Micro- and nano-fluidics involve a broad range of scales from the atomic scales to the continuum ones. A full molecular dynamics simulation is able to simulate the fluid flows at the micro- and nano-scales. However, it is computationally prohibitive due to the limitation of computer memory and computation time. On the other hand, a full continuum description, such as the Navier-Stokes equations, is computationally available but unable to describe the fluid flows in the region where the continuum assumption breaks down. A typical problem of this kind is the superhydrophobics: the patterned roughness on a hydrophobic solid surface enhances its hydrophobics and yields a large slip velocity at the solid surfaces. The superhydrophobics property is particularly attractive, since it may provide an efficient method for mass transport and drag reduction in micro- and nano-fluidics. An appropriate approach to simulate the superhydrophobics is to use the molecular dynamics in one region where the continuum assumption breaks down and use the Navier-Stokes equations in another region where the continuum assumption holds true, and those two descriptions are coupled in the overlap region. The computation time in the hybrid method is expected to be much less than that in the full molecular dynamics simulation. The challenge is how to couple the Navier-Stokes equations with the molecular dynamics simulation. In this talk, I will introduce our recent work on the dynamic coupling model (Chem. Eng. Sci. 62 3574-3579 2007) for the hybrid computation and use the hybrid simulation to study superhydrophobics. The numerical issue associated with the hybrid method will be discussed.

Scattering problems play an essential role in many scientific areas such as
radar and sonar (e.g., stealth aircraft design and submarine detection),
geophysical exploration (e.g., oil and gas exploration), medical imaging (e.g., breast cancer detection), and near-field optical microscopy (e.g., imaging of small scale biological samples). In this talk, we consider the scattering problem of a time-harmonic plane wave incident on a heterogeneous medium consisting of isotropic point (small scale) scatterers and an extended (wavelength comparable) obstacle scatterer in three dimensional space. The motivation arises from the near-field imaging, which is a vigorously developed research field because it provides an effective approach to break the diffraction limit and obtain images with subwavelength resolution.

The classical Foldy-Lax method provides an efficient approach to compute the scattered field from the interaction between the incident wave and the point scatterers; while boundary integral equation methods have been well studied for solving the scattering problem solely involving extended obstacle scatterers. It is a challenging two-scale multiple scattering problem when both the point scatterers and the extended obstacles are present. We developed a generalized Foldy-Lax method to fully take account of the multiple scattering in the heterogeneous medium. Two different but consistent formulations will be introduced: a series solution formulation and an integral equation formulation. The series solution formulation will be shown as an efficient iterative scheme to the integral equation formulation. The convergence of the scattered fields and the far-field patterns from the series solution formulation will be characterized in terms of scattering coefficients. Numerical experiments will be presented to show the agreement and the effectiveness of the proposed two approaches.

I will present a second order accurate, geometrically flexible and easy to implement method for solving the variable coefficient Poisson equation with interfacial discontinuities on an irregular domain. We discretize the equations using an embedded approach on a uniform Cartesian grid employing virtual nodes at interfaces and boundaries. A variational method is used to define numerical stencils near these special virtual nodes and a Lagrange multiplier approach is used to enforce jump conditions and Dirichlet boundary conditions. Our combination of these two aspects yields a symmetric positive definite discretization. In the general case, we obtain the standard 5-point stencil away from the interface. For the specific case of interface problems with continuous coefficients, we present a discontinuity removal technique that admits use of the standard 5-point finite difference stencil everywhere in the domain. Numerical experiments indicate second order accuracy in L-infinty.

This lecture will describe the mathematical and statistical methods of estimating the errors in the optimal assessment of the past climate change, quantifying the uncertainties in the climate change detection, and analyzing the main uncertainty sources for climate predictions. Empirical orthogonal functions are extensively used to deal with spatial inhomogeneity. Temporal non-stationarity and model nonlinearity will be discussed. Detailed error analyses of the annual mean global and regional averages of the surface air temperature since 1861 will be presented.

This talk is dedicated to the studies of a three-dimensional, nonisothermal, anisotropic, two-phase transport model of proton exchange membrane fuel cell (PEMFC) and its efficient numerical method. Besides addressing the conservation equations of mass, momentum, species, charge and energy in view of the nonisothermality, anisotropy and multiphase flow in PEMFC model, from an efficient numerical method's point of view, we present some new formulations for species equations in the interests of the interactions among the species. In a framework of finite element-upwind finite volume method, some efficient numerical methods are designed and investigated in order to achieve fast and convergent numerical simulation for this PEMFC model.

Numerical implementation is done by using a novel automated finite element/finite volume program generator (FEPG). By virtue of a high-level algorithm description language (script), component programming and human intelligence technologies, FEPG can quickly generate finite element/finite volume source code for PEMFC simulation. Thus, one can focus on the efficient algorithm research without being distracted by the tedious computer programming on finite element/finite volume methods. The 3D numerical simulations demonstrate that a convergent and reasonable physical solution can be attained within 100 steps or so, comparing to the oscillating and nonconvergent nonlinear iterations conducted by the standard finite element/finite volume method. Numerical success demonstrates that FEPG is an efficient tool for both algorithm research and software development of a 3D multiphysics PEMFC model.