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11
Hyperboloidal layers for hyperbolic equations on unbounded domains
, 2010
"... We show how to solve hyperbolic equations numerically on unbounded domains by means of compactification, thereby avoiding the introduction of an artificial outer boundary. The essential ingredient is a suitable transformation of the time coordinate in combination with compactification. Based on this ..."
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We show how to solve hyperbolic equations numerically on unbounded domains by means of compactification, thereby avoiding the introduction of an artificial outer boundary. The essential ingredient is a suitable transformation of the time coordinate in combination with compactification. Based on this idea, we present a new layer method, called hyperboloidal layers. Accuracy and efficiency of this method is demonstrated by numerical tests including the one dimensional Maxwell equations using finite difference methods, and the three dimensional scalar wave equation with and without nonlinear source terms using spectral methods.
Multidomain spectral method for the helically reduced wave equation
 In grqc/0702050
, 2007
"... We consider the 2+1 and 3+1 scalar wave equations reduced via a helical Killing field, respectively referred to as the 2–dimensional and 3–dimensional helically reduced wave equation (HRWE). The HRWE serves as the fundamental model for the mixed–type PDE arising in the periodic standing wave (PSW) a ..."
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We consider the 2+1 and 3+1 scalar wave equations reduced via a helical Killing field, respectively referred to as the 2–dimensional and 3–dimensional helically reduced wave equation (HRWE). The HRWE serves as the fundamental model for the mixed–type PDE arising in the periodic standing wave (PSW) approximation to binary inspiral. We present a method for solving the equation based on domain decomposition and spectral approximation. Beyond describing such a numerical method for solving strictly linear HRWE, we also present results for a nonlinear scalar model of binary inspiral. The PSW approximation has already been theoretically and numerically studied in the context of the post–Minkowskian gravitational field, with numerical simulations carried out via the “eigenspectral method. ” Despite its name, the eigenspectral technique does feature a finite–difference component, and is lower–order accurate. We intend to apply the numerical method described here to the theoretically well–developed post–Minkowski PSW formalism with the twin goals of spectral accuracy and the coordinate flexibility afforded by global spectral interpolation.
Perfectly matched layers for timeharmonic second order elliptic problems
 ARCHIVES OF COMPUTATIONAL METHODS IN ENGINEERING
"... The main goal of this work is to give a review of the Perfectly Matched Layer (PML) technique for timeharmonic problems. Precisely, we focus our attention on problems stated in unbounded domains, which involve second order elliptic equations writing in divergence form and, in particular, on the H ..."
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The main goal of this work is to give a review of the Perfectly Matched Layer (PML) technique for timeharmonic problems. Precisely, we focus our attention on problems stated in unbounded domains, which involve second order elliptic equations writing in divergence form and, in particular, on the Helmholtz equation at low frequency regime. Firstly, the PML technique is introduced by means of a simple porous model in one dimension. It is emphasized that an adequate choice of the so called complex absorbing function in the PML yields to accurate numerical results. Then, in the twodimensional case, the PML governing equation is described for second order partial differential equations by using a smooth complex change of variables. Its mathematical analysis and some particular examples are also included. Numerical drawbacks and optimal choice of the PML absorbing function are studied in detail. In fact, theoretical and numerical analysis show the advantages of using nonintegrable absorbing functions. Finally, we present some relevant real life numerical simulations where the PML technique is widely and successfully used although they are not covered by the standard theoretical framework.
Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme Optimizing Perfectly Matched Layers in Discrete Contexts
"... Perfectly Matched Layers (PMLs) are widely used for the numerical simulation of wavelike problems defined on large or infinite spatial domains. However, for both the timedependent and the timeharmonic cases, their performance critically depends on the socalled absorption function. This paper dea ..."
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Perfectly Matched Layers (PMLs) are widely used for the numerical simulation of wavelike problems defined on large or infinite spatial domains. However, for both the timedependent and the timeharmonic cases, their performance critically depends on the socalled absorption function. This paper deals with the choice of this function when classical numerical methods are used (based on finite differences, finite volumes, continuous finite elements and discontinuous finite elements). After reviewing the properties of the PMLs at the continuous level, we analyse how they are altered by the different spatial discretizations. In the light of these results, different shapes of absorption function are optimized and compared by means of both one and twodimensional representative timedependent cases. This study highlights the advantages of the socalled shifted hyperbolic function, which is efficient in all cases and does not require the tuning of a free parameter, by contrast with the widely used polynomial functions. Copyright c © 2014 John Wiley &
Chapter 5 Evaluation of the Transient Performance of SuperWideband PrintedCircuit Antennas Using TimeDomain
"... Abstract A timedomain electromagnetics code is used to evaluate the transient and radiation performances of three printedcircuit antennas for superwideband (SWB) monitoring applications. For two antennas, one in microstrip and one in coplanar technologies, operating between 3 and 30 GHz with a re ..."
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Abstract A timedomain electromagnetics code is used to evaluate the transient and radiation performances of three printedcircuit antennas for superwideband (SWB) monitoring applications. For two antennas, one in microstrip and one in coplanar technologies, operating between 3 and 30 GHz with a return loss of 10 dB, it is demonstrated that the vertically polarized omnidirectional radiation characteristics in the lower frequency band change to a more directional pattern at higher frequencies and that the crosspolar field component increases with frequency and gives rise to possible dualpolarized applications for the microstrip antenna. In comparison, the coplanar antenna shows slightly better performance, especially with respect to its transient response. Its groupdelay variation is only 180 ps compared to 250 ps of the microstrip antenna, and its amplitude response provides better polarization purity. The evaluation of the coplanar concept is extended to cover a bandwidth between 3 and 60 GHz. The timedomain evaluation, as validated by a frequencydomain technique, demonstrates that bandwidths in extent of decade bandwidths are possible with simple printedcircuit antennas. Characteristics and performances are presented for possible applications in future SWB monitoring systems, radar technology, throughwall imaging systems, and other future wireless services. Antenna dimensions are provided for future comparisons with improved and/or multilevel electromagnetics codes.
SPECTRAL APPROXIMATION OF TIMEHARMONIC MAXWELL EQUATIONS IN THREEDIMENSIONAL EXTERIOR DOMAINS
"... Abstract. We develop in this paper an efficient and robust spectralGalerkin method for solving the threedimensional timeharmonic Maxwell equations in exterior domains. We first reduce the problem to a bounded domain by using the capacity operator which characterizes the transparent boundary condi ..."
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Abstract. We develop in this paper an efficient and robust spectralGalerkin method for solving the threedimensional timeharmonic Maxwell equations in exterior domains. We first reduce the problem to a bounded domain by using the capacity operator which characterizes the transparent boundary condition (TBC). Then, we adopt the transformed field expansion (TFE) approach to reduce the problem to a sequence of Maxwell equations in a spherical shell. Finally, we develop an efficient spectral algorithm by using Legendre approximation in the radial direction and vector spherical harmonic expansion in the tangential directions. Key words. Maxwell equations, exterior problems, transparent boundary conditions, vector spherical harmonics, Legendre spectral method. 1.