PDE & Numerical Methods Seminar - Fall 2005

Location and time: Penn State University
Department of Mathematics
11:15am --- 12:15am
Monday (odd weeks)
216 McAllister Building

09/12/05 Zhengfu Xu
10/10/05 Wen Shen
Optimal Tracing of Viscous Shocks in Solutions of Viscous Conservation Laws

Abstract: This paper contains a qualitative study of a scalar conservation law with viscosity: $$ u_t+f(u)_x=u_{xx}\,.$$ We consider the problem of identifying the location of viscous shocks, thus obtaining an optimal finite dimensional description of solutions to the viscous conservation law. We introduce a nonlinear functional whose minimizers yield the viscous travelling profiles which ``optimally fit'' the given solution. We prove that, outside an initial time interval and away from times of shock interactions, our functional remains very small, i.e.~the solution can be accurately represented by a finite number of viscous travelling waves.
10/24/05 David Trebotich
Big Physics in Small Spaces: Numerical Algorithms for Biological Flow at the Microscale

Abstract: Biological flow is complex, not well-understood and inherently multiscale due to the presence of macromolecules whose molecular weights are comparable to length scales in the typical flow geometries of microfluidic devices or critical anatomies. Modeling these types of flows such as DNA in solution or blood is a challenge because their constitutive behavior is not easily represented. For example, a highly concentrated solution of suspended polymer molecules may be represented at the system level with a continuum viscoelastic constitutive model. However, when geometry length scales are comparable to the inter-polymer spacing, a continuum approximation is no longer appropriate, but, rather, a discrete particle representation coupled to the continuum fluid is needed. Furthermore, fluid-particle methods are not without their issues as stochastic, diffusive and advective processes can result in disparate time scales which make stability difficult to determine while capturing all the relevant physics. At Lawrence Livermore National Laboratory we have developed advanced numerical algorithms to model particle-laden fluids at the microscale. We will discuss a new stable and convergent method for flow of an Oldroyd-B fluid which captures the full range of elastic flows including the benchmark high Weissenberg number problem. We have also fully coupled the Newtonian continuum method to a discrete polymer representation with constrained and unconstrained particle dynamics in order to predict the fate of individual DNA molecules in post microarrays. The method is capable of modeling short range forces and interactions between particles using soft potentials and rigid constraints. Our methods are based on higher-order finite difference methods in complex geometry with adaptivity. Our Cartesian grid embedded boundary approach to irregular geometries has also been interfaced to a fast and accurate level-set method for extracting surfaces from volume renderings of medical image data and used to simulate cardio-vascular and pulmonary flows in critical anatomies.
11/7/05 Luca Heltai
The Finite Element Immersed Boundary Method -- abstract -- The immersed boundary method is both a mathematical formulation and a numerical method. In its continuous version it is a fully non- linearly coupled formulation for the study of fluid structure interactions. This method was introduced by Peskin to study the blood flow in the heart and further applied to many situations where a fluid interacts with an elastic structure. Many numerical approaches have been proposed to reduce the difficulties related to the non-linear coupling between the structure and the fluid evolution, however numerical instabilities arise when explicit or semi-implicit methods are considered. A variational approach to this method will be presented, together with its finite element discretization and a stability analysis based on energy estimates. We use a linearization of the Navier-Stokes equations and a hyper elastic model and we prove the unconditional stability of a fully implicit discretization, achieved with the use of a backward Euler method for both the fluid and the structure evolution, and a CFL condition for a semi-implicit method where the fluid terms are treated implicitly while the structure is treated explicitly.
11/14/05 Kevin Chu
Princeton University
Title: Towards an Understanding of Nonlinear Charge Transport Phenomena Abstract: Charge transport plays a critical role in many colloidal, micro-electrochemical and biological systems subjected to applied voltages or electric fields. Studying the electrochemical transport properties of these systems is an important first step towards understanding their response to applied fields. However, due to the nonlinear coupling between the total electric field and ion concentration fields, analysis of charge transport problems have traditionally relied on simple circuit models which are only valid in highly restricted operating regimes. In this talk, we shall discuss recent advances in our theoretical understanding of the full nonlinear charge transport equations in the thin double layer limit. Our main results include the formalization and derivation of surface conservation laws and characterization of the structure of electric and ion concentration fields around spherical, metallic (i.e. highly-polarizable) colloid particles via numerical solution of the governing equations.
11/21/05 TBA
12/05/05 Livshits, Irene
Ball State University
Multigrid method for eigenvalue Shordinger problems

Previous schedules: Spring 2005, Fall 2004, Spring 2004, Fall 2003, Spring 2003, Fall 2002, Spring 2002,
Fall 2001, Spring 2001, Fall 2000, Fall 1999, Spring 1999, Spring 1998, Fall 1998.

For more information or to suggest speakers, please contact Chun Liu
< liu@math.psu.edu>.

Sponsored in part by CCMA and by individual faculty grants

Last modified: Tue 09/01/2005