Control algorithm for multiscale flow simulations of water

We present a multiscale algorithm to couple atomistic water models with continuum incompressible flow simulations via a Schwarz domain decomposition approach. The coupling introduces an inhomogeneity in the description of the atomistic domain and prevents the use of periodic boundary conditions. The use of a mass conserving specular wall results in turn to spurious oscillations in the density profile of the atomistic description of water. These oscillations can be eliminated by using an external boundary force that effectively accounts for the virial component of the pressure. In this Rapid Communication, we extend a control algorithm, previously introduced for monatomic molecules, to the case of atomistic water and demonstrate the effectiveness of this approach. The proposed computational method is validated for the cases of equilibrium and Couette flow of water.

Figure 1

Schematic representation of the control algorithm for reducing density fluctuations. The controller uses the error $e_i$, which is the difference between the target density ${\rho }_i^t$ and the computed density ${\rho }_i^m$, to obtain a correction to the boundary force $\Delta F_i$, which is applied to the MD boundary to produce the correct density and virial component of the pressure; the procedure is iterated until the error is smaller than a prescribed tolerance.

Figure 2

(a) The external boundary force computed after applying dynamic control theory. (b) The corresponding reduced density $\left({\rho }^{+}=\rho /{\rho }_{\text{bulk}}\right)$ values, without the control (zero external boundary force, dotted line) and with the control (solid line). For these simulations, we used $k_p={\text{nm}}^{2.5}\text{kJ}/\text{amu}\text{mol}$ and both the force and density were sampled over 0.36 ns.

Figure 3

The probability distribution of the cosine of the angle $\varphi$ between the dipole of each water molecule with the normal to the $x$ direction at distances from the boundary: (a) 0.1 nm, (b) 0.2 nm, (c) 0.4 nm, and (d) 0.6 nm.

Figure 4

(Color online) Schematic representation of the hybrid simulation for the Couette flow, indicating the atomistic region treated with MD, the continuum region, and the motion of the boundary far from the atomistic region (with velocity $\pm v$).

Figure 5

Velocity profiles sampled over 1.8 ns in the $x$ direction in the case of Couette flow, with the hybrid method and the control algorithm applied to the atomistic region (fluctuating black line) compared to the continuum case (dashed line). In the hybrid approach, the region $-3\text{nm}\le x\le 3\text{nm}$ is treated atomistically and the regions $-6\text{nm}<x<-1\text{nm}$ and $1\text{nm}<x<6\text{nm}$ are treated by a continuum approach.