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Some comments on different exact computational methods

Different classifications
There are several classifications of the exact methods. The simplest one is based on the kind of the problem formulation (differential, surface or volume integral equations) and is made above. A more physical classification is that of Th.Wriedt (1998) where he devides the methods in 3 groups: Mie theory expansions (SVM), surface-based (PMM, EBCM, GMT, MoM) and volume-based (FDTDM, TLMM, FEM and various volume IEMs - CDM, DDA, etc.) methods.

Features of the methods Among the methods only SVM and EBCM are capable to provide very accurate results (5 and more correct decimals) (Mishchenko et al., 2000).

The SVM and EBCM can be applied to particles of any shapes (refs), but look to be efficient only for spheres - cylinders - spheroids (1-2D) and axisymmetric (2D) particles, respectively [ref]. The same is with FIEM (ref).

Relationships of the methods a) The SVM for spheroids is shown to be equivalent to the EBCM formulated in spheroidal coordinates (Schmidt et al., 1998).

b) A similarity of the FEM and the FDM is in the space discretization, but whereas in the FDM a scattarer is descretized on a cubic grid which leads to a staircase approximated boundary, in the FEM the scatterer boundary can be represented by using elements of different shapes (Wriedt, 1998).

c) The FDM group splits in the FDTDM and the TLM which differ as follows. Whereas in the FDTDM the time and space domains are simply discretized with some steps, in the TLM the medium including the scatterer is discretized into cells and these cells representing elementary multiports are connected to neibouring cells by transmission lines, just forming a 3D network (Wriedt, 1998).

d) Earlier the TMM was used as a synonim of the EBCM (Varadan & Varadan, 1980). Now the T-matrix is obtained within different methods (e.g., in SVM by Schulz et al., 1998, in GMT by ref?) and the usage of the term TMM became confusing.

e) The GMT can be considered as a further generalization of the PMM (Mishchenko et al., 2000, p.36).

f) The MoM is hard to be classified as it can be presented in differential (?) and both surface and volume integral forms (ref?, Wriedt, 1998).

g) The MoM (in its vI form) and the CDM belong to the same group of methods called the volume integral methods where the former is an actual field formulation and the latter is based on the concept of exited dipoles (Wriedt, 1998).

h) There are many modifications of the volume integral methods (CDM, MoM, etc) which differ in the way of treatment of the self-interaction of elementary scatterer parts (Mishchenko et al., 2000).

i) There are some other methods besides mentioned ones (see, e.g., Mishchenko et al, 2000, p.43).

Codes available See Wriedt's site for many TMM, FEM, FDM, PMM, MoM, CDM, GMT codes, Flatau's site for SVM codes and Macke's page for RT/MCM code. A collection of various SVM codes is available at our site.

Comparison of the methods The best sofar made comparison work is that of Wriedt & Comberg (1998), where they compared DDA, EBCM, GMT and FDTD for 3D scatterers. Other papers on the comparison are cited, for instance, by Mishchenko et al. (2000, p.43).

About the method reviews There are several recent reviews of the methods. We can refer, for example, to such ones:

a) Mishchenko M.I. et al. (2000) in Mishchenko et al. (eds) Light Scattering ...

b) Wriedt Th. (1998) Part. Part. Charact., v.15, p.67-74.

c) Jones A.R. (1999) Prog. Energy Combust. Sci., v.25, p.1-53.

d) Kahnert M. (2003) J. Quant. Spectr. Rad. Transf., v.79-80, p.775-824.

All these reviews are in English and during last 10 years there were nothing of this kind in Russian. In our database we provide the reader with the very good review prepared by N.G.Khlebtsov (see HTML and PostScript versions). The main features of this paper are its historical correctness, completness (over 500 references) and consideration of works made in the former Soviet Union, which is absent in other sources. Later we will also present another more suitable for students review by N.G.Khlebtsov (in form of a PDF-file). The English translations of these reviews are in preparation.
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Created by V.I.
Last modified: 11/05/03, V.I.