Modeling and Simulation of High-Dimensional Stochastic Multiscale PDE Systems

In order to capture the small-scale heterogeneity, two computational multiscale algorithms are developed. The first approach is a stochastic multiscale method. This new method ensures both local and global mass conservation. Starting from a specified covariance function, the stochastic log-permeability is discretized in the stochastic space by using a truncated Karhunen-Loeve expansion with several random variables. Because of the small correlation length of the covariance function, a high stochastic dimensionality often results. Therefore, a newly developed adaptive high-dimensional model representation technique (HDMR) is used in the stochastic space. HDMR decomposes the high-dimensional problem into a set of low-dimensional stochastic subproblems, which are efficiently solved by using an adaptive sparse grid collocation method. Numerical examples are presented to show the accuracy and efficiency of the developed stochastic multiscale method. The second approach is a hybrid method by integrating first-principles kinetic Monte Carlo (KMC) simulation with a continuum model. We demonstrate this integrated computational framework by applying it to study the effects of heat and mass transfer on the heterogeneous reaction kinetics. The hybrid reaction model consists of a surface phase domain where catalytic surface reactions occur and a gas-phase boundary layer domain imposed on the catalyst surface where the temperature and pressure gradients exist. The surface phase is described by using site-explicit first-principles KMC simulations. The heat and mass transfer in the gas-phase boundary layer domain, which is represented by thermal and molecular diffusion, is characterized by using the continuum-based method. Simulation results indicate that heat and mass transfer in the gas-phase boundary layer could dramatically affect the observed reaction kinetics at nominal operating reaction condition.