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On Energy Preserving Consistent Boundary Conditions for the Yee Scheme in 2D
Univ Texas Austin.
KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30). (Swedish e-Science Research Center (SeRC)ORCID iD: 0000-0002-6321-8619
2012 (English)In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 52, no 3, 615-637 p.Article in journal (Refereed) Published
Abstract [en]

The Yee scheme is one of the most popular methods for electromagnetic wave propagation. A main advantage is the structured staggered grid, making it simple and efficient on modern computer architectures. A downside to this is the difficulty in approximating oblique boundaries, having to resort to staircase approximations. In this paper we present a method to improve the boundary treatment in two dimensions by, starting from a staircase approximation, modifying the coefficients of the update stencil so that we can obtain a consistent approximation while preserving the energy conservation, structure and the optimal CFL-condition of the original Yee scheme. We prove this in L_2 and verify it by numerical experiments.

Place, publisher, year, edition, pages
Springer, 2012. Vol. 52, no 3, 615-637 p.
Keyword [en]
Yee scheme, FDTD, Computational electromagnetics
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-73240DOI: 10.1007/s10543-012-0376-2ISI: 000308234600006Scopus ID: 2-s2.0-84865704007OAI: oai:DiVA.org:kth-73240DiVA: diva2:488690
Funder
Swedish e‐Science Research Center
Note

QC 20120919

Available from: 2012-02-02 Created: 2012-02-02 Last updated: 2017-12-08Bibliographically approved
In thesis
1. Modified Stencils for Boundaries and Subgrid Scales in the Finite-Difference Time-Domain Method
Open this publication in new window or tab >>Modified Stencils for Boundaries and Subgrid Scales in the Finite-Difference Time-Domain Method
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis centers on modified stencils for the Finite-Difference Time-Domain method (FDTD), or Yee scheme, when modelling curved boundaries, obstacles and holes smaller than the discretization length.  The goal is to increase the accuracy while keeping the structure of the standard method, enabling improvements to existing implementations with minimal effort.

We present an extension of a previously developed technique for consistent boundary approximation in the Yee scheme.  We consider both Maxwell's equations and the acoustic equations in three dimensions, which require separate treatment, unlike in two dimensions.

The stability properties of coefficient modifications are essential for practical usability.  We present an analysis of the requirements for time-stable modifications, which we use to construct a simple and effective method for boundary approximations. The method starts from a predetermined staircase discretization of the boundary, requiring no further data on the underlying geometry that is being approximated.

Not only is the standard staircasing of curved boundaries a poor approximation, it is inconsistent, giving rise to errors that do not disappear in the limit of small grid lengths. We analyze the standard staircase approximation by deriving exact solutions of the difference equations, including the staircase boundary. This facilitates a detailed error analysis, showing how staircasing affects amplitude, phase, frequency and attenuation of waves.

To model obstacles and holes of smaller size than the grid length, we develop a numerical subgrid method based on locally modified stencils, where a highly resolved micro problem is used to generate effective coefficients for the Yee scheme at the macro scale.

The implementations and analysis of the developed methods are validated through systematic numerical tests.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2012. xi, 34 p.
Series
Trita-CSC-A, ISSN 1653-5723 ; 2012:07
Keyword
FDTD, Yee, Staircasing
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-95510 (URN)978-91-7501-417-3 (ISBN)
Public defence
2012-06-15, F3, Lindstedtsvägen 26, KTH, Stockholm, 10:00 (English)
Opponent
Supervisors
Funder
Swedish e‐Science Research Center
Note

QC 20120530

Available from: 2012-05-30 Created: 2012-05-28 Last updated: 2013-04-09Bibliographically approved

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Publisher's full textScopusThe final publication is available at www.springerlink.com

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Runborg, Olof

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Engquist, BjörnHäggblad, JonRunborg, Olof
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