Analysis of a Spacetime Discontinuous Galerkin Method for Systems of Conservation Laws

The focus of this dissertation is analysis of a causal spacetime discontinuous Galerkin (CSDG) method for one-dimensional hyperbolic systems of conservation laws. The CSDG method is based on causal spacetime discretizations, and the Galerkin basis herewith consists of piecewise constant functions. The method can be used on layered as well as on unstructured spacetime grids, and is well-suited for adaptive meshing and parallelization. Its formulation is consistent with the weak formulation of conservation laws and it naturally suggests a discrete version of the Lax entropy condition. The CSDG method is an explicit method, that is, a direct element-by-element solution procedure is possible. We investigate existence, uniqueness and continuous dependence on the initial data of a CSDG solution for general hyperbolic systems of conservation laws with Lipschitz continuous spatial flux function. We prove that if a CSDG solution exists, then it must satisfy discrete entropy inequalities. Our main result is for genuinely nonlinear Temple class systems. We show that given a causal spacetime mesh, a unique CSDG solution satisfying local Riemann invariant bounds exists. This enables us to prove convergence of the CSDG method to a weak solution which obeys certain entropy inequalities.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2015

Author dc:creator

• Jegdic, Katarina S.

Contributors dc:contributor

• Robert Jerrard

• Applied Mechanics

Language dc:language

eng

Identifier

(MiAaPQ)AAI3153332

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/86837