Implementation of Fourier Expansion Based Differential Quadrature Method (FDQM) and Polynomial Based Differential Quadrature Method (PDQM) for the 2D Helmholtz Problem

Ahmet R. G., Sukru O., Selcuk H.

SCIENTIFIC RESEARCH AND ESSAYS, cilt.8, sa.34, ss.1670-1682, 2013 (Hakemli Dergi)

Özet

The polynomial-based differential quadrature method (PDQM) and Fourier expansion-based differential quadrature method (FDQM) are examined to solve 2D Helmholtz problem expressing definition of many electromagnetic problems. PDQM and FDQM solution of the problems were calculated and compared, analytically. While grid node and wave number are steady, the PDQM and FDQM were applied to 2D Helmholtz problem for calculation and comparison. While the grid node is equal to 7 (N=M=7) and the wave number is equal to 1 (k=1),  (maximum absolute error between the numerical result and the exact solution) for PDQM and for FDQM has been found as 2.758×10-5 and 2.921×10-6 respectively. Then wave number was increased in order to examine the effects of its variation to the accuracy, while grid node was kept in steady state. While grid node was steady and equal to 7 and the wave number was increased from 1 to 5,  for FDQM has been found as 2.921×10-6, 7.995×10-5, 1.020×10-4, 0.1825 and 1.779 respectively. It has been found that the FDQM is more suitable than PDQM for 2D Helmholtz problem having harmonics, and it has been observed that the wave number increases, the accuracy of FDQM results gradually decreases in the case of fixed mesh size and computational domain.