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

FRANCISCO MARCELO FERNÁNDEZ; PAOLO AMORE; BORIS RÖSLER; JOHN. P. BOYD

JOURNAL OF COMPUTATIONAL PHYSICS

ACADEMIC PRESS INC ELSEVIER SCIENCE

Lugar: Amsterdam; Año: 2016 vol. 312 p. 252 - 271

0021-9991

We apply second order finite differences to calculate the lowest eigenvalues of theHelmholtz equation, for complicated non-tensor domains in the plane, using different gridswhich 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 eigenvaluescorresponding to different grids allow us to obtain extremely precise values. When possible we have assessed the precision of our extrapolations comparing them with the highlyprecise results obtained using the method of particular solutions. Our empirical findingssuggest an asymptotic nature of the FD series. In all the cases studied, we are able toreport numerical results which are more precise than those available in the literature.