Applied Numerical Mathematics, (Special Issue in Honor of Viktor Ryaben'kii 90th Birthday), 2015 PDF and online
Current Projects and New Results
The Method of Difference Potentials and Models for Material Interfaces
The Difference Potentials Method (DPM) is a general framework for approximating the solution to boundary value problems.
DPM is applied here to approximate the solution to an interface problem for a second-order elliptic equation in 2D. It is a numerical
challenge to accurately resolve the solution near the interface, especially as it becomes more elongated like in the results above.
For details about recent extensions of DPM to parabolic (time-dependent) interface models see our recently accepted paper online
Shallow Water Wave Modeling and Central-Upwind Schemes
Tsunamis are triggered by seismic events on the ocean bottom appear as small perturbations on the surface of the ocean.
The figures above highlight the performance of the central-upwind scheme on a simple benchmark test for the 2D
(Saint-Venant) shallow water equations. The left-moving wave interacts with an exposed cylindrically-shaped island.
This project is joint work with Yekaterina Epshteyn and Alexander Kurganov.
Another test showing the water surface as a right-moving wave interacts with submerged bottom topography (not shown).