Organizing Committee
Abstract

The workshop will be devoted to the analysis of wave phenomena from different perspectives: mathematical modeling and analysis, experimental physics, and numerical analysis. One of the goals of this event is to gather scientists coming from a priori distant communities but sharing a common interest in wave propagation phenomena in a broad sense (fluid mechanics, quantum mechanics, plasma physics, rigorous analysis). We plan to focus on various themes representing topical problems in these fields, from experimental reproduction of physical phenomena, numerical issues, to the most recent rigorous mathematical results.

In experimental physics, several topics will be addressed, from rogues waves and wave breaking phenomena, vortex filaments, to wave turbulence in fluids or in acoustics. The analysis of observational and experimental data, combined with PDE physical models also yields the question of data assimilation and machine learning technics in the context of wave propagation. The workshop will address some of these questions in particular in oceanography, turbulence modeling and stochastic forcing, and statistical analysis of waves.

The notion of formation of singularities will also be an important topic of this workshop, from the theoretical point of view (blow-up phenomena, filament reconnexion) and by using computer-assisted proof technics or high-performance numerical simulation.

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Confirmed Speakers & Participants

  • Speaker
  • Poster Presenter
  • Attendee
  • Virtual Attendee

Workshop Schedule

Monday, September 20, 2021
  • 9:50 - 10:00 am EDT
    Welcome
    11th Floor Lecture Hall
    • Brendan Hassett, ICERM/Brown University
  • 10:00 - 10:40 am EDT
    Quasilinear Diffusion of magnetized fast electrons in a mean field of quasi-particle waves packets
    11th Floor Lecture Hall
    • Speaker
    • Irene Gamba, University of Texas at Austin
    • Session Chair
    • Luis Vega, Basque Center for Applied Mathematics (BCAM)
    Abstract
    Quasi-linear diffusion of magnetized fast electrons in momentum space results from stimulated emission and absorption of waves packets via wave-particle resonances. Such model consists in solving the dynamics of a system of classical kinetic diffusion processes described by the balance equations for electron probability density functions (electron pdf) coupled to the time dynamics waves (quasi-particles) in a quantum process of their resonant interaction. Such description results in a Mean Field model where diffusion coefficients are determined by the local spectral energy density of excited waves whose perturbations depend on flux averages of the electron pdf.
    We will discuss the model and a mean field iteration scheme that simulates the dynamics of the space average model, where the energy spectrum of the excited wave time dynamics is calculated with a coefficient that depends on the electron pdf flux at a previous time step; while the time dynamics of the quasilinear model for the electron pdf is calculated by the spectral average of the quasi-particle wave under a classical resonant condition where the plasma wave frequencies couples the spectral energy to the momentum variable of the electron pdf. Recent numerical simulations will be presented showing a strong hot tail anisotropy formation and stabilization for the iteration in a 3 dimensional cylindrical model.
    This is work in collaboration with Kun Huang, Michael Abdelmalik at UT Austin.
  • 10:55 - 11:15 am EDT
    Coffee Break
    11th Floor Collaborative Space
  • 11:15 am - 12:00 pm EDT
    Modeling inviscid water waves
    11th Floor Lecture Hall
    • Virtual Speaker
    • Christophe Lacave, Universite Grenoble Alpes
    • Session Chair
    • Luis Vega, Basque Center for Applied Mathematics (BCAM)
    Abstract
    We consider numerical strategies to handle two-dimensional water waves in a fully non-linear regime. The free-surface is discretized via lagrangian tracers and the numerical strategy is constructed carefully to include desingularizations, but no artificial regularizations. We approach the formation of singularities in the wave breaking problem and also model solitary waves and the effect of an abruptly changing bottom. We present a rigorous analysis of the singular kernel operators involved in these methods.
  • 12:15 - 1:15 pm EDT
    Lunch/Free Time
  • 1:15 - 1:55 pm EDT
    Anomalous conduction in one dimensional chains: a wave turbulence approach.
    11th Floor Lecture Hall
    • Miguel Onorato, Università di Torino
    Abstract
    Heat conduction in 3D macroscopic solids is in general well described by the Fourier's law. However, low dimensional systems, like for example nanotubes, may be characterized by a conductivity that is size-dependent. This phenomena, known as anomalous conduction, has been widely studied in one dimensional chains like FPUT, mostly using deterministic simulations of the microscopic model. Here, I will present a mesoscopic approach based on the wave turbulence theory and give the evidence, through extensive numerical simulations and theoretical arguments, that the anomalous conduction is the result of the presence of long waves that rapidly propagate from one thermostat to the other without interacting with other modes. I will also show that the scaling of the conductivity with the length of the chain obtained from the mesoscopic approach is consistent with the one obtained from microscopic simulations.
  • 2:10 - 2:50 pm EDT
    On the asymptotic stability of shear flows and vortices
    11th Floor Lecture Hall
    • Alexandru Ionescu, Princeton University
    Abstract
    I will talk about some recent work on the global linear and nonlinear asymptotic stability of two families of solutions of the 2D Euler equations: shear flows on bounded channels and vortices in the plane. This is joint work with Hao Jia.
  • 3:00 - 4:30 pm EDT
    Reception
    Hemenway's Patio (weather permitting)
Tuesday, September 21, 2021
  • 10:00 - 10:40 am EDT
    Small scale formations in the incompressible porous media equation
    11th Floor Conference Room
    • Virtual Speaker
    • Yao Yao, Georgia Tech
    Abstract
    The incompressible porous media (IPM) equation describes the evolution of density transported by an incompressible velocity field given by Darcy’s law. Here the velocity field is related to the density via a singular integral operator, which is analogous to the 2D SQG equation. The question of global regularity vs finite-time blow-up remains open for smooth initial data, although numerical evidence suggests that small scale formation can happen as time goes to infinity. In this talk, I will discuss rigorous examples of small scale formations in the IPM equation: we construct solutions to IPM that exhibit infinite-in-time growth of Sobolev norms, provided that they remain globally smooth in time. As an application, this allows us to obtain nonlinear instability of certain stratified steady states of IPM. This is a joint work with Alexander Kiselev.
  • 10:55 - 11:15 am EDT
    Coffee Break
    11th Floor Collaborative Space
  • 11:15 am - 12:00 pm EDT
    PINNs and DeepOnets for Wave Systems
    11th Floor Lecture Hall
    • Virtual Speaker
    • George Karniadakis, Brown University
  • 12:15 - 1:45 pm EDT
    Lunch/Free Time
  • 1:45 - 2:25 pm EDT
    The second boundary value problem for a discrete Monge-Ampere equation
    11th Floor Conference Room
    • Gerard Awanou, University of Illinois, Chicago
    Abstract
    In this work we propose a natural discretization of the second boundary condition for the Monge-Ampere equation of geometric optics and optimal transport. It is the natural generalization of the popular Oliker-Prussner method proposed in 1988. For the discretization of the differential operator, we use a discrete analogue of the subdifferential. Existence, unicity and stability of the solutions to the discrete problem are established. Convergence results to the continuous problem are given.
  • 2:40 - 3:45 pm EDT
    Lightning Talks
    11th Floor Lecture Hall
  • 3:45 - 4:15 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Wednesday, September 22, 2021
  • 10:00 - 10:40 am EDT
    Efficient and accurate structure preserving schemes for complex nonlinear systems
    11th Floor Conference Room
    • Jie Shen, Purdue University
    Abstract
    Many complex nonlinear systems have intrinsic structures such as energy dissipation or conservation, and/or positivity/maximum principle preserving. It is desirable, sometimes necessary, to preserve these structures in a numerical scheme. I will present some recent advances on using the scalar auxiliary variable (SAV) approach to develop highly efficient and accurate structure preserving schemes for a large class of complex nonlinear systems. These schemes can preserve energy dissipation/conservation as well as other global constraints and/or are positivity/bound preserving, only require solving decoupled linear equations with constant coefficients at each time step, and can achieve higher-order accuracy.
  • 10:55 - 11:15 am EDT
    Coffee Break
    11th Floor Collaborative Space
  • 11:15 am - 12:00 pm EDT
    Energy growth for the Schrödinger map and the binormal flow
    11th Floor Lecture Hall
    • Valeria Banica, Sorbonne Université
    Abstract
    In this talk I shall present a result of blow up of a density energy associated to the Schrödinger map and the binormal flow, a classical model for the dynamics of vortex filaments in Euler equations. This is a joint work with Luis Vega.
  • 12:15 - 1:45 pm EDT
    Lunch/Free Time
  • 1:45 - 2:25 pm EDT
    Water Waves with Background Flow over Obstacles and Topography
    11th Floor Lecture Hall
    • Virtual Speaker
    • Jon Wilkening, University of California, Berkeley
    Abstract
    We present two accurate and efficient algorithms for solving the incompressible, irrotational Euler equations with a free surface in two dimensions with background flow over a periodic, multiply-connected fluid domain that includes stationary obstacles and variable bottom topography. One approach is formulated in terms of the surface velocity potential while the other evolves the vortex sheet strength. Both methods employ layer potentials in the form of periodized Cauchy integrals to compute the normal velocity of the free surface. We also propose a new algorithm to dynamically vary the spacing of gridpoints on the free surface to efficiently resolve regions of high curvature as they develop. We study singularity formation and capillary effects and compare our numerical results with lab experiments.
  • 2:45 - 3:15 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 3:15 - 3:55 pm EDT
    Topological Origin of Certain Fluid and Plasma Waves
    11th Floor Lecture Hall
    • Brad Marston, Brown University
    Abstract
    Symmetries and topology play central roles in our understanding of physical systems. Topology, for instance, explains the precise quantization of the Hall effect and the protection of surface states in topological insulators against scattering from disorder or bumps. However discrete symmetries and topology have so far played little role in thinking about the fluid dynamics of oceans and atmospheres. In this talk I show that, as a consequence of the rotation of the Earth that breaks time reversal symmetry, equatorially trapped Kelvin and Yanai waves emerge as topologically protected edge modes. The non-trivial structure of the bulk Poincare ́ waves encoded through the first Chern number of value 2 guarantees the existence of these waves. Thus the oceans and atmosphere of Earth naturally share basic physics with topological insulators. As equatorially trapped Kelvin waves in the Pacific ocean are an important component of El Niño Southern Oscillation and other climate oscillations, these new results demonstrate that topology plays a surprising role in Earth’s climate system. We also predict that waves of topological origin will arise in magnetized plasmas. A planned experiment at UCLA’s Basic Plasma Science Facility to look for the waves is described.
Thursday, September 23, 2021
  • 10:00 - 10:40 am EDT
    Resonances as a Computational Tool
    11th Floor Lecture Hall
    • Katharina Schratz, Heriot-Watt University
  • 10:55 - 11:15 am EDT
    Coffee Break
    11th Floor Collaborative Space
  • 11:15 am - 12:00 pm EDT
    Efficient time-stepping methods for the rotating shallow water equations
    11th Floor Lecture Hall
    • Virtual Speaker
    • Thi- Thao Phuong- Hoang, Auburn University
    Abstract
    Numerical modeling of geophysical flows is challenging due to the presence of various coupled processes that occur at different spatial and temporal scales. It is critical for the numerical schemes to capture such a wide range of scales in both space and time to produce accurate and robust simulations over long time horizons.
    In this talk, we will discuss efficient time-stepping methods for the rotating shallow water equations discretized on spatial meshes with variable resolutions. Two different approaches will be considered: the first approach is a fully explicit local time-stepping algorithm based on the strong stability preserving Runge-Kutta schemes, which allows different time step sizes in different regions of the computational domain. The second approach, namely the localized exponential time differencing method, is based on spatial domain decomposition and exponential time integrators, which makes possible the use of much larger time step sizes compared to explicit schemes and avoids solving nonlinear systems. Numerical results on various test cases will be presented to demonstrate the performance of the proposed methods.
  • 12:15 - 1:45 pm EDT
    Lunch/Free Time
  • 1:45 - 2:25 pm EDT
    Dynamics in particle suspension flow
    11th Floor Conference Room
    • Li Wang, University of Minnesota
    Abstract
    In this talk, I will consider two set up of particle suspension flow. One is a gravity driven flow down an incline, and the other is a pressure driven flow in a Hele-Shaw cell. In the former case, the interesting phenomena is the formation of singular shock that appears in the high particle concentration case that relates to the particle-rich ridge observed in the experiments. We analyze the formation of the singular shock as well as its local structure. In the latter case, we rationalize a self-similar accumulation of particles at the interface between suspension and air. Our results demonstrate that the combination of the shear- induced migration, the advancing fluid-fluid interface, and Taylor dispersion yield the self-similar and gradual accumulation of particles.
  • 2:45 - 3:15 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 3:15 - 3:55 pm EDT
    Separatrix crossing and symmetry breaking in NLSE-like systems due to forcing and damping
    11th Floor Conference Room
    • Debbie Eeltink, MIT
    Abstract
    The nonlinear Schrödinger equation (NLSE) is a workhorse for many different fields (e.g. optical fibers, Bose-Einstein condensates, water waves). It describes the evolution of the envelope of a field in time or space, taking into account the nonlinear interaction of the components of the spectrum of the envelope. While the NLSE is well-studied in its conservative form, a relevant question to ask is how does it respond to damping and forcing? Limiting the spectrum to only three components allows one to construct a phase-space for the NLSE, spanned by the relative phase of the sidebands, and the energy fraction in the sidebands. Using wave-tank measurements, we show that forcing and damping the NLSE induces separatrix crossing: switching from one solution-type to the other in the phase-space. Our experiments are performed on deep water waves, which are better described by the higher-order NLSE, the Dysthe equation. We, therefore, extend our three-wave analysis to this system. However, our conclusions are general as the dynamics are driven by the leading order terms. To our knowledge, it is the first phase evolution extraction from water-wave measurements. Furthermore, we observe a growth and decay cycle for modulated plane waves that are conventionally considered stable. Finally, we give a theoretical demonstration that forcing the NLSE system can induce symmetry breaking during the evolution.
Friday, September 24, 2021
  • 10:00 - 10:40 am EDT
    Extreme Wave Events in Reflective Environments
    11th Floor Lecture Hall
    • Amin Chabchoub, University of Sydney
    Abstract
    Waves dynamics in coastal zones is known to comprise incident and reflective wave motion. We report an experimental study in which several incident JONSWAP wave trains have been generated in a uni-directional water wave tank while the artificial beach inclination and its permeability have been varied to allow a variety of reflective wave conditions. Key statistical features obtained from an adaptive coupled nonlinear Schrödinger model simulations show an excellent agreement with the laboratory data collected near the beach.
  • 10:55 - 11:15 am EDT
    Coffee Break
    11th Floor Collaborative Space
  • 11:15 - 11:55 am EDT
    Symmetry in stationary and uniformly rotating solutions of the Euler equations
    11th Floor Lecture Hall
    • Javier Gomez Serrano, Princeton University
  • 12:10 - 12:50 pm EDT
    High order strong stability preserving multi-derivative implicit and IMEX Runge--Kutta methods with asymptotic preserving properties
    11th Floor Lecture Hall
    • Sigal Gottlieb, University of Massachusetts Dartmouth
    Abstract
    In this talk we present a class of high order unconditionally strong stability preserving (SSP) implicit two-derivative Runge--Kutta schemes, and SSP implicit-explicit (IMEX) multi-derivative Runge--Kutta schemes where the time-step restriction is independent of the stiff term. The unconditional SSP property for a method of order $p>2$ is unique among SSP methods, and depends on a backward-in-time assumption on the derivative of the operator. We show that this backward derivative condition is satisfied in many relevant cases where SSP IMEX schemes are desired. We devise unconditionally SSP implicit Runge--Kutta schemes of order up to $p=4$, and IMEX Runge--Kutta schemes of order up to $p=3$. For the multi-derivative IMEX schemes, we also derive and present the order conditions, which have not appeared previously. The unconditional SSP condition ensures that these methods are positivity preserving, and we present sufficient conditions under which such methods are also asymptotic preserving when applied to a range of problems, including a hyperbolic relaxation system, the Broadwell model, and the Bhatnagar-Gross-Krook (BGK) kinetic equation.

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Poster Session Gallery

Smooth Self-Similar Solutions to the Compressible Euler Equations

Anxo Biasi

Computer-assisted-proofs of dynamics in a nonconservative NLS

Jonathan Jaquette

Associated Semester Workshops

Hamiltonian Methods and Asymptotic Dynamics
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