Numerical Analysis Seminar

Date: November 15, 2023

Time: 3:00PM - 4:00PM

Location: BLOC 302

Speaker: Yulong Xing, Ohio State University

Title: High Order Structure Preserving Numerical Methods for Euler Equations with Gravitation

Abstract: Hydrodynamical evolution in a gravitational field arises in many astrophysical and atmospheric problems. In this presentation, we will talk about high order structure preserving methods for the Euler equations under gravitational fields, which can exactly preserve some fundamental continuum properties of the underlying problems in the discrete level. We consider the Euler–Poisson equations in spherical symmetry with an equilibrium state governed by the Lane–Emden equation, and design well-balanced (WB) and total-energy-conserving (TEC) discontinuous Galerkin finite element methods. High order semi-implicit well-balanced asymptotic preserving (AP) finite difference scheme, for all Mach Euler equations with gravitation, may also be discussed. Extensive numerical examples — including a toy model of stellar core-collapse with a phenomenological equation of state that results in core-bounce and shock formation — are provided to verify the well-balanced property, positivity-preserving property, high-order accuracy, total energy conservation and good resolution for both smooth and discontinuous solutions.