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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.