Sixth Order Compact Finite Difference Method for 2D Helmholtz Equations with Singular Sources and Reduced Pollution Effect

Abstract

Due to its highly oscillating solution, the Helmholtz equation is numerically challenging to solve. To obtain a reasonable solution, a mesh size that is much smaller than the reciprocal of the wavenumber is typically required (known as the pollution effect). High order schemes are desirable, because they are better in mitigating the pollution effect. In this paper, we present a sixth order compact finite difference method for 2D Helmholtz equations with singular sources, which can also handle any possible combinations of boundary conditions (Dirichlet, Neumann, and impedance) on a rectangular domain. To reduce the pollution effect, we propose a new pollution minimization strategy that is based on the average truncation error of plane waves. Our numerical experiments demonstrate the superiority of our proposed finite difference scheme with reduced pollution effect to several stateof-the-art finite difference schemes in the literature, particularly in the critical pre-asymptotic region where kh is near 1 with k being the wavenumber and h the mesh size.

Hybrid Finite Difference Schemes for Elliptic Interface Problems with Discontinuous and High-Contrast Variable Coefficients

“…We follow the same setup as in [6,7,8]. Let Ω = (l 1 , l 2 ) × (l 3 , l 4 ) and we assume l 4 − l 3 = N 0 (l 2 − l 1 ) for some N 0 ∈ N. For any positive integer N 1 ∈ N, we define N 2 := N 0 N 1 and so the grid size is h := (l 2 − l 1 )/N 1 = (l 4 − l 3 )/N 2 .…”

“…achieves sixth order of accuracy for [8] by (5.4), (5.5), (5.6) and (5.7), respectively. Moreover, the maximum accuracy order of a 4-point finite difference scheme for B 1 u = ∂u ∂ n + αu = g 1 and B 4 u = ∂u ∂ n + βu = g 4 at the point (x 0 , y N 2 ) with three smooth functions α(y), β(x) and a(x, y) is six, where α(y…”

“…) and (5.5). As in [6,7,8], near the point (x * i , y * j ), the parametric equation of Γ I can be written as:…”

“…In [8], we discussed the 6-point and 4-point finite difference schemes with sixth order accuracy for the side points and corner points of the Helmholtz equations respectively with a constant wave number k in a rectangle. In this paper, we also extend the above results in [8] to the elliptic equations with variable coefficients and mixed combinations of Dirichlet u| Γ i = g i , Neumann ∂u ∂ n | Γ j = g j and Robin ∂u ∂ n + αu| Γ k = g k with smooth functions α, g i , g j and g k , where Γ i /Γ j /Γ k for i, j, k = 1, 2, 3, 4 is one side of the rectangle (see Fig. 2 for an example of the mixed boundary conditions).…”

Compact 9-point finite difference methods with high accuracy order and/or M-matrix property for elliptic cross-interface problems

“…For elliptic problems with various boundary conditions in non-rectangular domains, [21] proposed a fourth-order augmented MIB with the FFT acceleration, [24] developed a second-order explicit-jump IIM, and [12,16,20] proposed third/fourthorder FDMs. In [4], we discussed sixth-order FDMs for various boundary conditions of the Helmholtz equation with a constant wave number k. In this paper, we consider the elliptic equation with the variable coefficient a and mixed combinations of Dirichlet, Neumann ∂u ∂⃗ n , and Robin ∂u ∂⃗ n + αu, ∂u ∂⃗ n + βu with variable functions α, β (see Fig. 1 for an example of the mixed boundary conditions).…”