Figure 1

(Color online) Schematic representation of the collision operation for MPC-$\mathrm{SR}\pm a$ and MPC-DR in two-dimensional (2D) space in the co-moving reference frame (with $\sum {\mathbf{v}}_i=0$) at $N_{\mathrm{c}}=3$ and $\theta =\pi ∕2$. Circles represent the positions of particles (○) and the center of mass (●). “old” indicates the velocities before the collision, “±a” and “DR” represent the velocities after the collision for MPC-$\mathrm{SR}\pm a$ and MPC-DR, respectively, and “$+a$-vs” indicates the velocities after the “$+a$” collision without velocity rescaling.Reuse & Permissions

Figure 2

(Color online) Radial distribution function $g\left(r\right)$ of MPC-$\mathrm{SR}+a$ (with $n=1$, $\Delta t^{*}=1$ and $n=10$, $\Delta t^{*}=0.1$) and MPC-$\mathrm{AT}+a$ (with $n=1$, $\Delta t^{*}=0.1$) in two dimensions. The inset shows the $n$ dependence of $g\left(r\right)$ of MPC-$\mathrm{SR}+a$ at $\Delta t^{*}=0.1$. Error bars are shown at several data points.Reuse & Permissions

Figure 3

(Color online) Dependence of the viscosity in MPC-$\mathrm{AT}\pm a$ on $\Delta t^{*}$ in two-or three-dimensional space for (a) $n=5$ and (b) $n=1$. Symbols represent the numerical data of MPC-$\mathrm{AT}+a$ in two dimensions (○, △) or three dimensions (●, ▼) and MPC-$\mathrm{AT}-a$ in two dimensions (◻, ◇) or three dimensions (×, +), respectively. Solid and dashed lines represent analytical results for MPC-$\mathrm{AT}+a$ and MPC-$\mathrm{AT}-a$, respectively. Error bars are smaller than the size of symbols.Reuse & Permissions

Figure 4

(Color online) Dependence in MPC-$\mathrm{AT}\pm a$ of the viscosities on the particle number per cell, $n$. (a),(c) ${\eta }_{\mathrm{kin}}$ and (b) ${\eta }_{\mathrm{col}}$. Symbols represent the numerical data of MPC-$\mathrm{AT}+a$ (○, △) and MPC-$\mathrm{AT}-a$ (◻, ◇) for $\Delta t^{*}=0.1$ and $\Delta t^{*}=1$ in two dimensions, respectively, and the numerical data of MPC-$\mathrm{AT}+a$ at $\Delta t^{*}=1$ (×) in two dimensions. In (a),(b), the viscosity is rescaled by $\Delta t^{*}$ and $1∕\Delta t^{*}$, respectively, in order to facilitate a presentation of data for different $\Delta t^{*}$ on the same scale. Solid and dashed lines in (a) and (b) represent analytical results for MPC-$\mathrm{AT}+a$ and MPC-$\mathrm{AT}-a$, respectively. Solid and dashed lines in (c) represent analytical results with or without the correction term $h_{\mathrm{m}}$, respectively. Error bars are smaller than the size of symbols.Reuse & Permissions

Figure 5

(Color online) Viscosity of MPC-$\mathrm{LD}\pm a$ as a function of (a) ${\gamma }^{*}$ and (b),(c) $n$ in three-dimensional space at (a) $n=3$ and $\Delta t^{*}=0.1$ and (b),(c) ${\gamma }^{*}=1$. Symbols represent the numerical data of MPC-$\mathrm{LD}+a$ (○, △) and MPC-$\mathrm{LD}-a$ (◻, ◇). Dashed and solid lines represent analytical results for (a) ${\eta }_{\mathrm{kin}}$ or ${\eta }_{\mathrm{col}}$, and (b),(c) $\Delta t^{*}=0.1$ or 1, respectively. Error bars are smaller than the size of symbols.Reuse & Permissions

Figure 6

(Color online) Viscosity $\eta$ in two-dimensional space as a function of $\Delta t^{*}$ for MPC-$\mathrm{SR}\pm a$ with $\theta =\pi ∕2$, MPC-$\mathrm{RA}\pm a$ with ${\theta }_0=\pi$, and MPC-DR with $n=1$ or $n=5$. Solid and dashed lines represent analytical results for $n=5$ and $n=1$, respectively. (a) Symbols represent the numerical data of MPC-$\mathrm{RA}+a$ (▲, △,+), MPC-DR (●,○,◁), MPC-$\mathrm{RA}-a$ (◇,×), MPC-$\mathrm{SR}-a$ (◻), and MPC-$\mathrm{SR}+a$ (◼). Error bars are smaller than the size of symbols.Reuse & Permissions

Figure 7

(Color online) Viscosity difference $\Delta {\eta }_{\mathrm{col}}={\overline{\eta }}_{\mathrm{col}}-{\check{\eta }}_{\mathrm{col}}$ of $\mathrm{MPC}\text{−}\mathrm{AT}-a$ in two and three dimensions. Symbols with dashed or solid lines represent the numerical data in two dimensions (○,◻) and three dimensions (◇, △) at $n=1$ or $n=5$, respectively. The inset shows the dependence of $\Delta {\eta }_{\mathrm{col}}$ on the average particle number $n$ per cell, for $\Delta t^{*}=0.1$ (◻) and $\Delta t^{*}=1$ (○).Reuse & Permissions

Figure 8

(Color online) Dependence of the diffusion constant $D$ of $\mathrm{MPC}\text{−}\mathrm{AT}\pm a$ on $\Delta t$ at $n=5$ in three dimensions. Symbols and lines represent numerical and analytical data, respectively. Error bars are smaller than the size of symbols.Reuse & Permissions

Figure 9

(Color online) Dependence of the viscosity $\eta$ on $\Delta t^{*}$ of $\mathrm{DPD}\pm a$ and $\mathrm{DPD}\text{−}\mathrm{MT}\pm a$ in three-dimensional space for $n{r}_{\mathrm{cut}}^3=3$ and $\gamma {\tau }_0∕m=9$. Symbols represent the numerical data of $\mathrm{DPD}+a$ (○), $\mathrm{DPD}-a$ (×,◻), $\mathrm{DPD}\text{−}\mathrm{MT}+a$ (△), and $\mathrm{DPD}-a$ (◇). Dashed and solid lines represent analytical results for $\mathrm{DPD}-a$ and other DPD methods, respectively. Error bars are smaller than the size of symbols.Reuse & Permissions