6.3 Systems of Equations
321
where s = U M , g = gk S t , and jj = H k S t . Let Ä be an eigenvalue of the
amplification matrix for this scheme, and define >:. = Äe- iks and c= .../iHk/).l.
Then
and
P = 1- 2c 2 ± 2ic (1 _c 2 Y /2 .
Two of the >:. are associated with gravity waves and two are associated with computational modes. The magnitude of Pis unity whenever Icl :::: 1, and since IÄI =
1>:' 1, it follows that the eigenvalues of the amplification matrix are bounded by
unity whenever Icl :::: 1. If the spatial derivatives are evaluated using the centereddifference operator 82x, this stability condition becomes .../iHM/ /).X < 1, where
strict inequality is required to ensure that the norm of the amplification matrix is
power bounded (see Section 3.1.1) . In contrast to the result obtained for an Eulerian scheme(3.!3), the stability of the leapfrog semi-Lagrangian approximation
(6.35)-(6.36) depends only on a Courant number defined with respect to the intrinsic gravity-wave phase speed and does not depend on the speed of the mean
flow.
An unconditionally stable method can be obtained if the forcing terms in the
linearized system are approximated by trapezoidal time-differencing to yield the
approximation
u+ - uO = _! [(ah)+ + (ah)OJ '
/).t
2
b
b
h+ - hO = _ H[(au)+ + (au)OJ .
/).1
2
ax
ax
(6.37)
(6.38)
The eigenvalues for the amplification matrix associated with this scheme are obtained by substituting solutions of the form
into the preceding to yield
(6.39)
This scheme is unconditionally stable, since IÄI = 1 for all M, and the norm of
the amplification matrix is power bounded because its eigenvectors are linearly
independent (and it can therefore be transformed into a diagonal matrix).
The right side of (6.38) becomes nonlinear if the preceding trapezoidal scheme
is generalized to approximate the nonlinear shallow-water equations. In order to
321
where s = U M , g = gk S t , and jj = H k S t . Let Ä be an eigenvalue of the
amplification matrix for this scheme, and define >:. = Äe- iks and c= .../iHk/).l.
Then
and
P = 1- 2c 2 ± 2ic (1 _c 2 Y /2 .
Two of the >:. are associated with gravity waves and two are associated with computational modes. The magnitude of Pis unity whenever Icl :::: 1, and since IÄI =
1>:' 1, it follows that the eigenvalues of the amplification matrix are bounded by
unity whenever Icl :::: 1. If the spatial derivatives are evaluated using the centereddifference operator 82x, this stability condition becomes .../iHM/ /).X < 1, where
strict inequality is required to ensure that the norm of the amplification matrix is
power bounded (see Section 3.1.1) . In contrast to the result obtained for an Eulerian scheme(3.!3), the stability of the leapfrog semi-Lagrangian approximation
(6.35)-(6.36) depends only on a Courant number defined with respect to the intrinsic gravity-wave phase speed and does not depend on the speed of the mean
flow.
An unconditionally stable method can be obtained if the forcing terms in the
linearized system are approximated by trapezoidal time-differencing to yield the
approximation
u+ - uO = _! [(ah)+ + (ah)OJ '
/).t
2
b
b
h+ - hO = _ H[(au)+ + (au)OJ .
/).1
2
ax
ax
(6.37)
(6.38)
The eigenvalues for the amplification matrix associated with this scheme are obtained by substituting solutions of the form
into the preceding to yield
(6.39)
This scheme is unconditionally stable, since IÄI = 1 for all M, and the norm of
the amplification matrix is power bounded because its eigenvectors are linearly
independent (and it can therefore be transformed into a diagonal matrix).
The right side of (6.38) becomes nonlinear if the preceding trapezoidal scheme
is generalized to approximate the nonlinear shallow-water equations. In order to