414
8. Nonreflecting Boundary Conditions
way to obtain a well-posed problem is by choosing (ll = (l2 = (l3 = (l4 = 0,
which specifies e and v at inflow and d at outflow.
Unlike the one-dimensional case, it is not clear what values of e, v, and d
should be prescribed to prevent waves impinging on the boundary from partially
reflecting into an inward-propagating mode. Suppose that the incoming variable
d is set to zero at x = L. Then the transformed equation for d reduces to
(U -
ad av
c)- - c- = 0
ax
ay
along x = L. Since ad/ay = 0 along x = L, the preceding implies that a
2v/ay2
is also zero along x = L and thus either f. = 0, or v must be zero throughout the
entire domain. The only nontrivial solution that is consistent with these conditions
(f. = 0 or v = 0) is a wave propagating exactly parallel to the x-axis. The condition d(L, t) = 0 must therefore produce spurious reflection whenever a wave
strikes the boundary at some angle other than 90°.
8.2.1 One-Way Wave Equations
Engquist and Majda (1977) suggested that an effective approximation to the true
radiation boundary condition in the two-dirnensional shallow-water system could
be obtained using a one-way wave equation whose solutions are waves with group
velocities directed outward through the boundary. In order to minimize spurious
reflection at the boundary, this one-way wave equation should be designed such
that its solutions approximate the outward-directed waves in the shallow-water
system as closely as possible.
Consider the linearized shallow -water equations for a basic state with no mean
flow, which reduce to
(8.26)
Substituting solutions of the form
TJ(x, y, t) = TJoei(kxHy- wl)
(where k and f. may be positive or negative, but w
(8.26) gives the shallow-water dispersion relation
0 to avoid redundancy) into
(8.27)
or equivalently,
w
k=±c (
cf. 2 2)1/2
1 - -
w 2
(8.28)
8. Nonreflecting Boundary Conditions
way to obtain a well-posed problem is by choosing (ll = (l2 = (l3 = (l4 = 0,
which specifies e and v at inflow and d at outflow.
Unlike the one-dimensional case, it is not clear what values of e, v, and d
should be prescribed to prevent waves impinging on the boundary from partially
reflecting into an inward-propagating mode. Suppose that the incoming variable
d is set to zero at x = L. Then the transformed equation for d reduces to
(U -
ad av
c)- - c- = 0
ax
ay
along x = L. Since ad/ay = 0 along x = L, the preceding implies that a
2v/ay2
is also zero along x = L and thus either f. = 0, or v must be zero throughout the
entire domain. The only nontrivial solution that is consistent with these conditions
(f. = 0 or v = 0) is a wave propagating exactly parallel to the x-axis. The condition d(L, t) = 0 must therefore produce spurious reflection whenever a wave
strikes the boundary at some angle other than 90°.
8.2.1 One-Way Wave Equations
Engquist and Majda (1977) suggested that an effective approximation to the true
radiation boundary condition in the two-dirnensional shallow-water system could
be obtained using a one-way wave equation whose solutions are waves with group
velocities directed outward through the boundary. In order to minimize spurious
reflection at the boundary, this one-way wave equation should be designed such
that its solutions approximate the outward-directed waves in the shallow-water
system as closely as possible.
Consider the linearized shallow -water equations for a basic state with no mean
flow, which reduce to
(8.26)
Substituting solutions of the form
TJ(x, y, t) = TJoei(kxHy- wl)
(where k and f. may be positive or negative, but w
(8.26) gives the shallow-water dispersion relation
0 to avoid redundancy) into
(8.27)
or equivalently,
w
k=±c (
cf. 2 2)1/2
1 - -
w 2
(8.28)