# Numerical methods for interfaces and regularizing effects in difference equations

Thursday, February 25, 2010 - 3:00pm - 3:45pm

EE/CS 3-180

J. Thomas Beale (Duke University)

*Keywords:*Navier-Stokes flow, Stokes flow, boundary integral,

stiff equations, fractional stepping, immersed interface,

immersed boundary, semigroups of operators,

finite difference methods, parabolic equations, diffusion,

regularity, stability, L-stable, A-stable, maximum norm

*Abstract:*We will discuss two related projects. Work with A. Layton has the goal

of designing a second-order accurate numerical method for viscous fluid

flow with a moving elastic interface with zero thickness, the original

problem for which Peskin introduced the immersed boundary method. We

will discuss some of the background for such numerical

methods. In our approach, we decompose the velocity in

the Navier-Stokes equations at each time into a part determined by the

(equilibrium) Stokes equations, with the interfacial force, and a

regular remainder which can be calculated without special treatment

at the interface. For the Stokes part we use the immersed interface

method or boundary integrals; for the regular part we use the

semi-Lagrangian method to

advance in time. Simple test problems indicate second-order accuracy

despite a first-order truncation error near the interface, as has come

to be expected with certain interfacial methods. We will describe

analytical results which partially justify this expectation. For a

fully discrete parabolic equation, we have proved a

regularizing effect: If we solve a nonhomogeneous heat equation with a

finite difference method, with L-stable temporal discretization, using

large time steps, then the solution and its first differences are

bounded uniformly by the maximum of the nonhomogeneity, and the second

differences are almost bounded. The proof uses the point of view of

analytic semigroups of operators.

MSC Code:

65Q10

Keywords: