A Robust Three-Dimensional Iterative Solver for the Time-Harmonic Wave Equation

2006 ◽  
Author(s):  
R. E. Plessix
Keyword(s):  
Wave Equation ◽  
Harmonic Wave ◽  
Three Dimensional ◽  
Iterative Solver ◽  
Time Harmonic

scholarly journals A new iterative solver for the time-harmonic wave equation

Geophysics ◽  
2006 ◽  
Vol 71 (5) ◽  
pp. E57-E63 ◽  
Author(s):  
C. D. Riyanti ◽  
Y. A. Erlangga ◽  
R.-E. Plessix ◽  
W. A. Mulder ◽  
C. Vuik ◽  
...  
Keyword(s):  
Wave Equation ◽  
Linear System ◽  
Time Domain ◽  
Multigrid Method ◽  
Harmonic Wave ◽  
Computational Time ◽  
Full Time ◽  
Iterative Solver ◽  
Time Harmonic ◽  
The Time Domain

The time-harmonic wave equation, also known as the Helmholtz equation, is obtained if the constant-density acoustic wave equation is transformed from the time domain to the frequency domain. Its discretization results in a large, sparse, linear system of equations. In two dimensions, this system can be solved efficiently by a direct method. In three dimensions, direct methods cannot be used for problems of practical sizes because the computational time and the amount of memory required become too large. Iterative methods are an alternative. These methods are often based on a conjugate gradient iterative scheme with a preconditioner that accelerates its convergence. The iterative solution of the time-harmonic wave equation has long been a notoriously difficult problem in numerical analysis. Recently, a new preconditioner based on a strongly damped wave equation has heralded a breakthrough. The solution of the linear system associated with the preconditioner is approximated by another iterative method, the multigrid method. The multigrid method fails for the original wave equation but performs well on the damped version. The performance of the new iterative solver is investigated on a number of 2D test problems. The results suggest that the number of required iterations increases linearly with frequency, even for a strongly heterogeneous model where earlier iterative schemes fail to converge. Complexity analysis shows that the new iterative solver is still slower than a time-domain solver to generate a full time series. We compare the time-domain numeric results obtained using the new iterative solver with those using the direct solver and conclude that they agree very well quantitatively. The new iterative solver can be applied straightforwardly to 3D problems.

A note on diffraction by a right-angled wedge

1965 ◽  
Vol 61 (1) ◽  
pp. 275-278 ◽  
Author(s):  
W. E. Williams
Keyword(s):  
Wave Function ◽  
Wave Equation ◽  
Surface Wave ◽  
Harmonic Wave ◽  
Boundary Value ◽  
Normal Derivative ◽  
Support Surface ◽  
Closed Form Solutions ◽  
Time Harmonic

It has been shown in recent years ((5)–(8), (10)) that it is possible to obtain closed form solutions for the time harmonic wave equation when a linear combination of the wave function and its normal derivative is prescribed on the surface of a wedge. Boundary-value problems of this type occur in the problem of diffraction by a highly conducting wedge or by a wedge whose surfaces are thinly coated with dielectric. In certain circumstances such surfaces can support surface waves and one important aspect of the solution of the boundary-value problem is the determination of the amplitude of the surface wave excited.

Uniqueness and non-uniqueness in acoustic tomography of moving fluid

2016 ◽  
Vol 24 (3) ◽  
Author(s):  
Alexey D. Agaltsov ◽  
Roman G. Novikov
Keyword(s):  
Wave Equation ◽  
Bounded Domain ◽  
Harmonic Wave ◽  
Acoustic Tomography ◽  
Time Harmonic ◽  
Moving Fluid

AbstractWe consider a model time-harmonic wave equation of acoustic tomography of moving fluid in an open bounded domain in ℝ

scholarly journals Finding scattering data for a time-harmonic wave equation with first order perturbation from the Dirichlet-to-Neumann map

2015 ◽  
Vol 23 (6) ◽  
Author(s):  
Alexey D. Agaltsov
Keyword(s):  
Wave Equation ◽  
Harmonic Wave ◽  
Order Perturbation ◽  
Scattering Data ◽  
Scalar Case ◽  
First Order ◽  
Time Harmonic ◽  
General Scalar ◽  
Dirichlet To Neumann Map ◽  
First Order Perturbation

AbstractWe present formulas and equations for finding scattering data from the Dirichlet-to-Neumann map for a time-harmonic wave equation with first order perturbation with compactly supported coefficients. We assume that the coefficients are matrix-valued in general. To our knowledge, these results are new even for the general scalar case.

scholarly journals Acoustic trapped modes in a three-dimensional waveguide of slowly varying cross section

2013 ◽  
Vol 469 (2149) ◽  
pp. 20120384 ◽  
Author(s):  
Simon N. Gaulter ◽  
Nicholas R. T. Biggs
Keyword(s):  
Cross Section ◽  
Turning Point ◽  
Harmonic Wave ◽  
Three Dimensional ◽  
Longitudinal Direction ◽  
Interior Layer ◽  
Oscillatory Behaviour ◽  
Trapped Modes ◽  
Trapped Mode ◽  
Time Harmonic

In this paper, we develop an asymptotic scheme to approximate the trapped mode solutions to the time harmonic wave equation in a three-dimensional waveguide with a smooth but otherwise arbitrarily shaped cross section and a single, slowly varying ‘bulge’, symmetric in the longitudinal direction. Extending previous research carried out in the two-dimensional case, we first use a WKBJ-type ansatz to identify the possible quasi-mode solutions that propagate only in the thicker region, and hence find a finite cut-on region of oscillatory behaviour and asymptotic decay elsewhere. The WKBJ expansions are used to identify a turning point between the cut-on and cut-off regions. We note that the expansions are non-uniform in an interior layer centred on this point, and we use the method of matched asymptotic expansions to connect the cut-on and cut-off regions within this layer. The behaviour of the expansions within the interior layer then motivates the construction of a uniformly valid asymptotic expansion. Finally, we use this expansion and the symmetry of the waveguide around the longitudinal centre, x =0, to extract trapped mode wavenumbers, which are compared with those found using a numerical scheme and seen to be extremely accurate, even to relatively large values of the small parameter.

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