Numerical methods for hyperbolic problems with uncertainty

The ultimate objective of numerical simulations is to predict physical events or the behaviors of engineering systems. In the past few decades, extensive efforts have been devoted to the development of accurate numerical algorithms. The goal is to make simulation predictions reliable in the sense that numerical errors are well under control and understood. Although this remains an active research area, there have been growing interests in understanding the impacts of uncertainty in data. The uncertainty may happen for parameter values, initial and boundary conditions. The study of uncertainty quantification is to provide more reliable predictions for real-life problems.

In this project, we will investigate uncertainty quantification for problems governed by partial differential equations, in particular for nonlinear conservation laws that have many important applications in practice. The random effect and the fact that nonlinear conservation laws admit discontinuous solutions yield great challenges for designing numerical algorithms and for the relevant numerical analysis.