Elsevier

Journal of Computational Physics

Volume 242, 1 June 2013, Pages 387-404
Journal of Computational Physics

A general approach for high order absorbing boundary conditions for the Helmholtz equation

https://doi.org/10.1016/j.jcp.2013.01.032Get rights and content

Abstract

When solving a scattering problem in an unbounded space, one needs to implement the Sommerfeld condition as a boundary condition at infinity, to ensure no energy penetrates the system. In practice, solving a scattering problem involves truncating the region and implementing a boundary condition on an artificial outer boundary. Bayliss, Gunzburger and Turkel (BGT) suggested an Absorbing Boundary Condition (ABC) as a sequence of operators aimed at annihilating elements from the solution’s series representation. Their method was practical only up to a second order condition. Later, Hagstrom and Hariharan (HH) suggested a method which used auxiliary functions and enabled implementation of higher order conditions.

We compare various absorbing boundary conditions (ABCs) and introduce a new method to construct high order ABCs, generalizing the HH method. We then derive from this general method ABCs based on different series representations of the solution to the Helmholtz equation – in polar, elliptical and spherical coordinates. Some of these ABCs are generalizations of previously constructed ABCs and some are new.

These new ABCs produce accurate solutions to the Helmholtz equation, which are much less dependent on the various parameters of the problem, such as the value of k, or the eccentricity of the ellipse. In addition to constructing new ABCs, our general method sheds light on the connection between various ABCs. Computations are presented to verify the high accuracy of these new ABCs.

Introduction

Describing the behavior of scattered waves about a body is essential in many fields, whether it be the reflections of sonar wave from a submarine, the reflection of radar waves from an airplane or the reflection of microwaves from a cellular phone. The behavior of the wave is described by the wave equation:ψtt=c22ψSeparating variables:ψ(x,t)=u(x)v(t)we get the equations:2u+k2u=0k=ωcvtt+ω2v=0

Eq. (1) is known as the Helmholtz equation. Thus, the Helmholtz equation represents the spatial behavior of a wave with frequency ω=ck. We note that given a condition for the wave equation, one can always formally get one for the Helmholtz equation by vtiωv. The reverse is true only if the operator is a rational function of iω. Otherwise, it leads to pseudo-differential operators.

We consider the following exterior problem: a wave propagates through a medium, impacts a given body and scatters. To model this, we consider an unbounded domain, X, with an inner boundary Γ0. We denote the propagating wave as w(x), and the scattered wave as u(x). Hence, we wish to solve the Helmholtz equation for u+w on X, using a boundary condition on Γ0 (we present a Neumann condition, but any boundary condition can be applied):[u+w]+k2(u+w)=0xX(u+w)n=0xΓ0where n is the normal to Γ0.

We consider a planar incident wave of the form:w=eikw·xwhere kw is the wave vector, with wave number kw and angle θw. This incident wave is a solution to the Helmholtz equation with wavenumber kw and thus we can write the scattered formulation of our problem:u+k2u=kw2-k2eikw·xxXun=-wnxΓ0

This formulation has only one boundary condition - the inner boundary condition (2). In order for the problem to be well-posed in the unbounded domain, we impose another boundary condition ”at infinity” - the Sommerfeld condition:ur-iku=o1rr2Dur-iku=o1rr3D

This condition implies that the solution involves only outgoing waves.

Section snippets

Exact solutions

We now present some exact solutions for the homogeneous Helmholtz equation. We recall that the scattering problem reduces to the homogeneous case if the incident wave has wavenumber kω=k. These solutions will allow us to compare the numerical results we will obtain later to exact solutions.

When the inner boundary is circular, it is natural to work with polar coordinates. The incident wave is given by:w=eikwrcosθ-θw

Separating variables, a series of Bessel functions results:w=m=-imJmkwreimθ-θw

Absorbing boundary conditions

For the wave equation an absorbing boundary condition was developed by Engquist and Majda [6] based on a splitting of the wave equation. This led to a pseudo-differential equation which was approximated by either a Taylor or rational approximation. This was later extended to higher order approximations by Hagstrom, Givoli et al. in a series of papers [7], [8], [11], [12]. These can all be formally extended to the Helmholtz equation by the formal substitution utiku. A different approach was

A general high order boundary condition

In order to construct an ABC for the Helmholtz equation, we first consider a more general case. When solving a PD Lu(α,β)=0 by separation of variables, one sets u(α,β)=A(α)B(β) and gets:-LA(α)A(α)=λ=LB(β)B(β)

Usually, λ has a countable set of values (λn). Then, the solution can be represented as a series:u(α,β)=j=0υjAj(α)Bj(β)We consider such a series whereLBj(β)=λjBj(β)It should be noted that L[·] is not the operator forming the PDE, but only an operator for which the Bj are its

Deriving ABCs for 2D Helmholtz equation

We now derive different ABCs based on the method proposed in the previous section. We examine different representations of the solution to the Helmholtz equation and derive different ABCs.

Polar spherical coordinates, using Hankel functions and spherical harmonics

We consider the solution of the 3D Helmholtz equation:u=j=0vjHj(kr)Yj(θ,ϕ)where Yj are spherical harmonics of degree j, and (Hj(r) solves the three dimensional Bessel equation and so we solve:r22Hjr2+2rHjr+r2-j(j+1)Hj=0To fit our notations we consider the following:αrβ(θ,ϕ)Aj(α)Hj(kr)Bj(β)Yj(θ,ϕ)L[·]=s[·]λj=-j(j+1)We derive the recursive functions:Aj(0)(r)=Hj(kr)Aj(1)=H0Hj(kr)r-H0(kr)rHj(kr)Aj(m+1)=-2mAm(m-1)Aj(m)-(m+j)(m-j-1)Am(m)Aj(m-1)We then construct the operators by

Numerical results

In this section we present numerical results of the various old methods and compare them to the new ABCs.

Conclusion

The Helmholtz equation describes wave propagation in the frequency domain. This is important both in acoustics and electromagnetism. Since the domain is unbounded, a Sommerfeld radiation condition is required to make it well-posed. For a numerical approximation we need to replace the unbounded domain by a finite domain bounded by an artificial boundary. One then needs to impose a boundary condition on that surface to make it both well-posed and accurate. One of the original approaches was that

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