# Interdisciplinary Applied Mathematics

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The physical quantities considered here are the average kinetic temperature

=swb (f>?}’    <i639)

and the normal component of the pressure tensor defined by

N

Pxx = p^2 ViX,    (16.40)

i=1

T, шах(ри — Pjj)

t    t

FIGURE 16.11. Time evolution of average kinetic temperature (approaching unity) and normal component of pressure (approaching zero) for fluid in equilibrium. Left: At = 0.01; Right: At = 0.05.

where a constant mass for all particles is assumed. When the system reaches equilibrium, the average kinetic temperature reaches unit value and the pressure approaches zero.

In Figure 16.11, we plot both quantities as a function of time for two different time steps. We use three different methods for integration, namely, the Euler, the    Verlet,    and the Lowe    methods.    We    see    that    at    At    = 0.01    all

methods converge to the correct value after some initial transients, whereas for At = 0.05 only Lowe’s method gives the correct value of temperature. In all methods, however, a stationary state is achieved, which implies that the simulated thermostat is effective in all cases.

In general, the splitting method gives similar results to those of Lowe’s method. (Nikunen et al., 2003) recommend the use of the splitting method and Lowe’s method, since they are more accurate and more efficient than several other methods that they tested, including the DPD-VV scheme. Their tests include problems with zero conservative forces as well as polymer flows where hard Lennard-Jones potentials combined with proper springs are employed to represent the polymer; the solvent is represented by soft potentials. Lowe’s method, in particular, which does not deal with the computations of dissipative or random forces, seems to be the most efficient. It is based on normally distributed random numbers, while other integration schemes employ random numbers, from a uniform distribution. Another important consideration is that with Lowe’s approach, one can achieve realistic values of the Schmidt number Sc ж 1000 for liquids, whereas with the other approaches the Schmidt number is of order one (Sc ж 1), which is clearly incorrect, since this corresponds to gases.

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