406
8. Nonreflecting Boundary Conditions
Solving for r, one obtains
ik A
k Sx
r = ie' ax tan - - ,
2
in which case
Irl = tan
I
k Sx I
-2- .
If the incident wave is well-resolved, then k b.x « 1, and there is very IittIe reflection . The magnitude of the reflection coefficient rises to unity for the 4b.x wave,
and it approaches infinity for a 2b.x wave. It may appear that the large reflection
coefficients associated with very short waves make the zero gradient condition unusable, but if (8.15) is applied at the outflow boundary, no waves shorter than 4b.x
will ever reach that boundary because the group velocities of these short waves are
directed upstream. The zero-gradient condition is therefore a possibly useful outflow boundary condition for the advection equation. On the other hand, the large
reflection coefficients associated with the shortest waves do render (8.15) unsuitable as an inflow boundary condition because waves with wavelengths between
2b.x and 4b.x propagate upstream and rapidly amplify when they encounter the
inflow boundary.
Nitta (1962) and Matsuno (l966a) used a similar procedure to analyze the reflections generated by a wide variety of boundary conditions. The magnitude of
the reflection coefficient generated by a fixed boundary value, such as f/Jo = 0,
is unity. Although fixed boundary values are not appropriate at outflow, they are
useful at the inflow boundary, where the only waves that encounter the boundary
will be low-amplitude computational modes. Second-order extrapolation,
f/Jo = 2f/JI - eh,
has reflection IrI = Itan k Sx /21
2 • The general nth-order extrapolation
(8.18)
L(-l)m
n
(n') '( )'
m=O
m . n m.
f/Jm = 0
reflects with amplitude IrI = Itan k Sx /2I
n • Higher-order extrapolation reduces
the reflection of well-resolved physical waves, but it increases the reflection of
computational modes.
As an alternative to extrapolation, one might employ a one-sided finite difference at an outflow boundary, replacing (8.14) by
df/Jo + U (f/JI - f/Jo) = O.
(8.19)
dt
b.x
The reflections generated by this scheme may once again be analyzed by substituting the general solution (8.17) into (8.19). The result is
-ieu(1 + r) + .!:!.-(e ikt1x - re- ikß x) - .!:!.-(l + r) = O.
b.x
b.x
8. Nonreflecting Boundary Conditions
Solving for r, one obtains
ik A
k Sx
r = ie' ax tan - - ,
2
in which case
Irl = tan
I
k Sx I
-2- .
If the incident wave is well-resolved, then k b.x « 1, and there is very IittIe reflection . The magnitude of the reflection coefficient rises to unity for the 4b.x wave,
and it approaches infinity for a 2b.x wave. It may appear that the large reflection
coefficients associated with very short waves make the zero gradient condition unusable, but if (8.15) is applied at the outflow boundary, no waves shorter than 4b.x
will ever reach that boundary because the group velocities of these short waves are
directed upstream. The zero-gradient condition is therefore a possibly useful outflow boundary condition for the advection equation. On the other hand, the large
reflection coefficients associated with the shortest waves do render (8.15) unsuitable as an inflow boundary condition because waves with wavelengths between
2b.x and 4b.x propagate upstream and rapidly amplify when they encounter the
inflow boundary.
Nitta (1962) and Matsuno (l966a) used a similar procedure to analyze the reflections generated by a wide variety of boundary conditions. The magnitude of
the reflection coefficient generated by a fixed boundary value, such as f/Jo = 0,
is unity. Although fixed boundary values are not appropriate at outflow, they are
useful at the inflow boundary, where the only waves that encounter the boundary
will be low-amplitude computational modes. Second-order extrapolation,
f/Jo = 2f/JI - eh,
has reflection IrI = Itan k Sx /21
2 • The general nth-order extrapolation
(8.18)
L(-l)m
n
(n') '( )'
m=O
m . n m.
f/Jm = 0
reflects with amplitude IrI = Itan k Sx /2I
n • Higher-order extrapolation reduces
the reflection of well-resolved physical waves, but it increases the reflection of
computational modes.
As an alternative to extrapolation, one might employ a one-sided finite difference at an outflow boundary, replacing (8.14) by
df/Jo + U (f/JI - f/Jo) = O.
(8.19)
dt
b.x
The reflections generated by this scheme may once again be analyzed by substituting the general solution (8.17) into (8.19). The result is
-ieu(1 + r) + .!:!.-(e ikt1x - re- ikß x) - .!:!.-(l + r) = O.
b.x
b.x