Acceleration of diffusive molecular dynamics simulations through mean field approximation and subcycling time integration

Diffusive Molecular Dynamics (DMD) is a class of recently developed computational models for the simulation of long-term diffusive mass transport at atomistic length scales. Compared to previous atomistic models, e.g., transition state theory based accelerated molecular dynamics, DMD allows the use of larger time-step sizes, but has a higher computational complexity at each time-step due to the need to solve a nonlinear optimization problem at every time-step. This paper presents two numerical methods to accelerate DMD based simulations. First, we show that when a many-body potential function, e.g., embedded atom method (EAM), is employed, the cost of DMD is dominated by the computation of the mean of the potential function and its derivatives, which are high-dimensional random variables. To reduce the cost, we explore both first- and second-order mean field approximations. Specifically, we show that the first-order approximation, which uses a point estimate to calculate the mean, can reduce the cost by two to three orders of magnitude, but may introduce relatively large error in the solution. We show that adding an approximate second-order correction term can significantly reduce the error without much increase in computational cost. Second, we show that DMD can be significantly accelerated through subcycling time integration, as the cost of integrating the empirical diffusion equation is much lower than that of the optimization solver. To assess the DMD model and the numerical approximation methods, we present two groups of numerical experiments that simulate the diffusion of hydrogen in palladium nanoparticles. In particular, we show that the computational framework is capable of capturing the propagation of an atomically sharp phase boundary over a time window of more than 30 seconds. The effects of the proposed numerical methods on solution accuracy and computation time are also assessed quantitatively.