Absorbing boundary conditions

Boundary procedures that are applied at the artificial numerical boundaries of a computational domain to miminize or eliminate the spurious reflections at these boundaries which occur in the simulations of wave propagation phenomena. They may also be called non-reflecting boundary conditions or radiating boundary conditions. There are different ways to design appropriate boundary procedures; below, only one approach is considered. References to other procedures are given.

Suppose that the solution of the acoustic wave equation,

$$ \tag{a1 } u _ {tt } = c ^ {2} ( u _ {xx } + u _ {yy } ) , $$

on an unbounded domain is desired. The solutions of this equation are plane waves, $ u = u ( x,y,t ) = e ^ {i ( \omega t + \xi x + \eta y ) } $, where $ \omega $ is the frequency and $ \xi $ and $ \eta $ are the spatial wave numbers, which travel in all directions. Computationally, the domain is of finite size and boundaries are introduced. Therefore, waves incident at the boundary will allow for non-physical reflections back into the domain. These reflections are to be minimized. A technique that has proven successful is the application of absorbing boundary conditions which have been derived from approximations to a one-way wave equation at the boundary [a3], [a5], [a6], [a10], [a8].

For (a1), a wave with wave numbers $ \xi $, $ \eta $ travels at the velocity

$$ ( c _ {x} ,c _ {y} ) = c \left ( - { \frac \xi \omega } , - { \frac \eta \omega } \right ) = c ( - \cos \theta, - \sin \theta ) , $$

where $ \theta $ is the angle measured counterclockwise from the negative $ x $- axis; waves with $ | \theta | < 90 ^ \circ $ travel to the left, $ c _ {x} < 0 $, and waves with $ | \theta | > 90 ^ \circ $ travel to the right, $ c _ {x} > 0 $. At an artificial numerical boundary $ x = 0 $, waves should travel to the left, $ c _ {x} \leq 0 $, or, equivalently, $ \xi $ and $ \omega $ are of the same sign. However, solutions of (a1) satisfy the dispersion relation

$$ \tag{a2 } \omega ^ {2} = c ^ {2} ( \xi ^ {2} + \eta ^ {2} ) , $$

and $ \xi $ and $ \omega $ are related by

$$ \tag{a3 } \xi = \pm { \frac \omega {c} } \sqrt {1 - \left ( { \frac{\eta c } \omega } \right ) ^ {2} } . $$

For the positive root, $ \xi $ and $ \omega $ are of the same sign and waves travel to the left only. Therefore, (a3) with the positive sign represents the dispersion relation for the appropriate boundary condition. This dispersion relation, however, corresponds to a pseudo-differential operator. To obtain a partial differential equation for waves which travel only to the left it is necessary to approximate the square root in (a3). The resulting equation is called a one-way wave equation. B. Engquist and A. Majda [a4] designed the paraxial one-way wave equations, based on Padé approximation to $ \sqrt {1 - s ^ {2} } $, $ s = { {\eta c } / \omega } $. For simple Padé approximation, $ 1 - ( {1 / 2 } ) s ^ {2} $, the one-way wave equation used as a boundary condition at $ x = 0 $ is

$$ c u _ {xt } = u _ {tt } - { \frac{1}{2} } c ^ {2} u _ {yy } . $$

A general reference on the derivation of one-way wave equations using rational approximants $ r ( s ) $ is given in [a5]. E.L. Lindman [a6] adopted a similar approach, which yields a system of equations at the boundary that can very easily be augmented to allow for approximations of higher order.

The potential effectiveness of the one-way wave equation is measured by its reflection coefficient

$$ \tag{a4 } R ( s ) = \left | { { \frac{r ( s ) - \sqrt {1 - s ^ {2} } }{r ( s ) + \sqrt {1 - s ^ {2} } } } } \right | , s \in [ - 1,1 ] . $$

For minimal reflection $ R ( s ) \ll1 $ is desirable. The implementation of the boundary condition cannot ensure that the reflection is minimal unless an appropriate numerical approximation can be determined. In particular, a stable finite difference approximation to the one-way wave equation is also required (cf. also Stability; Difference scheme). R.A. Renaut [a8] provides a standard approach by which the differential equation at the boundary can be discretized and gives a stable implementation. Numerical implementations for the solution of the acoustic wave equation and the reflection coefficients are studied in [a8]. For electromagnetics the standard approach uses the Mur boundary conditions, [a7]. In [a12] high-order boundary conditions are tested for the numerical solution of the Maxwell equations.

Finite-difference solutions of partial differential equations are usually local in space because only a few grid points on the computational grid are employed to derive approximations to the underlying partial derivatives in the equation. To obtain more accurate solutions either higher-order approximations can be derived or global solution techniques can be considered. The higher-order finite-difference approximations tend to make the design of boundary conditions more difficult because grid points near the boundary are not automatically defined. To absorb incident waves at the computational boundary one approach uses a damping region. In this case the computational domain is increased in size but the solution is accepted only on the smaller domain. Within the damping region the wave is progressively damped to zero, [a11]. The method is successful but suffers from the disadvantage of a computational overhead induced by the damping region, which is considerable for three dimensions.

Global approximations for partial differential equations as in pseudo-spectral methods are increasingly popular. Pseudo-spectral methods use global interpolation to approximate the unknown function and its derivatives on the computational domain. Implementation of boundary operators is not immediate, although damping regions have been used successfully, [a2]. Recently, absorbing boundary conditions derived from the one-way wave equations have also been successfully implemented for pseudo-spectral methods, [a9].

The perfectly matched layer introduced by J.P. Berenger, [a1], for Maxwell's equations involves the application of a non-physical absorbing material adjacent to the computational boundary. The method is implemented by splitting certain field components in the perfectly matched layer region into subcomponents which can be perfectly absorbed by the perfectly matched layer material. Numerical tests report that this approach is superior to the use of one-way wave equations for electromagnetics.

References

[a1]	J.P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves" J. Comp. Phys. , 114 (1994) pp. 185–200
[a2]	C. Cerjan, D. Kosloff, R. Kosloff, M. Reshef, "A non-reflecting boundary condition for discrete acoustic and elastic wave equations" Geophysics , 50 (1985) pp. 705–708
[a3]	R.W. Clayton, B. Engquist, "Absorbing boundary conditions for acoustic and elastic wave equations" Bull. Seis. Soc. Amer. , 67 (1977) pp. 1529–1540
[a4]	B. Engquist, A. Majda, "Radiation boundary conditions for acoustic and elastic wave calculations" Comm. Pure Appl. Math. , 32 (1979) pp. 313–357
[a5]	L. Halpern, L.N. Trefethen, "Wide-angle one-way wave equations" J. Acoust. Soc. Amer. , 84 (1988) pp. 1397–1404
[a6]	E.L. Lindman, "Free space boundary conditions for the time dependent wave equation" J. Comp. Phys. , 18 (1975) pp. 66–78
[a7]	G. Mur, "Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations" IEEE Trans. Electromagn. Compat. , 23 (1981) pp. 377–382
[a8]	R. A. Renaut, "Absorbing boundary conditions, difference operators and stability" J. Comp. Phys. , 102 (1992) pp. 236–251
[a9]	R.A. Renaut, J. Fröhlich, "A pseudospectral Chebychev method for the 2D wave equation with domain stetching and absorbing boundary conditions" J. Comp. Phys. , 124 (1996) pp. 324–336
[a10]	R.A. Renaut, J. Peterson, "Stability of wide-angle absorbing boundary conditions for the wave equation" Geophysics , 54 (1989) pp. 1153–1163
[a11]	A.C. Reynolds, "Boundary conditions for the numerical solution of wave propagation problems" Geophysics , 43 (1978) pp. 1099–1110
[a12]	P.A. Tirkas, C.A. Balanis, R.A. Renaut, "Higher order absorbing boundary conditions for the finite-difference time-domain method" IEEE Trans. Antennas and Propagation , 40 : 10 (1992) pp. 1215–1222