WPI-AIMR Math Unit

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AIMR Math Group Seminar 2024/10/15

Date : October. 15th (Tue) 2024 15:30-17:25
Venue: AIMR, common space in 3C

1. 15:35-16:15

Speaker: Ilaria Perugia (University of Vienna)

Title: Structure-preserving discretization of cross-diffusion systems

Abstract:

Several applications in physics, biology, and chemistry involve systems with multiple components, such as gas mixtures, competing population species, and reacting chemical substances. These problems are modeled with nonlinear reaction-diffusion equations that include cross-diffusion terms. Cross diffusion occurs when the flux of one component is driven by the gradient of another component. The main challenges in designing numerical methods for approximating nonlinear cross-diffusion systems are that the diffusion matrix may not be symmetric or positive semidefinite, and that a maximum principle may not be available. In this talk, we present numerical methods based on the boundedness-by-entropy framework introduced by A. Jüngel in 2015. Motivated by the inherent entropy structure of the PDE system, nonlinear transformations involving the entropy variable allow for the enforcement of positivity in the approximate solutions. Specifically, we focus on a Local Discontinuous Galerkin method. By appropriately introducing auxiliary variables, the problem is reformulated so that nonlinearities do not appear within differential operators or interface terms. This results in nonlinear operators that can be naturally evaluated in parallel. The method allows for arbitrary degrees of approximation in space, preserves boundedness of the physical unknowns without requiring postprocessing or slope limiters, and satisfies a discrete version of the entropy stability estimate of the continuous problem.

2. 16:35-17:15

Speaker: Hirofumi Notsu (Kanazawa University)

Title: Adaptive stabilized Lagrange-Galerkin methods: Theory and applications

Abstract:

In flow problems, we often employ small mesh sizes locally to deal with a boundary layer or an interface. Then, due to the CFL condition, we need a small time step size depending on the smallest mesh size. Similarly, when introducing an adaptive mesh refinement technique, we should consider the CFL condition, which requires a small time step size. To overcome this restriction, we employ the Lagrange-Galerkin (LG) method, i.e., a finite element method combined with the method of characteristics. Since the material derivative is discretized along the trajectory of a fluid particle and the function values at upwind points are directly used, the LG method is CFL-free, at least for linear convection-diffusion/Oseen equations, and we can employ a time step size independent of the mesh size. In this talk, we present error estimates of the LG methods for the linear equations and the Navier-Stokes equations. We also present numerical results of their applications to complex flow problems.

*Register here

*The seminar is co-hosted by MathCCS Seminar.

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AIMR Math Group Seminar

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Speaker: Dr. Kazuki Koga

Advanced Institute for Materials Research Tohoku University｜Mathematical Science Group

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AIMR Math Group Seminar 2024/10/15

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　▼Research Subjects 　▼Members 　▼Press Releases 　▼Seminar & Workshop 　▼Past Activities 　▼News & Informations 　▼Access & Contact 　▼Links