A parallel-in-time, multiscale interaction procedure is introduced for systems described at molecular scales by time-dependent random variables that obey Langevin dynamics. At larger, kinetic scales, the system is described by a probability distribution function that obeys an associated Fokker-Planck equation. It is assumed that the main quantity of interest is the kinetic-scale probability distribution function, and that molecular time scales are much smaller than those of interest at the kinetic scale. The overall multiscale interaction procedure is an iterative refinement predictor-corrector algorithm similar to the parareal method, but with different physical models used in each stage. In the predictor stage, a partial differential equation solver is used to obtain estimates of the probability distribution function on a subdivision of a time interval. At the starting time of each subinterval, the predicted probability distribution function is used to initialize an ensemble of random variables that are subsequently evolved forward in time by the Langevin dynamics. A probability distribution function estimate is constructed from the Langevin time evolution at the molecular scale and compared to that from the kinetic scale Fokker-Planck equation to determine whether further iterative refinement is needed. The molecular-scale computations are executed on highly parallel graphics processor units, while the kinetic-scale simulation is running on central processing units. The performance of the algorithm is tested on the simple dumbbell model of polymer flow.
Mitran, S. (2010). Time parallel kinetic-molecular interaction algorithm for CPU/GPU computers. In Procedia Computer Science (Vol. 1, pp. 745–752). Elsevier B.V. https://doi.org/10.1016/j.procs.2010.04.080