Accurate radiation boundary conditions for the time-dependent wave equation on unbounded domains

Abstract

Asymptotic and exact local radiation boundary conditions (RBC) for the scalar time-dependent wave equation, ÿrst derived by Hagstrom and Hariharan, are reformulated as an auxiliary Cauchy problem for each radial harmonic on a spherical boundary. The reformulation is based on the hierarchy of local boundary operators used by Bayliss and Turkel which satisfy truncations of an asymptotic expansion for each radial harmonic. The residuals of the local operators are determined from the solution of parallel systems of linear ÿrstorder temporal equations. A decomposition into orthogonal transverse modes on the spherical boundary is used so that the residual functions may be computed e ciently and concurrently without altering the local character of the ÿnite element equations. Since the auxiliary functions are based on residuals of an asymptotic expansion, the proposed method has the ability to vary separately the radial and transverse modal orders of the RBC. With the number of equations in the auxiliary Cauchy problem equal to the transverse mode number, this reformulation is exact. In this form, the equivalence with the closely related non-re ecting boundary condition of Grote and Keller is shown. If fewer equations are used, then the boundary conditions form highorder accurate asymptotic approximations to the exact condition, with corresponding reduction in work and memory. Numerical studies are performed to assess the accuracy and convergence properties of the exact and asymptotic versions of the RBC. The results demonstrate that the asymptotic formulation has dramatically improved accuracy for time domain simulations compared to standard boundary treatments and improved e ciency over the exact condition.

Exact solutions for transient spherical radiation

“…This is constructed by superposition of multipole components and forms a transient analogue of the steady harmonic test problem used to check transient computations at large times. 27,28 For all of the exact solutions presented here, close agreement is demonstrated both with steady time harmonic solutions at large times and with accurate high-order numerical schemes at finite elapsed times. Figure 1͑a͒ shows the geometry of the problem to be considered.…”

“…The steady harmonic solution for a piston in a spherical baffle has been used extensively for this purpose. 27,28 Such checks do not of course provide any assurance of the accuracy or correctness of a transient solution at finite elapsed times.…”

Local nonreflecting boundary condition for time-dependent multiple scattering

“…However, this new boundary condition does not require any spherical harmonics or inner products with them; hence, it is somewhat easier and cheaper to implement. The usefulness and accuracy of the NBC (21) was illustrated via numerical experiments in [28]. It was also recently extended to Maxwell's equations in three space dimensions [16].…”

“…In particular, Hagstrom and Hariharan [25] derived a new formulation of the classical Bayliss and Turkel [6] conditions of arbitrarily high order, yet without high order derivatives. It holds for B a sphere and is local both in space and time, while its high accuracy and efficiency in computations was shown in [28]. Similarly, Givoli and Neta [15] and Hagstrom and Warburton [26] each proposed a reformulation of the Higdon [27] conditions without high order derivatives for rectangular B -see [14] for a recent review.…”

Computational simulation of wave propagation problems in infinite domains

“…For example, several kinds of wave absorbing boundaries such as the viscous damping boundary [1,2], wave transmitting boundary [3], consistent boundary [4], boundary elements [5,6], perfectly matched layer (i.e. PML) absorbing boundary [7,8] and high-order accurate wave absorbing boundary [9][10][11][12][13][14][15][16][17][18] have been proposed to absorb wave energy at the common boundary between the near field and the far field. The basic idea behind the PML absorbing boundary is that a special layer of finite thickness is used to replace the far field of an infinite domain.…”