8.4 Wave-Absorbing Layers
435
A second solution, indicated by the thin solid line in Fig. 8.7, was obtained using
the linear extrapolation boundary condition (8.18) at x = L instead of using a
wave-absorbing layer.
Figure 8.7 focuses on the trailing edge of the disturbance in the subdomain
0.7 ::: x ::: 1. The time shown is 1 =
at which time three-quarters of the
initial pulse has passed into the wave-absorbing layer. The interface between the
absorbing layer and the interior domain coincides with the right edge of each plot.
The horizontal grid spacing for the simulation shown in Fig. 8.7a is 1/32 and cx
is chosen such that RN 6.1 is unity. Considerable reflection is produced by the
absorbing layer, which is only four grid points wide. Weaker reflection is also
produced by the extrapolation boundary condition. Figure 8.7b shows the solutions to the same physical problem obtained after halving Sx, which increases
the width of the absorbing layer to eight grid points. Since the Courant number
is fixed, 6.1 is also halved, and the maximum value of Rn6.1 is reduced to onehalf. Both solutions are improved by this increase in resolution, but the sponge
layer continues to produce significantly more reflection than that generated by the
extrapolation boundary condition. The grid spacing is reduced by an additional
factor of four in Fig. 8.7c, so that 6.x = 1/256. This increase in numerical resolution continues to improve the solution obtained with the extrapolation boundary
condition, but does not improve the solution obtained with the wave-absorbing
layer. Further increases in the resolution do not make the solution computed with
the wave-absorbing layer converge toward the correct solution. Such convergence
is, however, exhibited by the solution obtained using the extrapolation boundary
condition.
ing to the correct solution within the absorbing layer as Sx
0 and 6.1
0, the
error outside the absorbing layer remains small and may be acceptable in some
applications . Of course, the use of a Rayleigh-damping absorber is not actually
recommended in situations, such as this, where an exact open boundary condition can be formulated for the original partial differential equation. It is harder to
make a clear-cut recommendation in the vast majority of cases, for which exact
open boundary conditions are not available. It is certainly possible that over some
range of numerical parameters, the errors generated by the ill-posed Rayleigh
damping absorber can be less than those obtained using well-posed numerical
approximations to overly reflective boundary conditions. Indeed, the Rayleighdamping absorber appears to have been used successfully in a variety of studies.
435
A second solution, indicated by the thin solid line in Fig. 8.7, was obtained using
the linear extrapolation boundary condition (8.18) at x = L instead of using a
wave-absorbing layer.
Figure 8.7 focuses on the trailing edge of the disturbance in the subdomain
0.7 ::: x ::: 1. The time shown is 1 =
at which time three-quarters of the
initial pulse has passed into the wave-absorbing layer. The interface between the
absorbing layer and the interior domain coincides with the right edge of each plot.
The horizontal grid spacing for the simulation shown in Fig. 8.7a is 1/32 and cx
is chosen such that RN 6.1 is unity. Considerable reflection is produced by the
absorbing layer, which is only four grid points wide. Weaker reflection is also
produced by the extrapolation boundary condition. Figure 8.7b shows the solutions to the same physical problem obtained after halving Sx, which increases
the width of the absorbing layer to eight grid points. Since the Courant number
is fixed, 6.1 is also halved, and the maximum value of Rn6.1 is reduced to onehalf. Both solutions are improved by this increase in resolution, but the sponge
layer continues to produce significantly more reflection than that generated by the
extrapolation boundary condition. The grid spacing is reduced by an additional
factor of four in Fig. 8.7c, so that 6.x = 1/256. This increase in numerical resolution continues to improve the solution obtained with the extrapolation boundary
condition, but does not improve the solution obtained with the wave-absorbing
layer. Further increases in the resolution do not make the solution computed with
the wave-absorbing layer converge toward the correct solution. Such convergence
is, however, exhibited by the solution obtained using the extrapolation boundary
condition.
ing to the correct solution within the absorbing layer as Sx
0 and 6.1
0, the
error outside the absorbing layer remains small and may be acceptable in some
applications . Of course, the use of a Rayleigh-damping absorber is not actually
recommended in situations, such as this, where an exact open boundary condition can be formulated for the original partial differential equation. It is harder to
make a clear-cut recommendation in the vast majority of cases, for which exact
open boundary conditions are not available. It is certainly possible that over some
range of numerical parameters, the errors generated by the ill-posed Rayleigh
damping absorber can be less than those obtained using well-posed numerical
approximations to overly reflective boundary conditions. Indeed, the Rayleighdamping absorber appears to have been used successfully in a variety of studies.