Kinetic models provide the bridges between microscopic and macroscopic descriptions for physical systems; they have a wide range of applications, including astronautics, nuclear engineering, plasma physics, and semiconductor device modeling. These models often involve multiple time and spatial scales, which pose tremendous difficulties in numerical simulations. Moreover, since kinetic models arise from approximations, there are intrinsic uncertainties in the equations employed and the data (initial conditions and boundary values) used. This project aims to develop efficient numerical methods and to conduct analysis for multiscale and uncertain kinetic equations. The questions under study concern fundamental issues in scientific and engineering computation in the modern age -- multiscale modeling and simulation and uncertainty quantification. Some of the research results are expected to provide excellent additions for graduate courses in applied mathematics and scientific computing, thus contributing to training of the future generation of researchers in modern applied mathematics and scientific computing.
The project aims to develop and analyze numerical methods for multiscale kinetic equations with uncertainties. The work addresses the numerical challenges of multiple time and spatial scales as well as intrinsic model uncertainties in collision kernels, scattering coefficients, initial and boundary data, forcing and source terms, etc. The investigator plans to tackle these numerical challenges via several computational and analytical tools: asymptotic-preserving schemes to deal with multiple scales; polynomial chaos expansion and stochastic Galerkin (and other non-intrusive) methods for the random uncertainties; and hypocoercivity theory to study the regularity, stability, sensitivity, and long-time behavior of these methods. The hypocoercivity analysis provides new numerical analysis tools to study a wide class of physically important nonlinear partial differential equations in mathematical physics.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
