Applied Math And Analysis Seminar
Monday, September 7, 2009, 4:30pm, 119 Physics
Tom Beale
Numerical Methods for Interfaces in Viscous Fluid Flow and Regularizing Effects in Parabolic Difference Equations

Abstract:

We will discuss two related projects, one algorithmic and the other analytical, both with the goal of designing second-order accurate numerical methods for viscous fluid flow with a moving elastic interface with zero thickness, the original problem for which Peskin introduced the immersed boundary method. In recent work with Anita Layton we simplify the problem in Navier-Stokes flow by decomposing the velocity at each time into a ``Stokes'' 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. 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. In the second part,we describe an analytical result which partially justifies this expectation by giving a regularizing effect for fully discrete parabolic equations: 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. The relevance to interfaces will be explained. (Note to graduate students: many of the terms used here will be explained as part of the talk.)

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