Structure-Preserving Methods for Nonlinear Hyperbolic Waves

Applied and Computational Mathematics Seminar
Monday, February 5, 2024 - 2:00pm for 1 hour (actually 50 minutes)
Skiles 005 and
Philippe G. LeFloch – Sorbonne University and CNRS – pglefloch@gmail.com
Yingjie Liu

Many numerical methods have been developed in the past years for computing weak solutions (with shock waves) to nonlinear hyperbolic conservation laws. My research, specifically, concerns the design of well-balanced numerical algorithms that preserve certain key structure of these equations in various applications, including for problems involving moving phase boundaries and other scale-dependent interfaces. In particular, in this lecture, I will focus on the evolution of a compressible fluid in spherical symmetry on a Schwarzschild curved background, for which I have designed a class of well-balanced numerical algorithms up to third-order of accuracy. Both the relativistic Burgers-Schwarzschild model and the relativistic Euler-Schwarzschild model were considered, and the proposed numerical algorithm took advantage of the explicit or implicit forms available for the stationary solutions of these models. The schemes follow the finite volume methodology and preserve the stationary solutions and, most importantly, allow us to investigate the global asymptotic behavior of such flows and determine the asymptotic behavior of the mass density and velocity field of the fluid. Blog: