way: column density of the gas disc in a
sub-parsec resolution simulation. The colour table only applies to the main
panel: the table has been changed in each zoom-in view to enhance contrast.
Figure adapted from Renaud et al. (2013) with permission from the authors.

Modelling star formation is therefore fundamentally a multi-scale, multi-
physics problem, and a very difficult one. Past or present day simulations
always rely on some compromises. We can decompose them into several
categories, corresponding roughly to the adopted box size and initial
conditions. First, full galaxy simulations, where spiral waves and spiral
shocks are used as the seed for turbulence. The box size must be of several
tens of kiloparsecs to contain the entire rotating disk and possibly the
galactic corona. The gas is compressed through the spiral shocks or various
colliding flows. Thermal and gravitational instabilities fragment the gas into
small molecular clouds. The spatial resolution never really drops below 0.1 or
1 pc in these simulations, so that stars cannot be modelled individually, at
least for massive galaxies like the Milky Way (see Fig. 4 for illustration).
Simulators rely on simplified, sub-grid star formation recipes based on
stochastically spawning star cluster particles of the same mass, or on the
sink particle technique where in this case sink particles represent a star
cluster, rather than a single star. Bournaud et al. (2010) studied the
properties the ISM substructure and turbulence in galaxy simulations with
resolutions up to 0.8 pc and $5\times 10^{3}$ ${\rm M}_{\odot}$ with RAMSES.
(Renaud et al., 2013; Kraljic et al., 2014) were able to capture the
transition from turbulence-supported to self-gravitating gas with resolution
up to 0.05 pc in simulations of Milky Way-like galaxies using RAMSES. Hopkins
et al. (2012) presented SPH simulations dwarfs galaxies and Milky Way (MW)
analogues to massive star-forming galactic discs with pc-scale resolution.
Later, Hopkins et al. (2014) wrapped their implementation of feedback in the
FIRE (Feedback In Realistic Environments) simulations suite, aiming, among
others objectives, at resolving the formation of giant molecular clouds and
the multi-phase interstellar medium (ISM). More recently, Hopkins et al.
(2018) updated the FIRE implementation in the GIZMO code and performed sub-
parsec resolution simulations of galactic discs. We also refer readers to the
work of Dobbs et al. (2018) using SPH, Tasker (2011) using ENZO and Smith et
al. (2018) using AREPO. In addition, we note that for dwarf galaxies, it
becomes possible to model much smaller scales so that at least massive stars
can be modelled individually: Hu et al. (2016); Hu (2019) using SPH and a
resolution of $1-5$ ${\rm M}_{\odot}$ by gas particle, Emerick et al. (2018,
2019) using ENZO with a maximum resolution of 1.8 pc.

The second approach relies on simulating only a small portion of vertically
stratified galactic disks. The box size is usually around 1 kilo-parsec, with
periodic boundary conditions in the direction of the disk plane and outflow
conditions in the direction perpendicular to the disk plane. The geometry is
clearly heavily constrained by this elongated, vertical and stratified layer
of gas but it captures most of the phenomenon at work locally in the disk.
Using the shearing box technique, one can also add more realistic shearing
conditions, so that turbulence can be maintained both by stellar feedback and
a large scale galactic shearing flow (Kim et al., 2002). In this case, the
resolution can drop significantly below 0.1 parsec, may be 0.01 parsec or even
less for the highest resolution simulations. Unfortunately, it is still
impossible to model individual stars at this resolution so that here again
most papers are based on the sink particle techniques for representing star
clusters. It is however almost possible to resolve massive molecular cores
(and their associated sink particles) individually, so that individual
supernovae and HII regions can be resolved. These simulations are probably the
most realistic models of the interstellar medium of the Galaxy, although still
far from resolving the entire stellar population. In standard setups, the
models account for the thermal instability and feedback (mechanical by SN and
radiative by massive stars). First grid-based models were presented in Korpi
et al. (1999) and included magnetic fields, SN heating and radiative cooling.
3D MHD AMR models of a kpc-scale ISM driven by SNe were presented in de
Avillez and Breitschwerdt (2005) and Joung and Mac Low (2006). Currently,
numerous studies and projects are based on a similar setup, with increasing
physics put on over several years (SNe, radiative feedback, cosmic-rays,
shear, etc…): the series of papers by Hennebelle and Iffrig (2014); Iffrig and
Hennebelle (2017); Colling et al. (2018) as well as the work of Martizzi et
al. (2016) and Butler et al. (2015, 2017) using RAMSES, by Kim et al. (2011,
2013); Kim and Ostriker (2015, 2017, 2018) using ATHENA, the SILCC project
papers using FLASH (Walch et al., 2015; Girichidis et al., 2016; Gatto et al.,
2017; Peters et al., 2017; Girichidis et al., 2018).

The third approach aims at simulating individual molecular cores. The box size
ranges from 100 parsec down to 0.1 parsec for the highest resolution
simulations. The mass typically ranges from 50 to a few $10^{3}$ ${\rm
M}_{\odot}$. Boundary conditions are either periodic or isolated. In the
latter case, the cloud as a whole can collapse, while the former set of
simulations represents a more uniform background with collapsing regions only
at very small scales. The smallest box sizes represents internal regions of
larger clouds and are the only simulations for which the entire star
population, down to the brown dwarf limit, can be modelled. These simulations
are directly tackling the origin of the stellar IMF, and the formation of
proto-planetary disks. We mention in the following a non-exhaustive list of
the work done in this intense field of research. The series of papers by Bate
& collaborators have investigated the collapse of molecular clouds of mass
ranging from 50 to 500 ${\rm M}_{\odot}$ with increasing physics and numerical
resolution since their pioneer work (Bate et al., 2003) using SPH. In
particular, they have investigated the effect of radiative transfer (Bate,
2009, 2012) and magnetic fields (Price and Bate, 2008, 2009). The work of Dale
et al. (2007); Dale and Bonnell (2011); Dale et al. (2012) have also used SPH
simulations to study the effect of ionizing radiation from the forming massive
stars. AMR codes have also been widely used in this context, accounting for a
lot of physics and initial and boundary conditions. We note the work of
Girichidis et al. (2011, 2012a, 2012b) using FLASH who studied the importance
of isolated initial conditions in isothermal cloud core collapse (without
stellar feedback). Federrath et al. (2014) also used FLASH to perform
simulations of isolated cores with protostellar feedback (jets). Krumholz et
al. (2007, 2012); Hansen et al. (2012); Myers et al. (2013, 2014); Cunningham
et al. (2011); Li et al. (2018); Offner and Chaban (2017) presented 3D
collapse models using ORION with radiative transfer and/or ideal MHD, as well
as protostellar feedback (luminosity, outflows). RAMSES has also been
extensively used in this approach: Hennebelle et al. (2011); Commerçon et al.
(2011b) for 100 ${\rm M}_{\odot}$ isolated core collapse models with (R)MHD,
Lee and Hennebelle (2018b, a, c) focused on the peak of the IMF using (M)HD
1000 ${\rm M}_{\odot}$ isolated core collapse models, Geen et al. (2015,
2016); Gavagnin et al. (2017) studied the effect of ionizing radiation using
the M1 moment method for radiative transfer. Last, we note the work of Wang et
al. (2010) using ENZO with isolated boundary conditions, ideal MHD and
protostellar jets. This non-exhaustive list of works done using this approach
demonstrates the importance and the utility of such models. These simulations,
however, rarely follow the formation of the second Larson core. They still
rely on the sink particle technique, and on sub-grid models as well, when it
comes to modeling proto-stellar jet launching, energy budget at accretion
shocks and their associated radiation.

The last category of simulations deals with isolated collapsing low-mass dense
cores, with the goal of following the formation of protostellar disks and
ultimately the entire second collapse without relying on any sub-grid model.
This ambitious strategy has already been discussed at length in the previous
sections, so we will not describe it here again. We also refer the reader to
the review by Wurster and Li (2018) on the formation of protostellar disks.

Figure 5: Simulation of protostellar disk formation within molecular clouds.
The column density maps show the entire molecular (upper left panel) and
successive zoom within a star-forming dense core. Figure adapted from Zhang et
al. (2018) with permission from the authors.

An interesting intermediate strategy has been developed recently with the
zoom-in technique. This method is used now routinely in galaxy formation
simulations, for which only a single dark matter halo can be re-simulated at
much higher resolution, keeping the entire cosmological environment alive but
at a coarser resolution. Here, the idea is to model the turbulent molecular
cloud on large scales, say with a box of size 100 parsecs, to identify a
single collapsing core, and to zoom-in on the core with much higher resolution
and follow the first and possibly the second collapse and the formation of the
proto-planetary disk. For instance, Kuffmeier et al. (2017, 2018) presented
RAMSES AMR zoom-in calculations from an outer scale of 40 pc down to cell
sizes of 2 au to study the effect of the environment on the formation and the
evolution of protoplanetary disks. Zhang et al. (2018) performed high
resolution calculations of the formation and evolution of a star-forming core,
obtained by running larger scale calculations of molecular cloud formation
(Zamora-Avilés et al., 2018). They used FLASH with a maximum resolution of 25
AU and were able to cover scales from 256 pc to 25 AU (see Fig. 5). Mocz et
al. (2017) performed moving mesh AREPO calculations from a box size of about 5
pc down to a maximum resolution of $\simeq 4$ AU ($\simeq 5\times 10^{-5}$ pc)
to study the magnetic fields morphology from large to dense core scales. At
larger scale, Padoan et al. (2017) performed a simulation of supernova-driven
turbulence using RAMSES, with a box size of 250 pc and a maximum resolution of
$0.0076$ pc. At kpc scales, Hennebelle (2018) (FRIGG project with RAMSES,
simulation of the formation of self-gravitating cores) and Seifried et al.
(2017) (SLICC-Zoom project with FLASH, simulations of the formation of
molecular clouds) presented zoom calculations of stratified Galactic disks
down to resolution of $10^{-2}-10^{-3}$ pc. Thanks to the steady increase in
CPU power, we expect to see more and more work using the zoom-in technique in
the coming years.

These different strategies are all in different ways very ambitious and
address different problems in the theory of star formation. In addition, a
compromise needs to be found between physical realism and computational
efficiency. On one hand, we need to include magnetic fields, radiation fields
and complex chemistry, but on the other hand, we need many small resolution
elements and many time steps. Moreover, in the context of high performance
computing, modern supercomputers are very hard to use at full efficiency when
complex grid geometries is used in conjunction with expensive and demanding
algorithms. There is a clear tendency for simulation projects deployed on the
largest supercomputers in the world to use simple grid geometry, like
Cartesian meshes and periodic boxes, with a highly simplified physical model.
These large scale simulations are however very interesting to explore
statistical aspects and large inertial range for turbulent flows. For example,
Federrath (2013) performed $4096^{3}$ hydrodynamic isothermal turbulence
simulations using FLASH with periodic boundary condition for more than 40 000
time-steps on 32 768 CPU cores. In the framework of the magnetorotational
turbulence, Fromang (2010); Ryan et al. (2017) performed high resolution
simulations of isothermal shearing boxes using, respectively, ZEUS and a GPU
version of RAMSES.

In conclusion, all simulations of star formation published in the literature,
many of which have been discussed here, explore various corners of the
numerical parameter space: resolution versus box size, statistics versus
internal structure, physical realism versus computational speed. It is
interesting that at the smallest scales of interest here, a “first principle”
approach is in principle possible and pursued by several groups. At larger
scales, however, star formation simulations share the same kind of limitations
as galaxy formation simulations, or climate models and weather forecasting
simulations, namely a strong dependence of the results on small, unresolved
scales. Although bigger computers with more efficient, higher order codes and
more realistic models will certainly help shed light of the mysteries of star
formation, a robust methodology to implement sub-grid models and couple them
properly to the fluid equations still needs to be invented in our field.
Various attempts have been proposed in the context of unresolved turbulence
and star formation from analytical works (Krumholz and McKee, 2005; Hennebelle
and Chabrier, 2011; Padoan and Nordlund, 2011; Federrath and Klessen, 2012)
but the methodology, in the context of the full spectrum of required physical
processes, is still at its infancy.

## Acknowledgments

We would like to warmly thank our two referees, Richard Klein and Christoph
Federrath, for their help in reviewing the manuscript. We thank Matthew Bate,
Florent Renaud and Shangjia Zhang for providing adapted figures from their
work.

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