Math Colloquium - Spring 2024

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January 26

Chen Liu, Purdue University

Host: Lou Kondic

Positivity-Preserving and Conservative High-Order Accurate Explicit-implicit Schemes for Compressible Flow Simulation

The compressible Navier–Stokes (NS) equations are one of the most important and popular models for studying viscous gas dynamics in various fields of applied mechanics and engineering. It is still a challenging task to construct a conservative and positivity-preserving scheme with high-order accuracy and large time step size in the sense of standard hyperbolic CFL.    In this talk, we propose a fully discrete semi-implicit scheme for solving the compressible NS equations within the Strang splitting framework. Our proposed scheme preserves conservation of arbitrary high order in space. The positivity-preserving property for up to Q3 space discretization is rigorously provable by using the matrix monotonicity. For Qk (k ≥ 4) space discretization, our schemes are rendered bound-preserving without losing conservation and accuracy, by a constrained optimization-based postprocessing procedure in each time step. Such a constrained optimization can be formulated as a nonsmooth convex minimization problem, which can be efficiently solved by the generalized Douglas–Rachford splitting method with the optimal algorithm parameters. By analyzing the asymptotic linear convergence rate, optimal algorithm parameters can be approximately expressed as a simple function of the numerical solutions. Our scheme for solving compressible NS equations enjoys the standard hyperbolic CFL on time step size. Numerical experiments suggest that our scheme produces satisfactory non-oscillatory solutions when physical diffusion is accurately resolved. Therefore, it is highly preferred and well-suited for simulating realistic physical and engineering problems. I will conclude the talk with potential future research directions in computational methods for fluid dynamics and related applications.

February 2

Amir Sagiv, Columbia University

Host: Yassine Boubendir

Floquet Hamiltonians - Spectrum and Dynamics

The last decade has witnessed tremendous experimental progress in the study of "Floquet media," crystalline materials whose properties are altered by time-periodic parametric forcing. Theoretical advancements, however, have so far been achieved through discrete and approximate models. Understanding these materials from their underlying, first-principle PDE models, however, remains an open problem.

Specifically, semi-metals such as graphene are known to transform into insulators under periodic driving. While traditionally this phenomenon is modeled by a spectral gap, in PDE models no such gaps are conjectured to form. How do we reconcile these seemingly contradictory statements? We show that the driven Schrödinger equation possesses an “effective gap” – a novel and physically relevant relaxation of a spectral gap. Adopting a broader perspective, we study the influence of time-periodic forcing on a general band structure. A spectrally-local notion of stability is formulated and proven, using methods from periodic homogenization theory.

February 9

Paula Vasquez, University of South Carolina

Host: Yuan-Nan Young

Coupling Macro-Micro Simulations in Complex Fluids

Some of the most remarkable properties and functions served by complex fluids originate from the interplay between external fields and microstructural dynamics. From a computational point of view this generates a set of challenges related to the need of coupling dynamics at different length and times scales, sometimes spanning several orders of magnitude. Micro-macro simulations have gained a lot of recognition within the field because these methods allow capturing full dynamics at the macroscale without losing resolution at the microscale. In this talk, we will review our efforts to couple existing macroscopic solvers for the Navier-Stokes equations with microstructural dynamics described by Langevin-type equations. In particular, we will discuss dumbbells models -under viscometric and capillary thinning flows fields- and parallel computing using GPUs.

February 16

Manas Rachh, (Flatiron Institute)

Host: Travis Askham

Static Currents in Type-I Superconductors

In this talk, we describe the classical magneto-static approach to the theory of type-I superconductors. The magnetic field and the current in type-I superconductors are related by the London equations and tend to decay exponentially inside the superconducting material with support of the fields contained primarily in O(λL) neighborhood of the superconductor. We present a Debye source based integral representation for the numerical solution of the London equations, and demonstrate the efficacy of our approach for moderate values of λL on complex three dimensional geometries. However, for typical materials λL ∼ O(10−7), which makes the PDE and integral equation increasingly difficult to solve in the limit λL → 0 due to the presence of two different length scales in the problem. We derive a limiting PDE and a corresponding integral equation, and show that the solutions of this limiting PDE and integral equations are O(λL) accurate as compared to the corresponding solutions of the London equations and the Debye source integral equations respectively. We demonstrate the effectiveness of this asymptotic approach both in terms of speed and accuracy through several numerical examples.

February 23

Eric Hester, University of California, Los Angeles

Host: Wooyoung Choi

Modeling Multiphase Matter: from Microparticles to Mega-Icebergs

The world is multiphase. Water and ice, rock and lava, nucleus and cytoplasm. How can we model these systems, and simulate them efficiently? I'll start with three examples from my research, boat dynamics in dead water, melting icebergs in salty oceans, and phase-separating polymers in microfluidic experiments. The same patterns recur. A seemingly simple partition into PDEs and boundary conditions can miss the narrow transition between phases. This diffuse interface in turn motivates a host of new numerical schemes. The immersed-boundary method, volume-penalty techniques, and phase-field models are a handful of examples. The bulk of my talk will discuss the mathematical tools we need to understand and improve these methods. Signed-distance coordinates give a straightforward tensor calculus around arbitrary submanifolds, and multiple scales matched asymptotics describes the resulting solutions to arbitrary order. I'll also discuss efficient spectral discretizations of these schemes, emphasizing how our mathematical tools can improve accuracy and alleviate stiffness, before concluding with some bigger questions for multiphase methods.

March 1

Joshua Taylor, NJIT

Host: David Shirokoff

Convex Optimization of Biochemical Processes

In this talk, we begin by optimizing the gradostat, in which several bioreactors are interconnected by mass flow and diffusion. The gradostat is of interest both as a classical nonlinear system and because the basic network structure and nonlinearities appear in a broad range of bioprocesses, including wastewater treatment. We formulate a convex relaxation for optimizing the gradostat. The relaxation is exact under several conditions, for example, if the network is outflow connected and irreducible. When the microbial growth in the bioreactors is described by the Monod or Contois functions, the relaxation is a second-order cone program, which can be solved at scales of over 10^5 variables in minutes with commercial software. We present simulations based on the operation of a wastewater treatment network over a two-week period.

March 8

David Ambrose, Drexel University

Host: Mike Siegel

Some Results for Non-Decaying, Non-Periodic Fluid Flows

Much work in fluid dynamics considers situations in which flows decay at infinity or are spatially periodic.  There are, however, simple situations one might wish to consider which lack such structure, e.g. a noisy perturbation of a periodic flow.  Without periodicity or decay, the Biot-Savart law cannot be used to reconstruct velocity from vorticity.  We will describe how to generalize and replace the Biot-Savart law, and we will state some existence results which follow from this.  For the specific case of non-decaying, non-periodic vortex sheets, we will give an expression for the velocity field induced by the vortex sheet and the associated formula for the Birkhoff-Rott integral.  This includes joint work with several coauthors.

March 22

Yoichiro Mori, University of Pennsylvania

Host: Enkeleida Lushi

Inextensible Interface Problem in 2D Stokes Fluid

We consider the dynamics of a closed intextensible interface immersed in a 2D Stokes fluid, a model that has been used for 2D simulations of vesicle dynamics. In this model, a 1D closed interface exerts a bending force and the interface is subject to an inextensibility constraint. As part of the problem, one must solve for the unknown tension that ensures membrane inextensibility. Given a force exerted on the interface, we first show that the problem of determining the tension is solvable if and only if the interface is not a circle. Using this result, we prove local-in-time well-posedness for this problem. We will finally discuss open questions and future directions.

Academic Homepage: https://www.sas.upenn.edu/~y1mori/ 

April 5

Charles Semple, University of Canterbury, New Zealand

Host: Kristina Wicke

Tree Reconstruction from Multi-state Characters

A central task in evolutionary biology is the construction of phylogenetic trees to represent the ancestral history of a collection of present-day taxa. The data typically used for this task are characters, which describe the attributes of the taxa under consideration. To illustrate, the nucleotide at a particular position of aligned DNA sequences is an example of a character. Furthermore, the states of this character are the subsets of taxa taking a certain nucleotide. In the context of this task, the following question naturally arises. How many characters does it take to uniquely determine a phylogenetic tree? Does the answer depend on the topology of the tree? If the characters have a bounded number of states, how does the number of characters it takes grow with respect to this bound and the number of taxa? In this talk, we investigate these and other related questions.

April 12

Steffen Hardt, Technische Universität Darmstadt

Host: Lou Kondic

Transport Processes in Nanochannels Induced by Electric and Temperature Fields

Nanofluidics is an emerging field that has rapidly developed in the past few years, partially due to advances in nanofabrication technology. For a number of reasons, transport processes in nanochannels differ from transport through larger-scale channels. First, in very small-scale channels, continuum-mechanical models loose their validity. Second, overlapping electric double layers from opposing channel walls cause effects such as ion exclusion that are absent on larger scales. This presentation will be concerned with the second scenario and will discuss theoretical and simulation results that are based on a continuum-mechanical description for electrolyte-filled nanochannels.

Nanochannels are true multiphysics entities in which the coupling of a number of different effects needs to be taken into account. Specifically, flow and species transport can either be driven by electric fields or temperature gradients. With respect to electric-field induced driving, two special scenarios are presented: electroosmotic flow (EOF) through conical channels equipped with gate electrodes and EOF through uncharged channels. We show that in channels with gate electrodes, a sinusoidal voltage applied to the electrode results in a net flow, provided that the spatial symmetry is broken. Here, the symmetry breaking is due to the conical shape of the channel. The standard way to induce EOF, however, is to apply an external DC electric field between the channel ends. Here, the common understanding is that the channel walls need to be charged to produce a flow. We show, however, that there can be a significant net EOF even through uncharged channels, an effect that relies on a special type of symmetry breaking. In that case, the symmetry is broken if the anion has a different valence than the cation. By contrast to EOF, the flow induced in nanochannels by temperature gradients is usually rather weak. However, if the solid surfaces have a non-zero slip length, the situation changes. We show that very significant thermoosmotic flow velocities can be obtained in nanochannels with realistic slip-length values. Finally, thermoelectric energy conversion processes in nanochannels are considered, that is, the generation of an electric field and/or an electric current upon application of a temperature gradient along the channel. In the past few years, very large values of the thermoelectric field that are hard to explain theoretically have been reported in experiments. Inspired by that, we have considered novel scenarios of thermoelectric energy conversion in nanochannels. On the one hand, we show that large thermoelectric fields can be obtained in channels filled with ionic liquids. The underlying physical mechanism is thermally- activated charge carrier generation. On the other hand, we consider ion transport by thermally- activated jumping over a potential energy barrier, which also produces unusually large values of the thermoelectric field.

In all of the above-mentioned scenarios, the governing equations are solved numerically using a finite-element method. In addition, analytical approximations are presented that match well with the numerical results.

April 19

Anna Balazs, University of Pittsburgh

Host: Lou Kondic

Chemically Controlled Shape-morphing of Elastic Sheets

Two-dimensional responsive materials that change shape into complex three-dimensional structures are valuable for creating systems ranging from wearable electronics to soft robotics. Typically, the final 3D structure is unique and predetermined through the materials’ processing.  Using theory and simulation, we devise a distinctive approach for driving shape changes of 2D elastic sheets in fluid-filled microchambers. The sheets are coated with catalyst to generate controllable fluid flows, which transform the sheets into complex 3D shapes. A given shape can be achieved by patterning the arrangement of the catalytic domains on the sheet and introducing the appropriate reactant to initiate a specific catalytic reaction. Moreover, a single sheet that encompasses multiple catalytic domains can be transformed into a variety of 3D shapes through the addition of one or more reactants. Materials systems that morph on-demand into a variety of distinct structures can simplify manufacturing processes and broaden the utility of soft materials. 

April 26

Oswald Knoth, Leibniz Institute for Tropospheric Research

Host: Simone Marras

CGDycore.jl: A Testbed for Dynamical Cores (DyCore) for Numerical Weather Prediction in the Julia Language

There is ongoing research in the design of numerical methods for numerical weather prediction.

This is connected with an increase in spatial resolution and the intensive use of graphic processor units (GPU).

Most of the main weather services and big research institutes have their own numerical DyCore developed over decades, or started again from scratch to implement a new DyCore with programming tools, which are efficient on modern heterogenous computer architectures. The Julia language is designed for high performance computing, is supported by multiple platforms, is easy to install, and has a lot of more features. It is therefore a suitable programming environment to write fast prototyping code with features like parallelization and running on different processor architectures. In the talk I will describe the ingredients to implement different actual DyCores in one computing environment. Special attentiation is given to:

1. Grids on the sphere and extruding to the third dimension
2. Discretization methods and function spaces
3. Parallelization
4. Programming on GPU’s

Some numerical results are presented for a horizontal spectral continuous Galerkin method with finite differences in the vertical on conforming quad grids.

May 3

Igor Aronson, Pennsylvania State University

Host: Enkeleida Lushi

Confined Bacterial Suspensions

Previous experiments have shown [1,2] that the complex spatiotemporal vortex structures emerging in motile bacterial suspensions are susceptible to weak geometrical constraints. By a combination of continuum theory and experiments, we have shown how artificial obstacles guide the flow profile and reorganize topological defects, which enables the design of bacterial vortex lattices with tunable properties. In more recent studies, we observed the emergence of spatiotemporal chaos in a bacterial suspension confined in a cylindrical well. As the well radius increases, we observed a bifurcation sequence from a steady-state vortex to periodically reversing vortices, four pulsating vortices, and, finally, to spatiotemporal chaos (active turbulence). The results of experiments are rationalized by the analysis of the continuum model for bacterial suspensions based on the complex Swift-Hohenberg equations. Furthermore, the bifurcation sequence is explained by reduction to amplitude equations for the three lowest azimuthal modes. Equations of motion are then reconstructed from experimental data.  The results indicate that the vortex reversal precedes the onset of spatiotemporal chaos in confined active systems.

[1] D Nishiguchi, IS Aranson, A Snezhko, A Sokolov, Engineering bacterial vortex lattice via direct laser lithography, Nature communications 9 (1), 4486 115 (2018)

[2] H Reinken, D Nishiguchi, S Heidenreich, A Sokolov, M Bär, S. H. L. Klapp & I. S. Aranson,  Organizing bacterial vortex lattices by periodic obstacle arrays. Commun Phys 3, 76 (2020)

Homepage: https://sites.psu.edu/iaronson/home/

Updated: April 19, 2024