410
8. Nonreflecting Boundary Conditions
tions) admit no unforced waves with group velocities directed inward through the
boundaries of the domain .
In order to see how a numerical method might allow waves to spontaneously
propagate inward through a lateral boundary, suppose that the advection equation
(8.1) is to be solved on the semi-infinite domain 0 x 00, and that U < 0, so
x = 0 is an outflow boundary. The interior solution will be approximated using
leapfrog-time centered-space differencing
(8.21)
where J.L = U /lt / S» . The outflow boundary condition will be determined by
extrapolation, tl>ö = tI>' i. The numerical dispersion relation for the interior finitedifference scheme (8.21) is
sinw/lt = J.Lsink/lx.
(8.22)
Substituting a wave of the form (8.20) into the extrapolation boundary condition
yields
(8.23)
The free modes of this finite-difference IBVP consist of those waves for which
wand k simultaneously satisfy (8.22) and (8.23), and they inc1ude the mode
(w, k) = (rr / /lt , 0) , whose group velocity,
aw
- = U
ak
cosk/lx
cosw/lt
,
is - U . The mode (1l' / /lt, 0) is therefore a free mode of the discretized problem that can propagate inward through the downstream boundary, and according to Trefethen's interpretation of the GKS stability criteria, the scheme should
be unstable. This instability is demonstrated in Fig. 8.3, which shows a numerical integration of (8.21) on the interval 0
x
l. The initial data, plotted
in Fig. 6.3a, are random numbers with amplitudes between 0 and 0.2. The wind
speed is U = -I, so x = I is an inflow boundary at which tI>' ioo is fixed at
zero. The extrapolation condition tl>ö = tI>' i is applied at outflow; S» = 0.01, and
the time step is fixed such that the Courant number is 0.9. Figures 6.3b and 6.3c
show the solution at two adjacent time steps after the instability has developed at
the downstream boundary. As suggested by the preceding analysis, the growing
mode has aperiod of 2/lt and a long horizontal wavelength.
In order to stabilize the numerical solution, it is necessary to change either the
interior finite-difference equation or the boundary condition. Suppose the boundary condition is replaced by tl>ö =
which is a backwards extrapolation in
both space and time. Substitution of the wave solution (8.20) into this boundary
condition yields
or
k Sx + w/lt = Zntt ; n = 0, ± 1, ±2, ... .
(8.24)
8. Nonreflecting Boundary Conditions
tions) admit no unforced waves with group velocities directed inward through the
boundaries of the domain .
In order to see how a numerical method might allow waves to spontaneously
propagate inward through a lateral boundary, suppose that the advection equation
(8.1) is to be solved on the semi-infinite domain 0 x 00, and that U < 0, so
x = 0 is an outflow boundary. The interior solution will be approximated using
leapfrog-time centered-space differencing
(8.21)
where J.L = U /lt / S» . The outflow boundary condition will be determined by
extrapolation, tl>ö = tI>' i. The numerical dispersion relation for the interior finitedifference scheme (8.21) is
sinw/lt = J.Lsink/lx.
(8.22)
Substituting a wave of the form (8.20) into the extrapolation boundary condition
yields
(8.23)
The free modes of this finite-difference IBVP consist of those waves for which
wand k simultaneously satisfy (8.22) and (8.23), and they inc1ude the mode
(w, k) = (rr / /lt , 0) , whose group velocity,
aw
- = U
ak
cosk/lx
cosw/lt
,
is - U . The mode (1l' / /lt, 0) is therefore a free mode of the discretized problem that can propagate inward through the downstream boundary, and according to Trefethen's interpretation of the GKS stability criteria, the scheme should
be unstable. This instability is demonstrated in Fig. 8.3, which shows a numerical integration of (8.21) on the interval 0
x
l. The initial data, plotted
in Fig. 6.3a, are random numbers with amplitudes between 0 and 0.2. The wind
speed is U = -I, so x = I is an inflow boundary at which tI>' ioo is fixed at
zero. The extrapolation condition tl>ö = tI>' i is applied at outflow; S» = 0.01, and
the time step is fixed such that the Courant number is 0.9. Figures 6.3b and 6.3c
show the solution at two adjacent time steps after the instability has developed at
the downstream boundary. As suggested by the preceding analysis, the growing
mode has aperiod of 2/lt and a long horizontal wavelength.
In order to stabilize the numerical solution, it is necessary to change either the
interior finite-difference equation or the boundary condition. Suppose the boundary condition is replaced by tl>ö =
which is a backwards extrapolation in
both space and time. Substitution of the wave solution (8.20) into this boundary
condition yields
or
k Sx + w/lt = Zntt ; n = 0, ± 1, ±2, ... .
(8.24)