Vortrag im Oberseminar Numerik und Optimierung am 24. Oktober 2018
Im Rahmen des Oberseminares "Numerik und Optimierung" hält
Herr Jan Glaubitz (TU Braunschweig)
am Mittwoch, 24. Oktober 2018, einen Vortrag mit dem Titel
Shock capturing in high-order methods for conservation laws
Beginn: 16:30 Uhr, Raum 25.22.02.81
Alle Interessierten sind herzlich willkommen!
Many fundamental physical principles can be described by conservation laws. These types of time dependent partial differential equations can be used as models in fluid dynamics, electrodynamics, and space and plasma physics.
Traditionally, low-order numerical schemes, for instance classical Finite Volume (FV) methods, have been used to solve hyperbolic conservation laws, in particular in industrial applications. But since they become quite costly for high accuracy and long time simulations, there is a rising demand of high-order methods. Yet, all high-order methods for hyperbolic conservation laws will fail to provide physically reasonable solutions without some kind of shock capturing. Due to the Gibbs-Wilbraham phenomenon, the numerical approximation will be polluted by spurious oscillations, which produce unphysical numerical solutions and might finally blow up the computation.
In this talk we discuss artificial viscosity methods to stabilise high-order methods, in particular Discontinuous Galerkin methods. The idea behind these methods is to add dissipation to the numerical solution. This idea dates back to the pioneering work of von Neumann and Richtmyer during the Manhattan project in the 1940's at Los Almos National Laboratory. Since then, artificial viscosity methods have attracted the interest of many researchers. By thoughtfully revisiting the development of some of the most commonly used artificial viscosity methods, we are able to pinpoint certain drawbacks of these methods and to formulate precise criteria on the viscosity terms for properties such as conservation and entropy stability to hold. As a result, we construct a class of novel artificial viscosity methods with favorable properties. Finally, we close this talk by addressing inherent drawbacks of all artificial viscosity methods and provide a prospect of novel shock-capturing methods.