Annual Report 2008.pdf - SAMSI

viscoelastic creeping flow solutions with a detailed focus on experimentally measurable
signatures: of elastic and viscous responses
to steady and time-periodic driving forces; and of unsteady (inertial) effects. We also assess the
point force approximation for micron-size driven beads. Finally, we illustrate the generality in
source geometry by analyzing the linear response for a nonlocal, planar source of unsteady
stress.
Sheng Xu
Southern Methodist University
Department of Mathematics
sxu@smu.edu
“Singular Forces in the Immersed Interface Method to Enforce the Prescribed Boundary
Motions”
In the immersed interface/boundary method, boundaries in a fluid are generated by singular
forces in the Navier-Stokes equations. In this poster, I will present an inverse approach which
explicitly calculates singular forces to enforce the prescribed motions of rigid boundaries. The
inverse approach has no stiff solid models and no implicit or iterative treatments. It is stable,
efficient, and accurate.
Zhiliang Xu
University of Notre Dame
Department of Mathematics
Zxu2@nd.edu
“A Conservative Interface Tracking Method”
Discontinuities of the solutions to the nonlinear hyperbolic systems of conservation laws are a
primary difficulty for numerical simulations. ${\mathcal O}(1)$ errors exist near an interface. A
fully conservative front tracking algorithm was developed to solve the problem. This algorithm
provides a general framework which works in any dimension and successfully preserves
conservation across the interface in the context of interface tracking. We proved that near
tracked discontinuities in the solution, this algorithm has ${\mathcal O}(\Delta x)$ errors,
improving over ${\mathcal O}(1)$ errors commonly found near a discontinuity in most finite
difference schemes. The algorithm is derived from an integral formulation of the $PDE$s. The
key is to track fronts (or discontinuities) in spaces and time, then to construct a space-time finite
volume decomposition which respects the space-time interface.
Pengchong Yan
University of California-Davis
Department of Mathematics
pyan@math.ucdavis.edu
"Multi-Frequency Imaging of Multiple Targets in Rician Fading Channels: Stability and
Resolution"