Fast Algorithms for the Interaction between Charged Dielectric Spheres

Zecheng Gan

Michigan State University

We will discuss fast algorithms for the simulation of charged dielectric spheres. Specifically, we present efficient and higher order accurate methods for solving the electrostatic Poisson equation in the presence of multiple dielectric spherical interfaces. Involved techniques including the boundary element method, method of moments, and image method, etc. We further use our methods to explore the role of dielectric effect in the self-assembly of colloidal particles. And we show that the polarization effect can play an important role and lead to interesting phenomena such as like-charge attraction and varies self-assembled structures.

Fast Huygens Sweeping Methods for Time-Depdendent Schrodinger Equation with Perfectly Matched Layers

Songting Luo

Iowa State University

We present fast Huygens Sweeping methods for time-dependent Schrodinger equation with perfectly matched layers. The wavefunction is given as an integral with retarded Green's functions. For the Green's functions, asymptotic approaches are applied to approximate them such that the integral can be evaluated efficiently with fast Fourier transform (FFT). The wavefunction is then propagated by a short-period of time, and it can be repeated to reach any later time. Numerical experiments are performed to verify the efficiency and accuracy of the methods.

A Positivity Preserving Moving Mesh Finite Element Method for the Keller-Segel Chemotaxis Model

Mohamed Sulman

Wright State University

We present an efficient adaptive moving mesh finite element method for solving the Keller–Segel chemotaxis type models. The adaptive mesh is obtained by a coordinate transformation defined from the computational domain to the physical domain to concentrate the grid nodes in regions of large solution variations in the physical domain. A positivity preserving finite element scheme is used for spatial discretization of the Keller–Segel equations.

The numerical results show that the proposed method reduces the computational cost while improving the overall accuracy of the computed solutions of the Keller–Segel model.

A Variational Lagrangian Scheme for the Multidimensional Porous Medium Equation by a Discrete Energetic Variational Approach

Yiwei Wang

Illinois Institute of Technology

In this talk, we'll present a new variational Lagrangian scheme for the multidimensional porous medium equation (PME). Our scheme is based on the piecewise linear approximation to the flow map and a discrete Energetic Variational Approach (EnVarA), which inherits various properties from the continuous energy-dissipation law and is energy stable. Several numerical experiments demonstrate the accuracy of our numerical method as well as its ability in tracking the free boundary for PME. This methods can be adopted to a large class of partial differential equations with energy-dissipation law, such as nonlinear diffusion equations, phase-field equations, and equations for liquid crystals.

Boundary functions for biorthogonal multiwavelets and their properties

Ahmet Alturk

Iowa State University & Amasya University

Wavelets were originally constructed on the whole real line. They are, therefore, suitable for analyzing functions defined on the entire real line. In most practical cases, however, we work with functions defined on compact intervals. So, there is a need for adapting wavelets from the entire real line to a finite interval. There are several approaches to handle this problem in the literature. In this talk, we focus on boundary function approach for biorthogonal multiwavelets. In particular, we investigate boundary functions defined by recursion relations and derive the regularity properties of them directly from the recursion coefficients. This is a joint work with Prof. Fritz Keinert (Iowa State University).

A Roadmap for Discretely Energy-Stable Schemes for Dissipative Systems Based on a Generalized Auxiliary Variable with Guaranteed Positivity

Zhiguo Yang

Purdue University

In this talk, we present a framework for devising discretely energy-stable schemes for general dissipative systems based on a generalized auxiliary variable. The auxiliary variable, which is a scalar number, can be defined in terms of the energy functional by a general class of functions, not limited to the square root function as in previous approaches. The current method has another remarkable property: the computed values for the generalized auxiliary variable are guaranteed to be positive on the discrete level, regardless of the time step sizes or the external forces. It is noted that this property of guaranteed positivity is not available in previous approaches. The discrete energy stability of the proposed numerical scheme and the positivity of the computed auxiliary variable have been proved for general dissipative systems. The numerical scheme presented herein requires only the solution of linear algebraic equations within a time step. With appropriate choice of the operator in the algorithm, the resultant linear algebraic systems upon discretization involve only constant and time-independent coefficient matrices, which only need to be computed once and can be pre-computed.

A New Moving Mesh Method for Phase Field Model Based on Energetic Variational Approach

Qing Cheng

Illinois Institute of Technology

In this talk, a new moving method will be introduced based on Energetic Variational Approach (EnVarA) in Lagrangian Coordinate. The continue equation can be derived by Onsager Principle. And the numerical schemes for the phase field model can be proved to be unconditionally energy stable and unique solvable. The advanage of this method can capture the sharp interface more accurately, some numerical simulations will be shown in this talk.

Onsager-theory-based Tensor Model for Nematic Phases of Bent-core Molecules

Jie Xu

Purdue University

We present a tensor model for nematic phases of bent-core molecules (and rigid molecules of the same symmetry) derived from Onsager theory. The form of free energy is determined by molecular symmetry, which includes the couplings and derivatives of a vector and two second-order tensors, with the coefficients derived as functions of molecular parameters. The model builds a definite mapping from molecular architectures and macroscopic behaviors. We use the model to study the nematic phases resulted from different molecular architectures.

We also discuss other applications, and computational problems arising from the model.