| Office Location: | 217 Physics |
| Office Phone: | (919) 660-2839 |
| Email Address: | |
Teaching (Fall 2009):
Much of my work has to do with incompressible fluid flow, especially qualitative properties of solutions and behavior of numerical methods, using analytical tools of partial differential equations. Recently I have developed a general method for the numerical computation of singular or nearly singular integrals, such as layer potentials on a curve or surface, evaluated at a point on the curve or surface or nearby (partly with M.-C. Lai). After regularizing the integrand, a standard quadrature is used on overlapping coordinate grids, and analytical corrections are added. In work with J. Strain we have applied this approach to Stokes flow (viscosity-dominated fluid flow) with a moving elastic interface. Anita Layton and I have developed a relatively simple approach for the more general problem of Navier-Stokes flow with an interface. Jumps in velocity gradient and pressure at the interface have to be accounted for, but we avoid extra work near the interface. Another paper with A. Layton gives an analytical explanation, with applications, of the observed gain in order of accuracy in certain finite difference methods for computing boundary value problems with irregular boundaries using only regular grids (A. Mayo's method or the immersed interface method of R. LeVeque and Z. Li). In a recent paper I proved that such a gain in accuracy is also possible for (time-dependent) diffusion equations, with appropriate choice of time-stepping, as a consequence of regularity estimates for the discrete problem analogous to those for the exact differential equation. Related projects include computation of unstable fluid interfaces (with G. Baker) and the design of a boundary integral method for 3-D water waves which is numerically stable and proved to converge; the discretization of potentials on a moving surface is a central issue.