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Higher order finite difference methods for the Helmholtz equation
Karlstad University, Faculty of Technology and Science.
2002 (English)Other (Other academic)
Abstract [en]

Wave propagation is described by the wave equation, or in the time-periodic case, by the Helmholtz equation. For problems with small wavelengths, high order discretizations must be used to resolve the solution. Two different techniques for finding compact finite difference schemes of high order for the Helmholtz equation are studied and compared. The first approach is Numerov's idea of using the equation to transfer higher derivatives to lower order ones. The second principle is the method of deferred correction, where a lower order approximation is used for error correction.Sharp estimates for the error are derived, in order to compare the arithmetic complexity for both approaches with a non-compact scheme. The characteristics of the errors for fourth order as well as sixth order accuracy are demonstrated and the advantages and disadvantages of the methods are discussed.

Place, publisher, year, pages
2002.
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kau:diva-1844OAI: oai:DiVA.org:kau-1844DiVA: diva2:345
Note
Published as part of Lic. thesis IT 2002-001, Dept. of Information Technology, Uppsala Univ., Uppsala, Sweden, 2002.Available from: 2008-09-10 Created: 2008-09-10 Last updated: 2010-10-01Bibliographically approved
In thesis
1. Some numerical and analytical methods for equations of wave propagation and kinetic theory
Open this publication in new window or tab >>Some numerical and analytical methods for equations of wave propagation and kinetic theory
2008 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [en]

This thesis consists of two different parts, related to two different fields in mathematical physics: wave propagation and kinetic theory of gases. Various mathematical and computational problems for equations from these areas are treated.

 

The first part is devoted to high order finite difference methods for the Helmholtz equation and the wave equation. Compact schemes with high order accuracy are obtained from an investigation of the function derivatives in the truncation error. With the help of the equation itself, it is possible to transfer high order derivatives to lower order or to transfer time derivatives to space derivatives. For the Helmholtz equation, a compact scheme based on this principle is compared to standard schemes and to deferred correction schemes, and the characteristics of the errors for the different methods are demonstrated and discussed. For the wave equation, a finite difference scheme with fourth order accuracy in both space and time is constructed and applied to a problem in discontinuous media.

 

The second part addresses some problems related to kinetic equations. A direct simulation Monte-Carlo method is constructed for the Landau-Fokker-Planck equation, and numerical tests are performed to verify the accuracy of the algorithm. A formal derivation of the method from the Boltzmann equation with grazing collisions is performed. The linear and linearized Boltzmann collision operators for the hard sphere molecular model are studied using exact reduction of integral equations to ordinary differential equations. It is demonstrated how the eigenvalues of the operators are found from these equations, and numerical values are computed. A proof of existence of non-zero discrete eigenvalues is given. The ordinary diffential equations are also used for investigation of the Chapman-Enskog distribution function with respect to its asymptotic behavior.

 

Place, publisher, year, edition, pages
Karlstad: Karlstads universitet, 2008. 21 p.
Series
Karlstad University Studies, ISSN 1403-8099 ; 2008:33
Keyword
wave propagation, finite difference metods high order methods, Landau-Fokker-Planck equation, Monte-Carlo simulations, Boltzmann equation, hard sphere model, eigenvalue problem
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-1848 (URN)978-91-7063-192-4 (ISBN)
Public defence
2008-10-04, 21A 342, Karlstads universitet, Karlstad, 13:15 (English)
Opponent
Supervisors
Available from: 2008-09-10 Created: 2008-09-10 Last updated: 2011-12-19Bibliographically approved

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http://www.it.uu.se/research/publications/lic/2002-001/
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Citation style
  • apa
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