Partial Differential Equations Conference

Abstracts of Plenary Speakers

Alina Chertock

Asymptotic preserving numerical methods for singularly perturbed problems


Solutions of many nonlinear PDE systems reveal a multiscale character; thus, their numerical resolution presents some major difficulties. Such problems are typically characterized by a small parameter representing, say, a low Mach or Fraude number. In the limiting regimes, the propagation speeds are very low, and therefore the use of explicit numerical methods would require very restrictive time and space discretization steps due to the CFL condition and restrictions on the smallness of numerical diffusion. This becomes rapidly too costly from a practical point of view, and consequently, numerical solutions for small parameter values may be out of reach. Moreover, standard implicit schemes, which will be uniformly stable, may be inconsistent with the limiting problem and may provide a wrong solution in the zero limits. Thus, designing robust numerical algorithms whose accuracy and efficiency are independent of the values of the small parameter is an important and challenging task. A widely used numerical approach applicable in all-speed regimes is based on asymptotic preserving (AP) numerical methods. AP methods guarantee that for a fixed mesh size and time step, the numerical scheme should automatically transform into a consistent and stable discretization of the limiting system.

In this talk, we will present several AP schemes for Navier–Stokes–Korteweg equation, rotational shallow water equations with Coriolis, and, if time permits, kinetic equations with singular limits.

Irene Fonseca

Phase separation in heterogeneous media


Modern technologies and biological systems, such as temperature-responsive polymers and lipid rafts, take advantage of engineered inclusions, or natural heterogeneities of the medium, to obtain novel composite materials with specific physical properties. To model such situations by using a variational approach based on the gradient theory, the potential and the wells may to depend on the spatial position, even in a discontinuous way, and different regimes should be considered. In the critical case case where the scale of the small heterogeneities is of the same order of the scale governing the phase transition and the wells are fixed, the interaction between homogenization and the phase transitions process leads to an anisotropic interfacial energy. In the subcritical case with moving wells where the heterogeneities of the material are of a larger scale than that of the diffuse interface between different phases, it is observed that there is no macroscopic phase separation and that thermal fluctuations play a role in the formation of nanodomains. The supercritical case for fixed wells is also addressed, and a partial characterization of the limit energy is given.

This is joint work with Riccardo Cristoferi (Radboud University, The Netherlands) and Likhit Ganedi (Aachen University, Germany), based on previous results also obtained with Adrian Hagerty (USA) and Cristina Popovici (USA).

Peter Monk

Development of a coupled Trefftz and finite element method for approximating Maxwell’s equations

Abstract: The talk focuses on developing a discontinuous Galerkin method for the time harmonic Maxwell system. This method is based on the use of a finite element grid, but uses plane wave solutions of Maxwell’s equations on each element to approximate the global field. Because each basis function satisfies Maxwell’s equations, the problem can be reduced to a coupled linear system on the faces of the grid. Arbitrarily high order of convergence can be achieved by taking more planes waves in suitable directions element by element, although ill-conditioning must be carefully controlled. Unfortunately this method has severe deficiencies when applied to some problems involving screens and transmission lines. To remedy this, we have coupled polynomial finite element methods with the plane wave scheme. I shall report on the history of Trefftz-DG methods and the current state of this effort.

Chi-Wang Shu

Stability of time discretizations for semi-discrete high order schemes for time-dependent PDEs

Abstract: In scientific and engineering computing, we encounter time-dependent partial differential equations (PDEs) frequently.  When designing high order schemes for solving these time-dependent PDEs, we often first develop semi-discrete schemes paying attention only to spatial discretizations and leaving time $t$ continuous.  It is then important to have a high order time discretization to main the stability properties of the semi-discrete schemes.  In this talk we discuss several classes of high order time discretization, including the strong stability preserving (SSP) time discretization, which preserves strong stability from a stable spatial discretization with Euler forward, the implicit-explicit (IMEX) Runge-Kutta or multi-step time marching, which treats the more stiff term (e.g. diffusion term in a convection-diffusion equation) implicitly and the less stiff term (e.g. the convection term in such an equation) explicitly, for which strong stability can be proved under the condition that the time step is upper-bounded by a constant under suitable conditions, the explicit-implicit-null (EIN) time marching, which adds a linear highest derivative term to both sides of the PDE and then uses IMEX time marching, and is particularly suitable for high order PDEs with leading nonlinear terms, and the explicit Runge-Kutta methods, for which strong stability can be proved in many cases for semi-negative linear semi-discrete schemes.  Numerical examples will be given to demonstrate the performance of these schemes.

Qiang Du

Nonlocal modeling, analysis and computation: some recent development

Abstract:Nonlocality has become increasingly prominent in nature, leading to the development of new mathematical theories to model and simulate its impact. In this lecture, we will concentrate on nonlocal models that involve interactions with a finite horizon, examining their significance in understanding phenomena involving potential anomalies, singularities, and other effects that arise from nonlocal interactions. Furthermore, we will present recent analytical studies that explore nonlocal operators and function spaces, discussing how they contribute to the development of robust numerical algorithms.

Thomas Bartsch

Normalized solutions to nonlinear Schrödinger equations with potential

Abstract: The talk will be concerned with the existence of solutions of the non-linear Schrödinger equation –Δu + V(x)u + λu = f(u) on R^N or on domains Ω ⊂ R^N when the L^2-norm of the solution is prescribed. The problem of normalized solutions has received a lot of attention in recent years but still much less understood compared with the problem when the L^2 norms are free. We discuss this problem and present recent results. The talk is based on work with Riccardo Molle, Matteo Rizzi, Gianmaria Verzini, Shijie Qi and Wenming Zou.

Manuel Del Pino

Dynamics of concentrated vorticities in 2D and 3D Euler flows

Abstract: A classical problem that traces back to Helmholtz and Kirchhoff is the understanding of the dynamics of solutions to the Euler equations of an inviscid incompressible fluid when the vorticity of the solution is initially concentrated near isolated points in 2d or vortex lines in 3d. We discuss some recent results on these solutions’ existence and asymptotic behavior. We describe, with precise asymptotics, interacting vortices, and traveling helices, and extension of these results for the 2d generalized SQG. This is research in collaboration with J. Dávila, A. Fernández, M. Musso, and J. Wei.