University of Illinois Urbana-Champaign

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

Jegdic, Katarina S.

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Description

Title
Analysis of a Spacetime Discontinuous Galerkin Method for Systems of Conservation Laws
Author(s)
Jegdic, Katarina S.
Issue Date
2004
Doctoral Committee Chair(s)
Robert Jerrard
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Applied Mechanics
Language
eng
Abstract
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.
Graduation Semester
2004
Type of Resource
text
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