A numerical scheme for saturated flows based on convex optimization

Department of Hydrology and Atmospheric Sciences
Weekly Colloquium

Thursday, October 24, 2019
4:00 pm in Harshbarger 206 ~ Refreshments at 3:45 pm hosted by HASSA

Mikhail (Misha) Stepanov
Department of Mathematics & Applied Mathematics GIDP, University of Arizona


The vertical movement of soil moisture is overdamped and energy driven. A numerical scheme is suggested in which the water content at the next time step is obtained as a solution of a certain optimization problem. The functional being minimized is the total energy (determines the general direction of the water content change) plus a term containing hydraulic conductivity (controls the magnitude of the change). The motivation for the scheme was to handle saturated flows, with the energy of oversaturated flow due to incompressibility of water being set to infinity. The scheme is unconditionally stable (energy is non-increasing) even with explicit in time treatment of hydraulic conductivities, in which case the arising optimization problem is convex. This is the joint work, still in progress, with Gregory Johnson (Applied Math), Xubin Zeng (HAS), and Bo Guo (HAS).


Dr. Mikhail (Misha) Stepanov is an Associate Professor of Mathematics/Applied Mathematics at the University of Arizona. He received his Ph.D. in physics/optics from the Institute of Automation and Electrometry, Novosibirsk, Russia, where his research focused on strong field effects in nonlinear plasma spectroscopy. At the University of Arizona, he currently teaches courses in the Principles and Methods of Applied Mathematics. He has also taught courses in Applied Stochastic Processes and Mathematical Analysis for Engineers. Recent publications include Analysis of spatial correlations in a model two-dimensional liquid through eigenvalues and eigenvectors of atomic-level stress matrices (Physical Review), Predicting failures in power grids:The case of static overloads (IEEE Transactions on Smart Grid), and Rain initiation time in turbulent warm clouds (Journal of Applied Meteorology and Climatology). The subject of today's talk is a work in progress with UA collaborators Gregory Johnson (Applied Math) and Hydrology and Atmospheric Sciences faculty members Xubin Zeng and Bo Guo.