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Another kind of numerical instabilities of the integral approach to the interior boundary‐value problem for the two‐dimensional Helmholtz equation
Author(s) -
Mattioli Franco
Publication year - 1982
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1620180802
Subject(s) - helmholtz equation , mathematics , wavenumber , mathematical analysis , instability , helmholtz free energy , function (biology) , boundary value problem , domain (mathematical analysis) , integral equation , boundary (topology) , physics , mechanics , optics , quantum mechanics , evolutionary biology , biology
In a previous paper it has been proved by the author that the integral equations arising from the application of Green's formula to the Helmholtz equation in a limited domain can show a certain type of numerical instability, if a real Green's function is used. It has been also proved that such instabilities cannot arise if a complex Green's function is employed. However, it has been found in this latter case also that numerical instabilities can occur. This has been proved and thoroughly analysed for a circular domain, and a technique of avoiding these instabilities has been devised. Furthermore, when this technique is followed, very accurate results can be obtained, regardless of wavenumber used. Thus, only three or four segments are sufficient to describe a wavelength, contrary to what until now has been obtained, i.e. that at least six segments are always necessary. This last result has been shown to be valid also for geometries other than the circular one.