Structure preserving schemes and kinetic models for approximating measure valued solutions of hyperbolic equations

Gkanis, Ioannis (2021) Structure preserving schemes and kinetic models for approximating measure valued solutions of hyperbolic equations. Doctoral thesis (PhD), University of Sussex.

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Abstract

In this thesis we consider approximate schemes and models for hyperbolic conservation laws. Systems of conservation laws are fundamental mathematical models and have received a lot of attention from the point of view of analysis, modelling and computations. They include the wave equations in elastic media and fundamental equations in fluid mechanics. We consider structure preserving schemes and kinetic models for approximating measure valued solutions of hyperbolic equations. Such solutions are of interest given their application to problems in uncertainty quantification and in statistical inference. This thesis contains new results on (i) the design of new schemes for the computation of entropy consistent approximations, with particular emphasis on the consistency of the computational algorithms to entropic measure valued solutions for HCL, (ii) the introduction of discrete and generalised kinetic models designed to directly approximate measure valued solutions by using a combination of approximate Young measures and the kinetic formulation of the conservation law and (iii) stability analysis of generalised viscus kinetic models. We obtain uniqueness within a particular class of vanishing viscosity limits of these models and of their corresponding measure valued solutions.

Item Type: Thesis (Doctoral)
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Subjects: Q Science > QA Mathematics > QA0299 Analysis. Including analytical methods connected with physical problems > QA0371 Differential equations > QA0377 Partial differential equations (second and higher orders)
Depositing User: Library Cataloguing
Date Deposited: 25 Jan 2021 11:48
Last Modified: 03 Feb 2021 14:38
URI: http://sro.sussex.ac.uk/id/eprint/96731

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