Numerical solution of the Helmholtz equation with high wavenumbers

Abstract

SUMMARYThis paper investigates the pollution effect, and explores the feasibility of a local spectral method, the discrete singular convolution (DSC) algorithm for solving the Helmholtz equation with high wavenumbers. Fourier analysis is employed to study the dispersive error of the DSC algorithm. Our analysis of dispersive errors indicates that the DSC algorithm yields a dispersion vanishing scheme. The dispersion analysis is further confirmed by the numerical results. For one-and higher-dimensional Helmholtz equations, the DSC algorithm is shown to be an essentially pollution-free scheme. Furthermore, for large-scale computation, the grid density of the DSC algorithm can be close to the optimal two grid points per wavelength. The present study reveals that the DSC algorithm is accurate and efficient for solving the Helmholtz equation with high wavenumbers.

High-order FDTD methods via derivative matching for Maxwell's equations with material interfaces

“…It should be emphasized that for M = m = 16, the parameter r could be even larger than 2.8, if a smaller l is used. In fact, the range 2.3 ≤ r ≤ 2.8 is sufficiently large for the purpose of selecting optimal r values of the DSC algorithm with M = 16 for scientific computing [24,53]. Similar findings also hold for different M values.…”

“…In the DSC spatial approximation of the LSTD method, apart from the different half computational bandwidth M can be freely chosen, there is a parameter r in the DSC algorithm which can be adjusted to deliver higher accuracy for the same M [22]. In practice, one can select the desired DSC parameters M and r according to the nature of the problem under consideration by means of the discrete Fourier analysis [24,53]. In the present study, it is found that by using a quite small r, the stability constraint of the LSTD method could be the same after the IDM is carried out.…”

“…Although a small r could always be used in the LSTD method to avoid the stability problem in electromagnetic applications, these r values are actually smaller than the usual optimal r values of the DSC approximation for a fixed M [24,53]. Thus, the high accuracy and applicability of the IDM method with LSTD are reduced.…”

“…To this end, we present herein a series of novel hierarchical implicit derivative matching methods for interface modeling, which can greatly exceed fourth-order accuracy. Our schemes make use of FPs, a technique extensively used in our previous work [22][23][24][25][26][51][52][53][54], to locally modify the differential stencils near the interfaces, an idea was first proposed by Driscoll and Fornberg in their BPS methods [39,40]. However, only one single structured grid is assumed in the present work, similar to other high-order embedding FDTD methods [4,[47][48][49][50].…”

Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces

“…Aforementioned methods have found much success in scientific and engineering applications [6][7][8]15,18,20,[25][26][27][28]30,32,34,39,41,40,42,53,54,[57][58][59]. A possible further direction in the field could be the development of higher order interface methods [20,60,61] which are particularly desirable for problems involving both material interfaces and high frequency oscillations, such as the interaction of turbulence and shock, and high frequency wave propagation in inhomogeneous media [5].…”