138
ERIC BLAYO AND LAURENT DEBREU
must be approximated to give tractable local conditions. A strong interest of this approach is its sound mathematical foundation, and its
practical efficiency in several domains of applications. Several papers
have recently readdressed the derivation of absorbing BCs for the inviscid shallow water system, and obtain apparently quite good numerical
results (Lie, 2001; McDonald, 2002, 2003; Nycander and D¨ o¨ os, 2003).
3.2
An hyperbolic point of view
When attempting to draw some synthesis of the numerous previous
studies on OBCs, two keypoints stand out, which seem to be necessary
constituents for any good OBC. The first point is that good results are
obtained when taking primarily into account the hyperbolic part of the
dynamics, and therefore when working on incoming characteristic variables. The second point is that this must be associated with a consistent
use of some external data.
Incoming characteristic variables.
Let us first introduce some
standard definitions concerning hyperbolic systems. The general form
of such a system is
∂Φ
∂t
+ A(Φ)
∂Φ
∂x
= F
(20)
where Φ(x, t) is a vector of n functions, A(Φ) is a n × n matrix of
functions of Φ, and F is a forcing term. For the system to be hyperbolic,
A must have n real eigenvalues and n distinct eigenvectors. Let W k
the k
th left eigenvector of A, corresponding to the k
th eigenvalue λ k :
W T
k A = λ k W T
k . Multipliying (20) on the left by W T
k , one gets:
W
T
k
d k Φ
dt
= W
T
k F
with
d k
dt
=
∂
∂t
+ λ k
∂
∂x
(21)
The operator d k /dt represents a total (or directional) derivative in the direction defined by
dx
dt
= λ k . To the hyperbolic system (20) correspond n
such families of curves, which are called characteristic curves of the system. If the system (20) is linear with constant coefficients, i.e. if A is a
constant matrix, one can define the new variables w k (x, t) = W T
k Φ(x, t).
(20) is then equivalent to the system of n uncoupled transport equations:
∂w k
∂t
+ λ k
∂w k
∂x
= W
T
k F
k = 1, . . . , n
(22)
The characteristic curves in that case are the lines x − λ k t = constant,
along which the w k (called characteristic variables or Riemann invariants) are conserved. One can notice that, at a given boundary, these
ERIC BLAYO AND LAURENT DEBREU
must be approximated to give tractable local conditions. A strong interest of this approach is its sound mathematical foundation, and its
practical efficiency in several domains of applications. Several papers
have recently readdressed the derivation of absorbing BCs for the inviscid shallow water system, and obtain apparently quite good numerical
results (Lie, 2001; McDonald, 2002, 2003; Nycander and D¨ o¨ os, 2003).
3.2
An hyperbolic point of view
When attempting to draw some synthesis of the numerous previous
studies on OBCs, two keypoints stand out, which seem to be necessary
constituents for any good OBC. The first point is that good results are
obtained when taking primarily into account the hyperbolic part of the
dynamics, and therefore when working on incoming characteristic variables. The second point is that this must be associated with a consistent
use of some external data.
Incoming characteristic variables.
Let us first introduce some
standard definitions concerning hyperbolic systems. The general form
of such a system is
∂Φ
∂t
+ A(Φ)
∂Φ
∂x
= F
(20)
where Φ(x, t) is a vector of n functions, A(Φ) is a n × n matrix of
functions of Φ, and F is a forcing term. For the system to be hyperbolic,
A must have n real eigenvalues and n distinct eigenvectors. Let W k
the k
th left eigenvector of A, corresponding to the k
th eigenvalue λ k :
W T
k A = λ k W T
k . Multipliying (20) on the left by W T
k , one gets:
W
T
k
d k Φ
dt
= W
T
k F
with
d k
dt
=
∂
∂t
+ λ k
∂
∂x
(21)
The operator d k /dt represents a total (or directional) derivative in the direction defined by
dx
dt
= λ k . To the hyperbolic system (20) correspond n
such families of curves, which are called characteristic curves of the system. If the system (20) is linear with constant coefficients, i.e. if A is a
constant matrix, one can define the new variables w k (x, t) = W T
k Φ(x, t).
(20) is then equivalent to the system of n uncoupled transport equations:
∂w k
∂t
+ λ k
∂w k
∂x
= W
T
k F
k = 1, . . . , n
(22)
The characteristic curves in that case are the lines x − λ k t = constant,
along which the w k (called characteristic variables or Riemann invariants) are conserved. One can notice that, at a given boundary, these