Qianyun Miao, Changhui Tan, and Liutang Xue

Mathematical Models and Methods in Applied Sciences, Volume 31, No 3, pp. 473-524 (2021).

Abstract

We study one-dimensional Eulerian dynamics with nonlocal alignment interactions, fea- turing strong short-range alignment, and long-range misalignment. Compared with the well-studied Euler-alignment system, the presence of the misalignment brings different behaviors of the solutions, including the possible creation of vacuum at infinite time, which destabilizes the solutions. We show that with a strongly singular short-range alignment interaction, the solution is globally regular, despite the effect of misalignment.

	doi:10.1142/S021820252150010X
	Download the Published Version
	This work is supported by NSF grant DMS #1853001

 Speaker: Rihui Lan (University of South Carolina)

In this talk, two different moving interface problems are studied by different arbitrary Lagrangian-Eulerian (ALE) finite element methods. For the transient Stokes/parabolic coupling with jump coefficients—a linearized fluid-structure interaction (FSI) problem, a novel \(H^1\)-projection is defined to account for the mesh motion. The well-posedness and optimal convergence properties in both the energy norm and \(L^2\) norm are analyzed for this mixed-type \(H^1\)-projection, with which the stability and optimal error estimate in the energy norm are derived for both semi- and fully discrete mixed finite element approximations to the Stokes/parabolic interface problem. For the parabolic/mixed parabolic moving interface problem with jump coefficients, a stable Stokes-pair mixed FEM within a specific stabilization technique and a novel ALE time-difference scheme are developed to discretize this interface problem in both semi- and fully discrete fashion, for which the stability and optimal error estimate analyses are conducted.
 

Time: March 27, 2020 2:30pm-3:30pm October 2, 2020 2:30pm-3:30pm
Location: LC317R Virtually via Zoom
Host: Lili Ju

  Access to the video of the talk

 Speaker: Hailiang Liu (Iowa State University)

We are concerned with positive and energy-dissipating schemes for solving the time-dependent system of Poisson-Nernst-Planck (PNP) equations, which has found much use in the modeling of biological membrane channels and semiconductor devices. As a gradient flow in density space, this strongly coupled system of nonlinear equations can take long time evolution to reach steady states. Hence, designing efficient and stable methods is highly desirable. In this talk we shall present a class of methods with structure-preserving properties for the PNP system, and review advances around related models such as the quantum diffusion equation.
 

Time: February 21, 2020 2:30pm-3:30pm
Location: LC317R
Host: Changhui Tan

Yi Sun, and Changhui Tan

Physica D, Volume 413, 132663 (2020).

Abstract

This paper presents a new class of one-dimensional (1D) traffic models with look-ahead rules that take into account of two effects: nonlocal slow-down effect and right-skewed non-concave asymmetry in the fundamental diagram. The proposed 1D cellular automata (CA) models with the Arrhenius type look-ahead interactions implement stochastic rules for cars’ movement following the configuration of the traffic ahead of each car. In particular, we take two different look-ahead rules: one is based on the distance from the car under consideration to the car in front of it; the other one depends on the car density ahead. Both rules feature a novel idea of multiple moves, which plays a key role in recovering the non-concave flux in the macroscopic dynamics. Through a semi-discrete mesoscopic stochastic process, we derive the coarse-grained macroscopic dynamics of the CA model. We also design a numerical scheme to simulate the proposed CA models with an efficient list-based kinetic Monte Carlo (KMC) algorithm. Our results show that the fluxes of the KMC simulations agree with the coarse-grained macroscopic averaged fluxes for the different look-ahead rules under various parameter settings.

	doi:10.1016/j.physd.2020.132663
	Download the Published Version
	This work is supported by NSF grant DMS #1853001
	This work is supported by a UofSC VPR ASPIRE I grant

 Speaker: Xiangcheng Zheng (University of South Carolina)

We proved the wellposedness of variable-order mobile-immobile time-fractional diffusion equations and the regularity of their solutions. Optimal-order finite element approximation was presented and analyzed. Numerical experiments were carried out for demonstration.
 

Time: January 31, 2020 2:30pm-3:30pm
Location: LC317R

Speaker: Xiaolin Li (Stony Brook University)

In this talk, I will review the history of the Lagrangian front tracking method and the computational platform built on this methodology. I will review the front tracking in the study of fluid interface instabilities, including Rayleigh-Taylor instability, Richtmyer Meshkov instability and fluid mixing induced by these instabilities. I will also introduce a fully conservative front tracking scheme and its application in the phase transition problem.
 

Time: March 20, 2020 2:30pm-3:30pm
Location: LC317R
Host: Xinfeng Liu

 Speaker: Yuanwei Qi (University of Central Florida)

In this talk, I shall report some recent progress on the existence and multiplicity of traveling waves to one of the most important models in Turing Pattern Formation. In addition, I shall pose some questions which are wide open which demand new ideas and fresh approaches. This is a joint-work with Xinfu Chen et al.
 

Time: January 17, 2020 2:30pm-3:30pm
Location: LC317R
Host: Changhui Tan

In the MacOS system, the keychain password has to match the user password in order to use keychain appropriately. In my case, they have to match in order to allow Apple Watch to unlock the Mac.

The official way to change the keychain password can be found HERE. However, in my case, the "Change password for keychain login" option is in gray and can not be selected. One solution is the following:

Open terminal and type in

security set-keychain-password

Then, type in old password and then new password. The password is then changed. This does not require admin access.

Yongki Lee, and Changhui Tan

Communications in Mathematical Sciences, Volume 20, No. 4, pp. 1151-1172 (2022).

Abstract

We study a Lighthill-Whitham-Richards (LWR) type traffic flow model, with a nonlocal look-ahead interaction that has a slow-down effect depending on the traffic ahead. We show a sharp critical threshold condition on the initial data that distinguishes global smooth solutions and finite- time wave breakdown. It is well-known that the LWR model leads to a finite-time shock formation, representing the creation of traffic jams, for generic smooth initial data with finite mass. Our result shows that the nonlocal slowdown effect can help to prevent shock formations, for a class of subcritical initial data.

	doi:10.4310/CMS.2022.v20.n4.a9
	Download the Published Version
	This work is supported by NSF grants DMS #1853001 and DMS #2108264
	This work is supported by a UofSC VPR ASPIRE I grant