High order eigenvalues for the Helmholtz equation in complicated non-tensor domains through Richardson extrapolation of second order finite differences

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• 作者：Paolo Amore^a ; ^{paolo.amore@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; John P. Boyd^b ; ^{jpboyd@umich.edu" class="auth_mail" title="E-mail the corresponding author} ; ; Francisco M. Ferná ; ndez^c ; ^{fernande@quimica.unlp.edu.ar" class="auth_mail" title="E-mail the corresponding author} ; Boris Rö ; sler^d ; ^{info@boris.net" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Finite difference ; Richardson extrapolation ; Helmholtz equation
• 刊名：Journal of Computational Physics
• 出版年：2016
• 出版时间：1 May 2016
• 年：2016
• 卷：312
• 期：Complete
• 页码：252-271
• 全文大小：810 K

文摘

We apply second order finite differences to calculate the lowest eigenvalues of the Helmholtz equation, for complicated non-tensor domains in the plane, using different grids which sample exactly the border of the domain. We show that the results obtained applying Richardson and Padé–Richardson extrapolations to a set of finite difference eigenvalues corresponding to different grids allow us to obtain extremely precise values. When possible we have assessed the precision of our extrapolations comparing them with the highly precise results obtained using the method of particular solutions. Our empirical findings suggest an asymptotic nature of the FD series. In all the cases studied, we are able to report numerical results which are more precise than those available in the literature.