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April 12 Math Colloquium

He Yang, Augusta University

Discontinuous Galerkin Methods for Relativistic Vlasov-Maxwell System and Khokhlov-Zabolotskaya-Kuznetzov Equation

In this talk, I will introduce our proposed discontinuous Galerkin (DG) methods to solve Relativistic Vlasov-Maxwell (RVM) system and Khokhlov-Zabolotskaya-Kuznetzov (KZK) equation. The RVM system is a kinetic model that describes the dynamics of plasma when the charged particles move in the relativistic regime and their collisions are not important. In the first part of my talk, I will introduce our proposed methods which preserve the structural properties of the RVM system, i.e., positivity of the particle number density function, mass and energy conservation. I will introduce some theoretical results and implementation issues of our methods. Numerical experiments, including streaming Weibel instability and wakefield acceleration, will also be presented to demonstrate the performance of our methods.
KZK equation is a model that describes the propagation of the ultrasound beams in the thermo-viscous fluid. Accurate numerical methods to simulate the KZK equation are important to its broad applications in medical ultrasound simulations. In the second part of my talk, I will discuss our proposed local discontinuous Galerkin method to solve the KZK equation. I will show numerical stability and a series of numerical experiments including the focused circular short tone burst excitation and the propagation of unfocused sound beams, which show that our scheme leads to accurate solutions and outperforms the benchmark solutions in the literature.

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