Scaling relations between numerical simulations and physical systems
  they represent
  The dynamical equations describing the evolution of a physical system
generally have a freedom in the choice of units, where different choices
correspond to different physical systems that are described by the same
equations. Since there are three basic physical units, of mass, length and
time, there are up to three free parameters in such a rescaling of the units,
$N_f \leq 3$. In Newtonian hydrodynamics, e.g., there are indeed usually three
free parameters, $N_f = 3$. If, however, the dynamical equations contain a
universal dimensional constant, such as the speed of light in vacuum $c$ or the
gravitational constant $G$, then the requirement that its value remains the
same imposes a constraint on the rescaling, which reduces its number of free
parameters by one, to $N_f = 2$. This is the case, for example, in
magneto-hydrodynamics (MHD) or special relativistic hydrodynamics, where $c$
appears in the dynamical equations and forces the length and time units to
scale by the same factor, or in Newtonian gravity where the gravitational
constant $G$ appears in the equations. More generally, when there are $N_{udc}$
independent (in terms of their units) universal dimensional constants, then the
number of free parameters is $N_f = max(0,3-N_{udc})$. When both gravity and
relativity are included, there is only one free parameter ($N_f = 1$, as both
$G$ and $c$ appear in the equations so that $N_{udc} = 2$), and the units of
mass, length and time must all scale by the same factor. The explicit
rescalings for different types of systems are discussed and summarized here.
Such rescalings of the units also hold for discrete particles, e.g. in N-body
or particle in cell simulations. They are very useful when numerically
investigating a large parameter space or when attempting to fit particular
experimental results, by significantly reducing the required number of
simulations.
