Past Event: Oden Institute Seminar

Analysis of Two Schemes for Kinetic Equations

Andres Galindo-Olarte, O'Donell postdoctoral fellow at The Oden Institute

Abstract

This talk will be divided in two parts. In the first, we will establish negative-order estimates for the accuracy of discontinuous Galerkin (DG) approximations to smooth solutions for the Vlasov-Maxwell (VM) system of equations. For the approximated solutions, we are able to extract this “hidden accuracy” through the use of a Smooth-Increasing Accuracy-Conserving (SIAC) filter which is a convolution kernel that is composed of a linear combination of B-splines. We provide rigorous error estimates for the DG solutions that show improvement to (2k + 1/2)-th order in the negative-order norm.  

In the second part we will analyze a hybrid method for radiation transport.  The method that consists of (i) splitting the kinetic equation into collisional and non-collisional components; (ii) applying a high-order method to the non-collisional component and a low order method for the collisional component; and (iii) re-partitioning the kinetic distribution after each time step in the algorithm.  In practice the hybrid method yields a more efficient method and provides a level of accuracy that is comparable to a uniform high-order treatment of the entire system.  We provide for the first-time rigorous estimates for the hybrid method when we perform semi-discretization in angle in the collided component.

Biography

Andrés is an Odonell postdoctoral fellow working with Dr. Irene M Gamba at the Oden Institute for Computational Engineering and Sciences at the University of Texas at Austin. He obtained his Ph.D. in Applied Mathematics from Michigan State University under the direction of Dr. Yingda Cheng. His research aims to develop and analyze numerical methods for solving kinetic equations from plasma physics. Here, Andrés is currently working on developing a new discontinuous Galerkin method for the  Fokker-Planck-Landau equation.