Hybrid Spectral Difference Methods for Elliptic Equations on Exterior Domains with the Discrete Radial Absorbing Boundary Condition
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Abstract
The hybrid spectral difference methods (HSD) for the Laplace and Helmholtz equations in exterior domains are proposed. We consider the fictitious domain method with the absorbing boundary conditions (ABCs). The HSD method is a finite difference version of the hybridized Galerkin method, and it consists of two types of finite difference approximations; the cell finite difference and the interface finite difference. The fictitious domain is composed of two subregions; the Cartesian grid region and the boundary layer region in which the radial grid is imposed. The boundary layer region with the radial grid makes it easy to implement the discrete radial ABC. The discrete radial ABC is a discrete version of the Bayliss–Gunzburger–Turkel ABC without pertaining any radial derivatives. Numerical experiments confirming efficiency of our numerical scheme are provided.
Keywords
Absorbing boundary condition Cell finite difference Helmholtz equation Hybrid difference Interface finite differenceMathematics Subject Classification
65N30 65N38 65N501 Introduction
In this paper a new absorbing boundary condition (that is named the discrete radial ABC) is introduced for the Laplace and Helmholtz equations. To derive the discrete ABC we follow the idea of the Bayliss–Gunzburger–Turkel (BGT) [3] absorbing boundary condition. To the author’s knowledge there has not been much attention to the ABCs for frequency domain problems. However, the frequency domain approach on a parallel computing environment can provide very efficient numerical solvers. On the other hand there have been many efforts for construction of the ABCs for the time dependent wave equations. Engquist and Majda [5] made a pioneering work, in which the ABC is obtained by considering waves that enter or leave the domain and annihilating those scattered waves entering the domain from the outside. Bayliss and Turkel [2] constructed a sequence of absorbing boundary conditions for the wave equation, based on annihilation of first several negative radial power terms in the multipole expansion of a solution, and those idea were exploited later to derive the BGT condition with Gunzburger for the static elliptic and wave equations. The high order derivative term of the ABC in [2] can be impractical for implementation, and Hagstrom and Hariharan [7] developed the ABCs that contain only first order derivatives at a cost of auxiliary functions. Huan and Thompson [9] developed these ideas further to obtain the ABCs applicable in the finite element method in a symmetric way. Higdon [8] obtained absorbing boundary conditions by invoking to the finite time difference rather than the differential equation, which improved convergence quality of numerical solutions. Our discrete radial ABC is also obtained by treating the annihilation condition of the BGT condition directly in the radial finite difference equation. For absorbing boundary conditions with elliptic or general shaped obstacles we refer to [1, 13, 14]. For a brief review on the various ABCs we refer to [6] and references therein.
For a numerical method of partial differential equations we consider the hybrid spectral difference method (HSD). The HSD can be understood as the finite difference version of the hybridized Galerkin method. The HSD is simple to implement and it has several advantages over the conventional finite difference methods. The method can be applied to nonuniform grids, retaining the optimal order of convergence, and numerical methods with an arbitrarily highorder convergence can be obtained easily. Problems on a complicated geometry can be treated reasonably well, and the boundary condition can be imposed exactly on the exact boundary. The mass conservation property holds in each cell and flux continuity holds across intercell boundaries. Embedded static condensation property considerably reduces global degrees of freedom [10, 11, 12].
The aim of this paper is to introduce the discrete radial ABC and the hybrid spectral difference method for elliptic equations. The hybrid spectral difference method can manage reasonably well obstacles of a general shape in scattering problems. The fictitious boundary can be taken as a circle, independently of the shape of obstacles, which makes easy the implementation of the discrete radial ABC of a high order.
The paper is organized as follows. In Sect. 2 we review the Bayliss–Gunzburger–Turkel absorbing condition, and the discrete radial ABC is introduced by following the annihilation idea of them. The high order radial derivatives in the BGT condition may not be approximated efficiently in the finite difference setting when the wave number is large. With the discrete radial ABC an exact annihilation is accomplished. In Sect. 3 unique solvability and convergence analysis on an annulus region are presented, and the proofs follow closely the ideas in [3]. In Sect. 4 the HSD on the rectangular grid with the Cartesian coordinate is introduced. The HSD in the polar grid can be derived in a similar manner and we delete details. Section 5 is devoted to numerical experiments for the exterior harmonic and Helmholtz equations. Some comparisons between the BGT condition and discrete radial ABC are made. It is shown that the discrete radial ABC produces quite accurate numerical solutions even for the wave equation of a large wave number. A scattering problem with a scatterer of a general shape is tested, which justify that our new method can be an effective numerical solver for a class of scattering problems.
2 Absorbing Boundary Conditions on the Boundary Layer
In this section we review the Bayliss–Gunzbuger–Turkel (BGT) absorbing boundary condition, and a new absorbing boundary condition (the discrete radial ABC) is proposed. The discrete radial ABC bears the same idea of the BGT condition, and it can be more easily implemented for a high order method. Moreover, convergence looks better than that of the BGT condition in finite difference settings.
2.1 The BGT Absorbing Boundary Condition for the Laplace Equation
2.2 The BGT Absorbing Boundary Condition for the Helmholtz Eqution
2.3 The Discrete Radial Absorbing Boundary Condition
FD approximation of the BGT ABC: \(\omega =1\)
N  4  8  16  32 

\(  \mathcal B_2^h(e^{i\omega r} r^{1/2} ) \)  2.8616e−05  1.7746e−06  1.1075e−07  6.9197e−09 
\(  \mathcal B_2^h(e^{i\omega r} r^{3/2})  \)  1.9248e−01  1.9245e−01  1.9245e−01  1.9245e−01 
\(  \mathcal B_3^h(e^{i\omega r} r^{1/2}) \)  4.5465e−03  5.5894e−04  6.9440e−05  8.6565e−06 
\(  \mathcal B_3^h(e^{i\omega r} r^{3/2} ) \)  4.7575e−03  4.4202e−04  4.7811e−05  5.5631e−06 
FD approximation of the BGT ABC: \(\omega =10\)
N  4  8  16  32 

\( \mathcal B_2^h(e^{i\omega r} r^{1/2})  \)  3.4244e−00  2.6480e−01  1.7358e−02  1.0954e−03 
\(  \mathcal B_2^h(e^{i\omega r} r^{3/2} )\)  1.4095e−00  2.6483e−01  1.9408e−01  1.9245e−01 
\(  \mathcal B_3^h(e^{i\omega r} r^{1/2}) \)  5.6022e−02  8.4025e−01  1.0902e−01  1.3709e−00 
\( \mathcal B_3^h(e^{i\omega r} r^{3/2} ) \)  1.9977e−02  2.8842e−01  3.6758e−00  4.5824e−01 
FD approximation of the discrete radial ABC: \(\omega =10\)
N  4  8  16  32 

\(  \mathcal R_2^h(e^{i\omega r} r^{1/2} ) \)  5.1238e−14  1.4211e−13  1.2711e−13  2.5421e−13 
\(  \mathcal R_2^h(e^{i\omega r} r^{3/2})  \)  1.0000  1.0000  1.0000  1.0000 
\(  \mathcal R_3^h(e^{i\omega r} r^{1/2}) \)  1.8190e−12  8.1348e−12  2.1828e−11  6.5078e−11 
\(  \mathcal R_3^h(e^{i\omega r} r^{3/2} ) \)  2.5421e−13  2.0337e−12  7.2760e−12  2.0580e−11 
On the other hand the discrete ABC (2.2) approximate the absorbing boundary condition exactly as shown in Table 3.
Remark 2.1
3 Solvability and Convergence Analysis
In this section we provide the unique solvability and convergence analysis for the discrete radial ABC of the exterior Laplace equation on an annulus region. The analysis closely follows those ideas in [3]. It must be possible to extend those analysis to the case of the Helmholtz equation. For simplicity of analysis we consider the fictitious domain as an annulus, \( 1< r < r_0\).
Theorem 3.1
Proof
Remark 3.2
Theorem 3.3
Proof
4 Hybrid Finite Difference Methods
As mentioned in Sect. 1 the domain \(\Omega =\Omega _1 \cup \Omega _2\) is a multiply connected domain with the inner boundary \(\Gamma _1\), the outer boundary \(\Gamma _2\) and the interface \(\Gamma _{12} \). To define the hybrid difference method (HDM) we need a quasirectangular partition of the subdomain \(\Omega _1\). Here, the quasirectangle includes a rectangle, a trapezoid (Fig. 3) or a trapezoidlike mesh with one rounded side (Fig. 4). For the subdomain \(\Omega _2\) we consider a radial subdivision by polar rectangles. In this section we describe the HDM in the Cartesian coordinate system because the hybrid difference in the polar coordinate can be obtained analogously. The hybrid spectral difference method (HSD) is a high order version of the hybrid difference method.
Let \(K_h\) denote the skeleton of a mesh generation \(\mathcal {T}_h\) of \(\Omega \), and let \(\mathcal N(\Omega )\) and \(\mathcal N(K_h)\) denote the set of grid points in the closure of a domain and that on its skeleton, respectively. It is worth to note that the mesh is composed mostly of rectangles. However, we need trapezoidal meshes to match the interface \(\Gamma _{12}\) and we may need rounded trapezoidal meshes to match \(\Gamma _1\) if \(\Gamma _1\) is curved.
\(\{ u(\eta _1), u(\eta _3), u(\eta _5), u(\eta _8)\}\) for each cell. Then, the interface difference (4.2) yields the global stiffness system with unknowns \(\{ u (\eta ): \eta \in \mathcal N (K_h) \}\). Therefore, the static condensation property is naturally embedded in the HDM.
Reduction in degrees of freedom by static condensation
\(Q_2^*\hbox {grid}\)  \(Q_4^*\hbox {grid}\)  \(Q_k^*\hbox {grid}\)  

HDM  \(N^2+2N(N+1)\)  \(9N^2+6N(N+1)\)  \((k1)^2 N^2 + 2(k1)N(N+1)\) 
Condensed  \(2N(N+1)\)  \(6N(N+1)\)  \(2(k1)N(N+1)\) 
Reduction rate  \(\frac{2}{3}\)  \(\frac{2}{5}\)  \(\frac{2}{k+1}\) 
Remark 4.1
The subdomain \(\Omega _2\) can contain one or two layers of polar (quasi) rectangles. To combine a low order HDM and a high order ABC we need two layers, and just one layer is enough for combination of the \(Q_m^*\) grid HDM and the \(\mathcal R_n^h\) ABC if \(n \le m+1\).
Remark 4.2
Suppose a domain is the unit square with a \(N\times N\)rectangular partition. Table 4 highlights reduction in degrees of freedom by the static condensation. The reduction rate is computed asymptotically under the assumption that N is a big number.
5 Numerical Experiments

An annulus with \(1< r < \frac{3}{2}\),

A Chinese coin (see Fig. 1) bounded by \(\Gamma _1 = \{ \max \{x,y\}=1 \}\) and \(\Gamma _2 = \{ r=3\}\),

The domain “D” bounded by \(\Gamma _1 = \{ x=1, \ y = 1, \ x^2 + y^2 = 1 \}\) and \(\Gamma _2 = \{ r=3\}\).
Here, \(\mathcal R_m^h u =0\) and \(\mathcal B_m^h u=0\) are called the mth order ABC. The first observation was comparison of condition numbers for the discrete and BGT ABCs. Both methods yield similar condition numbers and we omit details.
Example 1
Example 2
Example 3
The scattering problem with the sound soft boundary condition is solved on the Chinese coin domain. Here, the grids are give as in Fig. 1. The incident wave \(u_{in} = e^{i \omega (x \cos \alpha +y \sin \alpha )}\) with the incident angle \(\alpha = 0\) and \(\omega =5\) is tested, and the scattered wave is plotted by increasing the degrees of freedom. The 4th order discrete radial ABC (\({\mathcal {R}}_4^{h}u =0\)) on the \(Q_4^*\) grid is used for numerical computation.
Example 4
The scatterer of the shape “D” is tested with the incident wave \(u_{in} = e^{i \omega (x \cos \alpha +y \sin \alpha )}\) and the incident angles \(\alpha = 0, \frac{\pi }{4}, \frac{\pi }{2}, \pi \). The same computational parameters are chosen as in Example 3.
As shown in Fig. 5 the discrete radial and BGT ABCs perform evenly well for the Laplace equation. The higher order ABC shows better convergence than the lower order ABC. As the mesh becomes finer the error seems to be dominated by the truncation error in the absorbing boundary condition. For the Helmholtz equation numerical examples only with the discrete radial ABC are presented. We observe that the BGT absorbing boundary condition produces spurious numerical solutions because of the poor finite difference approximation property as mentioned in Sect. 2, and we omit it here. As shown in Figs. 6 and 7 the discrete radial ABC performs quite robust even for a larger wave number (in view of the level of discretization). Here, the computation is performed on a \(N \times 2N\) mesh by doubling N up to \(N=64\) for the radial \(Q_2^*\) grid and up to \(N=32\) for the radial \(Q_4^*\) grid. When the wave number is small the higher order ABC performs clearly better than the lower order ABC. However, convergence property becomes similar regardless of orders of the discrete radial ABC when the wave number increases. Even the 2nd order method \(\mathcal R_2^h u=0\) performs the best for the cases, \(\omega =50, \ 100\). Nonetheless, the high order hybrid difference combined with the high order discrete radial ABC (for example, the \(Q_4^*\) grid method and \(\mathcal R_4^h=0\)) produces more reliable solutions for a wide range of wave numbers. From those observation the scattering problems in Examples 3 and 4 are solved by using the ABC, \(\mathcal R_4^h (u)=0\) on the \(Q_4^*\) grid. Figure 8 clearly shows how the scattered wave converges as the degrees of freedom (dof) increase. Figure 9 represents the real part of the scattered wave with various incident angles, and this example shows that our method can manage a complicated geometryproblem quite well. The numerical results accord well with those existing numerical results.
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