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Topics in Analysis and Computation of Linear Wave Propagation
KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
2008 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [en]

This thesis concerns the analysis and numerical simulation of wave propagation problems described by systems of linear hyperbolic partial differential equations.

A major challenge in wave propagation problems is numerical simulation of high frequency waves. When the wavelength is very small compared to the overall size of the computational domain, we encounter a multiscale problem. Examples include the forward and the inverse seismic wave propagation, radiation and scattering problems in computational electromagnetics and underwater acoustics. In direct numerical simulations, the accuracy of the approximate solution is determined by the number of grid points or elements per wavelength. The computational cost to maintain constant accuracy grows algebraically with the frequency, and for sufficiently high frequency, direct numerical simulations are no longer feasible. Other numerical methods are therefore needed. Asymptotic methods, for instance, are good approximations for very high frequency waves. They are based on constructing asymptotic expansions of the solution. The accuracy increases with increasing frequency for a fixed computational cost. Most asymptotic techniques rely on geometrical optics equations with frequency independent unknowns. There are however two deficiencies in the geometrical optics solution. First, it does not include diffraction effects. Secondly, it breaks down at caustics. Geometrical theory of diffraction provides a technique for adding diffraction effects to the geometrical optics approximation by introducing diffracted rays. In papers 1 and 2 we present a numerical algorithm for computing an important type of diffracted rays known as creeping rays. Another asymptotic model which is valid also at caustics is based on Gaussian beams. In papers 3 and 4, we present an error analysis of Gaussian beams approximation and develop a new numerical algorithm for computing Gaussian beams, respectively.

Another challenge in computation of wave propagation problems arises when the system of equations consists of second order hyperbolic equations involving mixed space-time derivatives. Examples include the harmonic formulation of Einstein’s equations and wave equations governing elasticity and acoustics. The classic computational treatment of such second order hyperbolic systems has been based on reducing the systems to first order differential forms. This treatment has however the disadvantage of introducing auxiliary variables with their associated constraints and boundary conditions. In paper 5, we treat the problem in the second order differential form, which has advantages for both computational efficiency and accuracy over the first order formulation.

Finally, paper 6 concerns the concept of well-posedness for a class of linear hyperbolic initial boundary value problems which are not boundary stable. The well-posedness is well established for boundary stable hyperbolic systems for which we can obtain sharp estimates of the solution including estimates at boundaries. There are, however, problems which are not boundary stable but are well-posed in a weaker sense, i.e., the problems for which an energy estimate can be obtained in the interior of the domain but not on the boundaries. We analyze a model problem of this type. Possible applications arise in elastic wave equations and Maxwell’s equations describing glancing and surface waves.

Place, publisher, year, edition, pages
Stockholm: KTH , 2008. , xiii, 36 p.
Series
Trita-CSC-A, ISSN 1653-5723 ; 2008:07
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-4715ISBN: 978-91-7178-961-7 (print)OAI: oai:DiVA.org:kth-4715DiVA: diva2:13586
Public defence
2008-05-20, Sal D2, KTH, Lindstedtsvägen 5, Stockholm, 10:15
Opponent
Supervisors
Note
QC 20100830Available from: 2008-04-30 Created: 2008-04-30 Last updated: 2010-08-30Bibliographically approved
List of papers
1. A fast phase space method for computing creeping rays
Open this publication in new window or tab >>A fast phase space method for computing creeping rays
2006 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 219, no 1, 276-295 p.Article in journal (Refereed) Published
Abstract [en]

Creeping rays can give an important contribution to the solution of medium to high frequency scattering problems. They are generated at the shadow lines of the illuminated scatterer by grazing incident rays and propagate along geodesics on the scatterer surface, continuously shedding diffracted rays in their tangential direction. In this paper, we show how the ray propagation problem can be formulated as a partial differential equation (PDE) in a three-dimensional phase space. To solve the PDE we use a fast marching method. The PDE solution contains information about all possible creeping rays. This information includes the phase and amplitude of the field, which are extracted by a fast post-processing. Computationally, the cost of solving the PDE is less than tracing all rays individually by solving a system of ordinary differential equations. We consider an application to mono-static radar cross section problems where creeping rays from all illumination angles must be computed. The numerical results of the fast phase space method and a comparison with the results of ray tracing are presented.

Keyword
creeping rays, high frequency wave propagation, scattering problems, numerical methods, geometrical theory of diffraction, eikonal equation, finite-difference calculation, high-frequency, wave-propagation, travel-time, level set, computation, equation, optics, rcs
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-16154 (URN)10.1016/j.jcp.2006.03.024 (DOI)000242332500018 ()2-s2.0-33750342435 (Scopus ID)
Note
QC 20100525Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2017-12-12Bibliographically approved
2. A multiple-patch phase space method for computing trajectories on manifolds with applications to wave propagation problems
Open this publication in new window or tab >>A multiple-patch phase space method for computing trajectories on manifolds with applications to wave propagation problems
2007 (English)In: Communications in Mathematical Sciences, ISSN 1539-6746, E-ISSN 1945-0796, Vol. 5, no 3, 617-648 p.Article in journal (Refereed) Published
Abstract [en]

We present a multiple-patch phase space method for computing trajectories on two-dimensional manifolds possibly embedded in a higher-dimensional space. The dynamics of trajectories are given by systems of ordinary differential equations (ODEs). We split the manifold into multiple patches where each patch has a well-defined regular parameterization. The ODEs are formulated as escape equations, which are hyperbolic partial differential equations (PDEs) in a three- dimensional phase space. The escape equations are solved in each patch, individually. The solutions of individual patches are then connected using suitable inter-patch boundary conditions. Properties for particular families of trajectories are obtained through a fast post-processing. We apply the method to two different problems : the creeping ray contribution to mono-static radar cross section computations and the multivalued travel-time of seismic waves in multi-layered media. We present numerical examples to illustrate the accuracy and efficiency of the method.

Keyword
ODEs on a manifold, phase space method, escape equations, high frequency wave, propagation, geodesics, creeping rays, seismic waves, travel-time, partial-differential equations, dielectric coated cylinder, high-frequency, creeping waves, travel-times, computation, rays, rcs
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-16973 (URN)000249723400006 ()2-s2.0-35349024163 (Scopus ID)
Note
QC 20100525Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2017-12-12Bibliographically approved
3. Taylor Expansion Errors in Gaussian Beam Summation
Open this publication in new window or tab >>Taylor Expansion Errors in Gaussian Beam Summation
(English)Manuscript (Other academic)
Keyword
wave propagation, high frequency, asymptotic approximation, summation of Gaussian beams, accuracy, error estimates
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-8298 (URN)
Note
QC 20100830Available from: 2008-04-30 Created: 2008-04-30 Last updated: 2010-08-30Bibliographically approved
4. A Wave Front-based Gaussian Beam Method for Computing High Frequency Waves
Open this publication in new window or tab >>A Wave Front-based Gaussian Beam Method for Computing High Frequency Waves
(English)Manuscript (Other academic)
Keyword
wave propagation, high frequency, asymptotic approximation, summation of Gaussian beams, wavefront methods
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-8299 (URN)
Note
QC 20100830Available from: 2008-04-30 Created: 2008-04-30 Last updated: 2010-08-30Bibliographically approved
5. Finite Difference Schemes for Second Order Systems Describing Black Holes
Open this publication in new window or tab >>Finite Difference Schemes for Second Order Systems Describing Black Holes
Show others...
2006 (English)In: Physical Review D. Particles and fields, ISSN 0556-2821, E-ISSN 1089-4918, Vol. 73, no 12, 124008-1-124008-14 p.Article in journal (Refereed) Published
Abstract [en]

In the harmonic description of general relativity, the principal part of Einstein's equations reduces to 10 curved space wave equations for the components of the space-time metric. We present theorems regarding the stability of several evolution-boundary algorithms for such equations when treated in second order differential form. The theorems apply to a model black hole space-time consisting of a spacelike inner boundary excising the singularity, a timelike outer boundary and a horizon in between. These algorithms are implemented as stable, convergent numerical codes and their performance is compared in a 2-dimensional excision problem.

Keyword
wave-equation, numerical relativity, general-relativity, approximations, stability
National Category
Astronomy, Astrophysics and Cosmology Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-8300 (URN)10.1103/PhysRevD.73.124008 (DOI)000238698900048 ()2-s2.0-33744907537 (Scopus ID)
Note
QC 20100830Available from: 2008-04-30 Created: 2008-04-30 Last updated: 2010-08-30Bibliographically approved
6. Hyperbolic Initial Boundary Value Problems which are not Boundary Stable
Open this publication in new window or tab >>Hyperbolic Initial Boundary Value Problems which are not Boundary Stable
(English)Manuscript (Other academic)
Keyword
partial differential equations, hyperbolic systems, boundary stable problems, Kreiss symmetrizers, pseudo-differential operators
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-8301 (URN)
Note
QC 20100830Available from: 2008-04-30 Created: 2008-04-30 Last updated: 2010-08-30Bibliographically approved

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