# Meeting Details

Title: Numerical methods for interfaces in fluids and regularizing effects in difference equations Department of Mathematics Colloquium Tom Beale, Duke University We will discuss a class computational methods for a moving interface in viscous fluid and related error analysis. We will begin with a description of the Navier-Stokes equations and the model of a moving interface which interacts with the fluid. (It could be a drastically simplified model of biological tissue on a small scale.) We will describe several numerical methods for such problems in which the fluid variables are calculated at grid points and the moving interface is represented separately. Accuracy is most difficult to maintain near the interface, but it has often been observed that the solution can be uniformly more accurate than would be expected from the truncation error in satisfying the equations. We will explain this gain with discrete, finite difference versions of regularity estimates for elliptic and parabolic differential equations, such as Laplace's equation and the heat equation, in maximum norm, which are almost sharp''. We will present a numerical method for the coupled motion of an elastic interface with Navier-Stokes flow, developed with Anita Layton, in which the velocity is decomposed into a Stokes velocity and a more regular part. The advantage is that the interaction of the interface with Stokes flow (dominated by viscosity) is much simpler to deal with than the full Navier-Stokes equations. Simple test problems and partial analysis indicate the method is second-order accurate. It can be modified to make the interface motion partially implicit, in order to allow larger time steps.