Abstract
In this work, we developed a geometric computational method for electromagnetic wave problems. The method unifies the spatial and temporal discretizations and produces a fourdimensional spacetime computational scheme, which treats spatial and time dimensions equally.
This new method seeks primarily to develop the current geometric methods, i.e., the finite integration technique and the cell method. We introduce some improvements to these methods and, additionally, develop tools for further development. For these, we present a couple of challenge problems and largely solve them with our method in this thesis.
There are several reasons why we focus on the mathematical structures underlying the spacetime and electromagnetic models. For example, first, fourdimensional fields cannot be modeled with elementary vector algebra, and, second, a functional computational method is usually based on a solid mathematical foundation. From the background of the theory of relativity, we develop a model for spacetime. Because this model includes an indefinite metric, all metricdependent concepts must agree with this nonRiemannian indefinite metric. Consequently, we utilize concepts new to electromagnetic computation. And because our target is electromagnetic computation, we choose a model that agrees well with computation: we model electromagnetic fields with cochains.
In the end, we introduce a computational method that makes use of a pair of fourdimensional Lorentzian orthogonal meshes. In developing it, we concentrate on the constitutive relations, boundary conditions, and gauging methods and implement the method and demonstrate it with several examples.
This new method seeks primarily to develop the current geometric methods, i.e., the finite integration technique and the cell method. We introduce some improvements to these methods and, additionally, develop tools for further development. For these, we present a couple of challenge problems and largely solve them with our method in this thesis.
There are several reasons why we focus on the mathematical structures underlying the spacetime and electromagnetic models. For example, first, fourdimensional fields cannot be modeled with elementary vector algebra, and, second, a functional computational method is usually based on a solid mathematical foundation. From the background of the theory of relativity, we develop a model for spacetime. Because this model includes an indefinite metric, all metricdependent concepts must agree with this nonRiemannian indefinite metric. Consequently, we utilize concepts new to electromagnetic computation. And because our target is electromagnetic computation, we choose a model that agrees well with computation: we model electromagnetic fields with cochains.
In the end, we introduce a computational method that makes use of a pair of fourdimensional Lorentzian orthogonal meshes. In developing it, we concentrate on the constitutive relations, boundary conditions, and gauging methods and implement the method and demonstrate it with several examples.
Original language  English 

Qualification  Doctor Degree 
Awarding Institution 

Supervisors/Advisors 

Award date  6 Apr 2011 
Publisher  
Print ISBNs  9789521525285 
Publication status  Published  11 Mar 2011 
MoE publication type  G4 Doctoral dissertation (monograph) 