25th Annual GAMM-Seminar Leipzig on

FEM and BEM for time-dependent wave problems

Lacunae-Based Stabilization of PMLs

Heydar Qasimov (MPI Leipzig)

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. We present 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. We construct a practical algorithm and prove a theorem 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.

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