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Accuracy of staircase approximations in finite-difference methods for wave propagation
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). KTH, Centres, SeRC - Swedish e-Science Research Centre.ORCID iD: 0000-0002-6321-8619
2014 (English)In: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 128, no 4, 741-771 p.Article in journal (Refereed) Published
Abstract [en]

While a number of increasingly sophisticated numerical methods have been developed for time-dependent problems in electromagnetics, the Yee scheme is still widely used in the applied fields, mainly due to its simplicity and computational efficiency. A fundamental drawback of the method is the use of staircase boundary approximations, giving inconsistent results. Usually experience of numerical experiments provides guidance of the impact of these errors on the final simulation result. In this paper, we derive exact discrete solutions to the Yee scheme close to the staircase approximated boundary, enabling a detailed theoretical study of the amplitude, phase and frequency errors created. Furthermore, we show how evanescent waves of amplitude occur along the boundary. These characterize the inconsistencies observed in electromagnetic simulations and the locality of the waves explain why, in practice, the Yee scheme works as well as it does. The analysis is supported by detailed proofs and numerical examples.

Place, publisher, year, edition, pages
2014. Vol. 128, no 4, 741-771 p.
Keyword [en]
FDTD, Yee, Staircasing
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-95502DOI: 10.1007/s00211-014-0625-1ISI: 000344752100005OAI: oai:DiVA.org:kth-95502DiVA: diva2:528685
Funder
Swedish e‐Science Research Center
Note

QC 20141212

Updated from manuscript to article in journal.

Available from: 2012-05-28 Created: 2012-05-28 Last updated: 2017-12-07Bibliographically 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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Runborg, Olof

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Numerical Analysis, NA (closed 2012-06-30)SeRC - Swedish e-Science Research Centre
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