Lacunae Based Stabilization of PMLs

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Title: Lacunae Based Stabilization of PMLs
Author: Qasimov, Heydar Rafiq oglu
Advisors: Pierre Gremaud, Committee Member
Shamim Rahman, Committee Member
Dmitri Zenkov, Committee Member
Ralph Smith, Committee Member
Semyon Tsynkov, Committee Chair
Abstract: Perfectly matched layers (PML) enclose the computational domain for simulating electromagnetic phenomena defined over unbounded regions. While being overall very successful, this procedure has sometimes been reported to develop instabilities and exhibit a physically unaccountable growth of the solution inside the layer over long integration times. In the thesis, we conduct a numerical as well as analytical study of the PML's response after a long time of integration. The physical and mathematical PMLs are implemented with several well-known schemes: Yee, leap-frog, Lax-Wendroff, and Runge-Kutta in time with central differences in space. Then, the eigen-structure of each discretization at quiescent state is studied to gain an insight into the nature of instability, the sources of growth of the solution and the potential contamination of the domain of interest. The results of this investigation provide useful information regarding the better and worse performers among the specific combinations of schemes and PMLs, yet they do not precisely identify the mechanism behind the growth of the solution. Therefore, the main focus of the thesis is to build a methodology that would inhibit the instability of the PML regardless of its source. The approach is based on the concept of numerical integration that exploits the presence of lacunae in the solutions. It applies to hyperbolic partial differential equations and systems that satisfy the Huygens' principle, in particular, the Maxwell's system of equations that governs the propagation of electromagnetic waves. The methodology does not modify the equations inside the layer and hence, while eliminating the undesirable growth, it fully preserves all the advantageous properties of a given PML, such as matching at the interface and the degree of absorption. A practical algorithm is constructed in the thesis, and a theorem is proved that guarantees a temporally uniform error bound over the domain of interest. The theoretical findings are corroborated numerically, and the potential for extending the methodology is discussed
Date: 2008-11-11
Degree: PhD
Discipline: Applied Mathematics

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