## Accuracy

## Energy Fluctuations versus Time Step Size

To study the accuracy of FEN ZI, we plot the
fluctuations in the total energy as a function of the time
step size in Figure 1. According to Allen and Tildesley
(*M. P. Allen and D. J. Tildesley, Computer Simulation of
Liquids. Oxford: Clarendon Press, 1987. *),
a plot of the energy fluctuations versus time step size
should follow an approximately logarithmic trend. In Figure 2 we
observe that, for large time step size (larger than 0.5 fs),
the fluctuations in total energy of MD
simulations are proportional to the time step size.

**Figure 1.** Plot of total energy fluctuations of DMPC 1x1 as a function
of time step size for different time step size - i.e., 0.125 fs, 0.25 fs, 0.5 fs, 1
fs, and 2 fs and single precision.

## FEN ZI vs. CHARMM

We use the code CHARMM as the reference code to validate the accuracy of FEN ZI.

The figure below shows the temperature (a function of kinetic energy), bond energies (i.e., bond-, angle,
Urey-Bradley-, improper-, and dihedral energies), and non-bond energies (i.e., VDW, electrostatic, and PME)
of a 3 ns DMPC simulation (17004 atoms, 14096 bonds, 19108 angles, and 22536 dihedral) for simulations in equilibrium
on CPU using the double-precision CHARMM code and on GPU using our single precision code.
The simulations are performed in constant temperature.

**Figure 1.** Comparison of the temperature and energy profiles for 3 ns of NVT MD simulation with 1fs step size for CHARMM on single core, 64 bits
and FEN ZI on GTX 480, 32 bits (*Click on the figure to enlarge it*).

We observe that the several quantities (i.e., temperature and energy) fluctuate, as expected, around the
same average value for both the double precision CPU simulation and single precision GPU simulation.
The FEN ZI initial energy (step 0) matches the CHARMM initial energy.
During the initial phase of the simulation, before the system reaches the equilibrium, we also observed
some energy drifting due to the different thermostats used
by the two codes (i.e., CHARMM uses Langevin thermostat
while FEN ZI uses velocity reassignment). Eventually the
two simulations converge, indicating that FEN ZI does in fact
produce results consistent with CHARMM and the use of
double precision is not needed on GPUs for this type of
simulations.