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8. Nonreflecting Boundary Conditions
inflow and no value at outflow. Each positive eigenvalue of the coefficient matrix will therefore be associated with aRiemann invariant requiring a boundary
value at x = 0, and each negative eigenvalue will necessitate the specification of
a boundary value at x = L.
8.1.2 The Radiation Condition
Radiation boundary conditions can be easily imposed in the one-dimensional
shallow-water system by transforming the equations to the diagonal form (8.3)
and setting the incoming Riemann invariant to zero at each boundary. Unfortunately, this approach does not easily generalize to two-dimensional shallowwater flow. In many practical problems itis simpler to retain the velocities and
the height (or pressure) field as the unknown prognostic variables and to develop
open boundary conditions involving these variables.
Consider, therefore, the problem of expressing the radiation boundary condit ion
at x = L in terms of u and TJ instead of the Riemann invariants in a case where
c » IUI. Since the incoming characteristic is zero at x = L for all t > 0, d(x, t)
must be zero throughout the wedge-shaped region of the x-t plane defined by the
inequalities (U - c)t + L < x < L. Thus for all t > 0,
TJ(x, t)
u(x,t) = - -
c
(8.5)
in a small neighborhood of the x = L boundary. In the same neighborhood of
x = L, the equation for the outward-directed characteristic is
= O.
c
8t
(u + c
+ (U + c) 8x (u +
(8.6)
Using (8.5) to eliminate TJ/c from the preceding equation yields the radiation
boundary condition for u at x = L,
8u
-
8u
+ (U + c)- = O.
(8.7)
8t
8x
An identical radiation boundary condition for TJ ,
8TJ
8TJ
at + (U + c) 8x = 0,
(8.8)
can be derived using (8.5) to eliminate u from (8.6). Similar conditions may also
be obtained x = 0; they are
8u
8t
-
8u
+(U -c)- =0
8x
(8.9)
and
8TJ
8TJ
-+(U-c)-=O.
8t
Bx
(8.10)