402
8. Nonreflecting Boundary Conditions
Solving the preceding for TJ (x, t) and substituting in the equation for the outwarddirected characteristic (8.6) yields
Bu
-+(U+c)-=au
1 (ade - + ( U + c ) -
ade) ,
(8.11)
or altematively,
at
ax
2 at
ax
au + (U + c) au = !... (u e _ ee) + (U + c)!..- (u _ ee) .
at
ax
at
2
ax e
2
This last form of the boundary condition shows that u(L , t) is equal to ue(L, t)
only in those special cases where ee is zero in the neighborhood of x = L, or
equivalently, where there are no waves in the large-domain solution propagating
away from the small domain in the neighborhood of the boundary.
If u is eliminated from (8.5), a comparable relation for the free surface displacement is obtained:
aTJ
-+(U+c)-=-- aTJ
c (ade - + ( U + c ) -
ade) .
(8.12)
at
ax
2 at
ax
As in (8.11), this relation simplifies to TJ(L, t) = TJe(L, t) only in the special case
where the solution on the large domain does not contain any waves that propagate
away from the small domain near the boundary at x = L.
8.1.4 Reflections at an Artificial BoundaryThe Continuous Case
Although exact nonreflecting boundary conditions were derived in Section 8.1.2
for the one-dimensional shallow-water problem, exact fonnulas for nonreflecting
boundary conditions are not generally available in more complicated problems.
Even in the relatively simple case of two-dimensional shallow-water flow some
approximation to the exact radiation boundary condition is required in order to obtain a useful relationship involving the prognostic variables and their derivatives at
the lateral boundary (see Section 8.2). As a consequence ofthese approximations,
errors are introduced in the mathematical model of the physical system before the
goveming equations are discretized. The errors generated by such approximations
are considered in this section.
Consider the one-dimensional shallow-water equations on the domain 0 x
L and suppose that a zero gradient condition is used to approximate the correct
radiation boundary condition at the lateral boundaries. The strength of the reflections generated at the x = 0 boundary may be analyzed by examining the behavior
of a unit-amplitude incident wave as it reflects off the boundary. Suppose that the
perturbation velocity has the form
u(x, t) = sin [ki(X - (U - c)t)] + r sin [kr(x - (U + c)t) + 4>].
8. Nonreflecting Boundary Conditions
Solving the preceding for TJ (x, t) and substituting in the equation for the outwarddirected characteristic (8.6) yields
Bu
-+(U+c)-=au
1 (ade - + ( U + c ) -
ade) ,
(8.11)
or altematively,
at
ax
2 at
ax
au + (U + c) au = !... (u e _ ee) + (U + c)!..- (u _ ee) .
at
ax
at
2
ax e
2
This last form of the boundary condition shows that u(L , t) is equal to ue(L, t)
only in those special cases where ee is zero in the neighborhood of x = L, or
equivalently, where there are no waves in the large-domain solution propagating
away from the small domain in the neighborhood of the boundary.
If u is eliminated from (8.5), a comparable relation for the free surface displacement is obtained:
aTJ
-+(U+c)-=-- aTJ
c (ade - + ( U + c ) -
ade) .
(8.12)
at
ax
2 at
ax
As in (8.11), this relation simplifies to TJ(L, t) = TJe(L, t) only in the special case
where the solution on the large domain does not contain any waves that propagate
away from the small domain near the boundary at x = L.
8.1.4 Reflections at an Artificial BoundaryThe Continuous Case
Although exact nonreflecting boundary conditions were derived in Section 8.1.2
for the one-dimensional shallow-water problem, exact fonnulas for nonreflecting
boundary conditions are not generally available in more complicated problems.
Even in the relatively simple case of two-dimensional shallow-water flow some
approximation to the exact radiation boundary condition is required in order to obtain a useful relationship involving the prognostic variables and their derivatives at
the lateral boundary (see Section 8.2). As a consequence ofthese approximations,
errors are introduced in the mathematical model of the physical system before the
goveming equations are discretized. The errors generated by such approximations
are considered in this section.
Consider the one-dimensional shallow-water equations on the domain 0 x
L and suppose that a zero gradient condition is used to approximate the correct
radiation boundary condition at the lateral boundaries. The strength of the reflections generated at the x = 0 boundary may be analyzed by examining the behavior
of a unit-amplitude incident wave as it reflects off the boundary. Suppose that the
perturbation velocity has the form
u(x, t) = sin [ki(X - (U - c)t)] + r sin [kr(x - (U + c)t) + 4>].