Spring 2023
Schedule
Time | Location | Speaker | Title |
---|---|---|---|
Jan 20, 11am | GP234 | Zheng Sun | On a numerical artifact of solving shallow water equations with a discontinuous bottom |
Feb 3, 11am | GP234 | Qiang Huang | Some numerical problems in semiconductor manufacturing |
Feb 24, 11am | GP234 | Xinwu Qian | Understanding and mitigating the spread of infectious disease in public transportation systems |
Mar 24, 11am | GP234 | Yiming Ren | FFT accelerated high order finite difference method for solving elliptic BVP and interface problem |
Mar 31, 11am | Zoom | Xinyue Zhao | Neural-network-based methods for solving free boundary problems |
Apr 14, 11am | GP234 | Miloud Sadkane | A Newton-type method for the eigenvalue problem of analytic matrix functions |
Apr 21, 11am | Zoom | Shibo Liu | Morse theory and its applications to nonlinear differential equations |
Apr 28, 11am | Zoom | Cory Hauck | Multi-species BGK models. |
May 10, 11am | GP234 | Dania Sheaib | Insights into the dynamics of buoyancy-induced flows under thermal stratification |
Abstracts
- Jan 20, 2023
Zheng Sun (Department of Mathematics, The University of Alabama)
Title: On a numerical artifact of solving shallow water equations with a discontinuous bottom
Abstract: The nonlinear shallow water equations are used to model the free surface flow in rivers and coastal areas for which the horizontal length scale is much greater than the vertical length scale. They have wide applications in oceanic sciences and hydraulic engineering. In this talk, we study a numerical artifact of solving the shallow water equations over a discontinuous riverbed. For various first-order methods, we report that the numerical solution will form a spurious spike in the numerical momentum at the discontinuous point of the bottom. This artifact will cause the convergence to a wrong solution in many test cases. We present a convergence analysis to show that this numerical artifact is caused by the numerical viscosity imposed at the discontinuous point. Motivated by our analysis, we propose a numerical fix which works for the nontransonic problems.
- Feb 3, 2023
Qiang Huang (Department of Chemical and Biological Engineering, The University of Alabama)
Title: Some numerical problems in semiconductor manufacturing
Abstract: A state-of-the-art semiconductor device is composed of multiple chips manufactured on different platforms for different functionalities. The inter-connections between these chips are formed using electrochemical deposition process, where metals and alloys grow in a microscale confined geometry. However, as the semiconductor technology continues to advance, such connecting structures further scale and the fabrication process faces critical challenges. The growth rate, shape, and uniformity of the metal structures need to be precisely controlled to ensure the proper communication between chips. Experimental approaches are typically used to explore different chemistries and processes. However, given the large parameter space and extremely high cost, such experimental explorations are often in a try-and-error fashion, providing not only delayed but also inconsistent and non-optimized solutions. This talk will mainly introduce this engineering challenge to the Mathematics community and will briefly discuss some preliminary efforts to tackle this challenge using a numerical approach.
- Feb 24, 2023
Xinwu Qian (Department of Civil, Construction and Environmental Engineering, The University of Alabama)
Title: Understanding and mitigating the spread of infectious disease in public transportation systems
Abstract: The urban transportation system catalyzes disease outbreaks since it provides the general public the mobility to participate in intensive urban activities. And the high passenger volume and long commuting time of the transportation system further facilitate the spread of infectious diseases. This talk will discuss the spreading dynamics of infectious diseases in urban transportation systems based on mathematical modeling, complex network theory, and validation with real-world trip data. Topics that will be covered in this talk include (1) the linkage between disease dynamics and the structure of physical contact networks, (2) the metapopulation model for understanding and controlling disease outbreaks by considering activity and travel contagions, and (3) a new model and algorithm to minimize the risk of disease for vulnerable populations accessing healthcare services. Case studies will also be presented for the early outbreak of COVID-19 in New York City. the contact networks in the subway systems of three large cities, and a local experiment for paratransit service in Birmingham, Alabama.
- Mar 24, 2023
Yiming Ren (Department of Mathematics, The University of Alabama)
Title: FFT accelerated high order finite difference method for solving elliptic BVP and interface problem
Abstract: In this talk, An augmented matched interface and boundary method is introduced for solving elliptic boundary value problems and interface problems. First, A new finite difference method is proposed based on the matched interface and boundary (MIB) and fast Fourier transform (FFT) schemes, which achieves a fourth order convergence and O(NlogN) efficiency. The method involves a ray-casting MIB scheme to handle different types of boundary conditions and an augmented MIB formulation for efficient inversion of the discrete Laplacian using the FFT algorithm. The accuracy and efficiency of the proposed method are numerically examined for various elliptic BVPs in two and three dimensions. Second, A FFT acclerated finite difference method is proposed for solving a three-dimensional elliptic interface problem involving a smooth material interface inside a cuboid domain. The method can handle different types of boundary conditions and complex geometry using a fourth order ray-casting MIB scheme and Cartesian derivative jumps as auxiliary variables. The proposed method is efficient and accurate, achieving an overall efficiency on the order of O(n^3logn) for a n ×n ×n uniform grid.
- Mar 31, 2023
Xinyue Zhao (Department of Mathematics, Vanderbilt University)
Title: Neural-network-based methods for solving free boundary problems
Abstract: Free boundary problems (the time-dependent versions are also often known as moving boundary problems) deal with systems of partial differential equations (PDEs) where the domain boundary is apriori unknown. Due to this special characteristic, it is challenging to solve free boundary problem numerically, and most studies in this field lack convergence proofs for the numerical methods. In this talk, I will present novel approaches based on neural networks to study two types of free boundary problems: 1) the classical obstacle problem, and 2) a modified Hele-Shaw problem. For each method we proposed, we established the convergence of the scheme and theoretically derived the convergence rate with the number of neurons. Several simulation examples are used to demonstrate the feasibility and capability of the proposed methods.
- Apr 14, 2023
Miloud Sadkane (CNRS - UMR 6205, Laboratoire de Mathématiques de Bretagne Atlantique, Université de Brest)
Title: A Newton-type method for the eigenvalue problem of analytic matrix functions
Abstract: In this talk, a Newton-type method for the eigenvalue problem of analytic matrix functions is proposed. The method finds the eigenvalue and eigenvector respectively as a point in the level set of the smallest singular value function and the corresponding right singular vector. The algorithmic aspects are discussed and illustrated numerically.
- Apr 21, 2023
Shibo Liu (Department of Mathematical Sciences, Florida Institute of Technology)
Title: Morse theory and its applications to nonlinear differential equations
Abstract: In the first part of this talk we quickly review some basic concepts in Morse theory, then in the setting of saddle point reduction, we investigate the relation between the critical groups of the original functional and the reduced functional. The abstract results are used to get multiple solutions for elliptic BVPs over bounded domain. In the second part, using Morse theory we discuss the existence of nonzero solutions for stationary nonlinear Schrodinger type equations with indefinite Schrodinger operator. Finally, by applying a variant of the Clark’s theorem we study a quasilinear Schrodinger equation whose nonlinear term can grow super critically. Infinitely many solutions with negative energy are obtained.
- Apr 28, 2023
Cory Hauck (Computer Science and Mathematics Division, Oak Ridge National Laboratory)
Title: Multi-species BGK models
Abstract: BGK (Bhatnagar–Gross–Krook) models have recently become a popular tool for simulating dilute, multi-species gases and plasmas. The primary motivation for their development is to avoid computing with the Boltzmann collision operator, whose evaluation often dominates the cost of a kinetic calculation. However, unlike in the single-species case, there is not an obvious choice of model parameters that recovers the desired conservation and entropy dissipation properties in the multi-species setting. In this talk, I will present a few such models, the numerical tools used to compute them, and a few results to demonstrate their value. This presentation includes joint work with Jeff Haack (Los Alamos), Christian Klingenberg (Würzburg), Michael Murillo (Michigan State), Sandra Warnecke (Würzburg), and Marlies Pirner (Würzburg).
- May 10, 2023
Dania Sheaib (Department of Mathematics, Florida Gulf Coast University)
Title: Insights into the dynamics of buoyancy-induced flows under thermal stratification
Abstract: Natural or free convection is a mechanism in which the motion of the fluid is not generated by an external source (such as a turbine or a fan) but only by density variations resulting from temperature gradients within the fluid. The differences in fluid density give rise to the driving force of natural convection known as buoyancy. Given the rising demand for sustainable energy resources, one interesting application of natural convection that has attracted research attention in recent years is the adoption of solar chimneys as an anergy-efficient means for building ventilation. A solar chimney may be described as an asymmetrically heated air channel in which air flow is due to the buoyancy force generated by solar heating. Driven by this application, we consider the flow dynamics of a stably stratified fluid in a vertical channel with time-periodic temperature variations on the sidewalls. In this talk, we will describe the problem set up and discuss the temperature and velocity profiles for the one-dimensional time-dependent flow. We will then explore the following question: Is it possible to observe resonance of the fluid flow with the periodic oscillations of the externally supplied temperatures at the walls?