Spectral Schemes for Stochastic Partial Differential Equations for Surface Processes

Scientific problems coming from a wide variety of disciplines including materials science, polymer science, and fluid turbulence are commonly described as multiscale problems since small scale, molecular level behavior can greatly impact large scale, observable features of such systems. A common means of incorporating these small scale effects in mathematical model equations involving larger scale quantities is through the use of stochasticity or randomness to represent the small scale quantities. One increasingly important class of such random models are stochastic partial differential equations. Such equations have proved to be difficult to solve using mathematical analysis; thus computational methods will be valuable tools in the study of stochastic partial differential equations. The main goal of this research is to develop new accurate and efficient spectral numerical schemes to study stochastic partial differential equations. This new computational method will be applied to the class of stochastic partial differential equation models known as mesoscopic models for surface processes, with special emphasis on problems from engineering and biophysics. In particular, a better understanding of the effect on a reaction of the distribution of a catalyst on a surface can lead to improvements in the overall efficiency of catalytic surface reactors. The labeling of the positions and types of proteins on the surface of a biomaterial is important in determining the structure of subcellular components. The application of the spectral scheme to the mesoscopic models for these problems will make possible numerical studies of a much greater size and scope than is currently feasible using existing numerical techniques. Given the nature of these applications, the further growth and development of interdisciplinary collaborations is an important aspect of this research.