# Fluid Mechanics and Waves Seminar - Fall 2023

Seminars are held on Mondays from 2:30 - 3:30PM in CULM 611, unless otherwise noted.

For questions about the seminar schedule, please contact Thi Phong Nguyen.

**September 25**

**Manas Rachh**, Flatiron Institute

**Location: **CULM 611

**Edge Effects at Insulator Interfaces**

In this talk, we will discuss edge effects at insulator interfaces for two models: Klein-Gordon singular waveguides and the Dirac equation. In many physical regimes of interest, these setups tend to have waves that propagate along the interface of two bulk materials, even when these materials are insulating, i.e. waves in the bulk are evanescent. The interface waves present difficulties in the form of non-uniqueness for equivalent integral equation formulations. We will discuss two different methods to mediate this issue: a preconditioner based on a surface Helmholtzian, and a complex scaling approach which deforms the integral equation into the complex plane. We show that both of these approaches result in modified integral equations whose solutions are numerically compactly supported and thus can be efficiently discretized using standard numerical solvers. We demonstrate the efficiency of both these methods through several numerical examples implemented in chunkie: an easy-to-use software package in MATLAB with state-of-the-art integral equation solvers.

**October 9**

**Arnold D. Kim, **UC Merced

**Location: **Webex

**Quantitative SAR Imaging of Dispersive Targets**

We introduce a dispersive point target model and extend the synthetic aperture radar (SAR) imaging problem to (i) identify and locate these targets and (ii) recover their frequency-dependent reflectivities. We show that Kirchhoff migration (KM) is able to identify dispersive point targets in an imaging region. However, KM predicts target locations that are shifted in range from their true locations. Because of this range shift, we cannot recover the complex-valued frequency dependent reflectivity, but we can recover its absolute value and hence its radar cross-section (RCS). After showing results for an unbounded homogeneous medium, we consider dispersive targets situated below an interface separating two adjacent half spaces as a model for the buried landmine problem.

**October 23**

**Thi-Thao-Phuong Hoang, **Auburn University

**Location: **Zoom

**Domain Decomposition Methods for Dimensionally Reduced Models of Flow and Transport in Fractured Porous Media**

The numerical simulation of flow and transport in a porous medium with fractures is challenging due to the presence of multiple spatial and temporal scales and strong heterogeneity of the domain of calculation. A fracture may have much higher or much lower permeability than that in the surrounding medium, thus the time scales in the fractures and in the rock matrix may vary significantly. In addition, the width of the fracture is much smaller than any reasonable parameter of spatial discretization. To tackle those challenges, we first consider a dimensionally reduced fracture model, where the fracture is treated as a hypersurface embedded in the porous medium. Then we develop domain decomposition (DD) methods that allow different time step sizes in the subdomains and in the fracture. For each DD method, a space-time interface problem is formulated, based on either the physical or optimized transmission conditions imposed on the fracture-interface. Such an interface problem is solved iteratively and globally in time. The proposed methods are fully implicit and are designed to converge fast while preserving the accuracy in time with nonconforming time grids. Convergence analysis as well as numerical results will be presented for the flow problem of a single phase, compressible fluid, and the linear advection-diffusion equation with mixed formulations.

**November 6**

**Montanelli Hadrien**, INRIA and Institute Polytechnique de Paris (France)

**Location:** Zoom

**The Linear Sampling Method for Random Sources**

We present in this talk an extension of the linear sampling method for solving the sound-soft inverse acoustic scattering problem with randomly distributed point sources. The theoretical justification of our method is based on the Helmholtz–Kirchhoff identity, the cross-correlation between measurements, and the volume and imaginary near-field operators. Implementations in MATLAB using boundary elements, the SVD, Tikhonov regularization, and Morozov’s discrepancy principle will also be discussed.

**November 20**

**Ruming Zhang,** TU Berlin

**Location: **Zoom

**Monotonicity-based Shape Reconstruction for an Inverse Scattering Problem in a Waveguide**

We consider an inverse medium scattering problem for the Helmholtz equation in a closed cylindrical waveguide with penetrable compactly supported scattering objects. We develop novel monotonicity relations for the eigenvalues of an associated modified near field operator, and we use them to establish linearized monotonicity tests that characterize the support of the scatterers in terms of near field observations of the corresponding scattered waves. The proofs of these shape characterizations rely on the existence of localized wave functions, which are solutions to the scattering problem in the waveguide that have arbitrarily large norm in some prescribed region, while at the same time having arbitrarily small norm in some other prescribed region. As a byproduct we obtain a uniqueness result for the inverse medium scattering problem in the waveguide with a simple proof. Some numerical examples are presented to document the potentials and limitations of this approach.

**November 27**

**Ianto Cannon,** Okinawa Institute of Science & Technology

**Location: **CULM 611 and Zoom

**Morphology of Droplets in Turbulent Flows**

Using simulations, we look at the shapes of surfactant-laden droplets in homogeneous isotropic turbulence. Particularly in relation to the Hinze scale; the length scale at which turbulent fluctuations balance surface tension forces. We find that droplets smaller than the Hinze scale have ellipsoid-like shapes, while larger droplets are long and filamentous. To extend our analysis of the filamentous droplets, we measure the Euler characteristic of their surfaces.

**December 4**

**Steven Roberts, **LLNL

**Location: **CULM 611 and Zoom

**The Order of Runge-Kutta Methods in Theory and Practice**

Runge-Kutta methods are one of the most popular families of integrators for solving ordinary differential equations, essential in simulating dynamic systems arising in physics, engineering, biology, and various other fields. Unfortunately, classical error analysis for Runge-Kutta methods relies on assumptions that rarely hold when solving stiff ordinary differential equations (ODEs): an asymptotically small timestep and a right-hand side function with a moderate Lipschitz constant. Without idyllic assumptions, Runge-Kutta methods can experience a problematic degradation in accuracy known as order reduction. While high stage order remedies order reduction, it is only viable for expensive, fully implicit Runge-Kutta methods. In this talk, I will discuss some recent advancements in deriving practical Runge-Kutta methods that avoid order reduction. Initially, the focus will be on explicit methods applied to linear ODEs, where we have found a systematic approach to construct schemes of arbitrarily high order. Then, we will expand to classes of nonlinear ODEs where I will present new, stiff order conditions with a rich and interesting connection to rooted trees.

*Updated November 20, 2023*