On some differential equations arising in time domain scattering problems for a dissipative wave equation

1991 ◽  
Vol 20 (1) ◽  
pp. 29-54 ◽  
Author(s):  
David J. N. Wall
Keyword(s):  
Wave Equation ◽  
Differential Equations ◽  
Time Domain ◽  
Scattering Problems ◽  
Time Domain Scattering ◽  
Dissipative Wave Equation

scholarly journals Convolution quadrature methods for time-domain scattering from unbounded penetrable interfaces

2019 ◽  
Vol 475 (2227) ◽  
pp. 20190029 ◽  
Author(s):  
Ignacio Labarca ◽  
Luiz M. Faria ◽  
Carlos Pérez-Arancibia
Keyword(s):  
Integral Equation ◽  
Frequency Domain ◽  
Time Domain ◽  
Window Size ◽  
Boundary Integral ◽  
Complex Frequency ◽  
Scattering Problems ◽  
Convolution Quadrature ◽  
Transmission Problems ◽  
Time Domain Scattering

This paper presents a class of boundary integral equation methods for the numerical solution of acoustic and electromagnetic time-domain scattering problems in the presence of unbounded penetrable interfaces in two spatial dimensions. The proposed methodology relies on convolution quadrature (CQ) schemes and the recently introduced windowed Green function (WGF) method. As in standard time-domain scattering from bounded obstacles, a CQ method of the user's choice is used to transform the problem into a finite number of (complex) frequency-domain problems posed, in our case, on the domains containing unbounded penetrable interfaces. Each one of the frequency-domain transmission problems is then formulated as a second-kind integral equation that is effectively reduced to a bounded interface by means of the WGF method—which introduces errors that decrease super-algebraically fast as the window size increases. The resulting windowed integral equations can then be solved by means of any (accelerated or unaccelerated) off-the-shelf Nyström or boundary element Helmholtz integral equation solvers capable of handling complex wavenumbers with large imaginary part. A high-order Nyström method based on Alpert's quadrature rules is used here. A variety of CQ schemes and numerical examples, including wave propagation in open waveguides as well as scattering from multiple layered media, demonstrate the capabilities of the proposed approach.

The Maxwell Equations and Constitutive Relations

2012 ◽  
Author(s):  
G. F. Roach ◽  
I. G. Stratis ◽  
A. N. Yannacopoulos
Keyword(s):  
Differential Equations ◽  
Flux Density ◽  
Time Domain ◽  
Maxwell Equations ◽  
Constitutive Relations ◽  
Magnetic Flux Density ◽  
Scattering Problems ◽  
Stochastic Problems ◽  
Time Harmonic ◽  
Electric Flux

This chapter first introduces the constitutive relations which are commonly used in electromagnetic theory for the mathematical modelling of complex electromagnetic media. These constitutive relations are to be understood as operators connecting the electric flux density and the magnetic flux density with the electric and the magnetic fields. When they are introduced into the Maxwell equations, this chapter obtains differential equations (PDEs) that govern the evolution of the electromagnetic fields. This chapter also seeks to formulate and discuss the scope of the various problems related to the Maxwell equations that will be treated in this volume. It introduces and formulates in terms of differential equations various problems of interest related to the Maxwell equations: time-harmonic problems, scattering problems, time-domain evolution problems, random and stochastic problems, controllability problems, homogenisation problems, and others.

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