Applied Seminar: Dr. Ignacio Tomas
Title: Solving hyperbolic systems of Conservation laws: fundamentals in the shock-hydrodynamics regime.
Author: Ignacio Tomas, CCR, Sandia National Laboratories.
Abstract: In this talk, we will discuss the fundamentals behind the numerical solution of hyperbolic systems of conservation laws in more than one space dimension with emphasis on the shock-hydrodynamics regime. The first step in the development of a high-order scheme is the development of a robust first-order method (ideally) endowed with a sound theoretical basis. We develop a general framework of first-order fully-discrete numerical schemes that are guaranteed to preserve every convex invariant of the hyperbolic system and satisfy every entropy inequality. This framework is not tied to any particular space discretization technique. We then proceed to present a new flux-limiting (hybridization) technique. This technique does not preserve or enforce bounds on conserved variables, but rather bounds on quasi-convex/concave functionals of the conserved variables. This flux-limiting technique, is suitable to enforce/preserve convex constraints of the numerical solution which are natural to hyperbolic systems. Examples of such bounds are positivity of the internal energy and the minimum principle of the specific entropy in the context of Euler’s equations. The resulting second-order schemes are theoretically guaranteed to work always, all the time, no exception. The presentation is will be geared towards non-experts and graduate students.
Contact Name: Jacob Schroder