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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.
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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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.
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.
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.
