7.3 Fractional-Step Methods
361
pressible two-dimensional Boussinesq equations, which could be split into the
form (7.67) by defining
x
a
.c.=
.
)0
0
v - V
0
0
0
0
0
v ·V
-1
(
N 2 0
Oz )
o .
c:ox c:o z 0 0
0
v·V
0
0
0
0
0
0
r = ( u w b P {,
(7.73)
Suppose that N and C s are constant and that the full nonlinear system is linearized
about a reference state with a mean horizontal wind U (z) , The operators associated with this linearized system will not commute unless dU [dz is zero.
As in the one-dimensional shalIow-water system, the advection operator .cl can
be approximated using the third-order Runge-Kutta method (7.68)-(7.70). The
second fractional step may be integrated using trapezoidal differencing for the
terms governing the vertical propagation of sound waves and fotward-backward
differencing for the terms governing horizontal sound-wave propagation and buoyancy oscillations. The resulting scheme is
u m+ 1 - um opm
- - - - + - - = 0 ,
OX
m+1 _ u/"
0 (pm+1 + pm)
w
- - - - + -
-bm=O,
oz
2
pm+1 _ pm
b m + 1 - b"
- - - - + N
2 w
m
+
1 = 0,
2 oum+ 1
2 0 (w m+ 1 + wm)
- - - - - + C s - - + C s -
=0,
OX
oz
2
(7.74)
(7.75)
(7.76)
(7.77)
This approximation to exp( r .c2) is stable and nondamping, provided that the
number max(csK,
r is less than 2. The trapezoidal approximation of the
terms involving vertical derivatives does not significantly increase the computations required on each small time step because it generates a simple tridiago -
nal system of algebraic equations for the w m +I throughout each vertical column
within the domain. If the horizontal resolution is very coarse, so that K « N / cs,
further efficiency can be also obtained by treating the terms involving buoyancy
oscillations with trapezoidal differencing. Since these terms do not involve derivatives, the resulting implicit algebraic system remains tridiagonal.
As an alternative to the trapezoidal method, the terms involving the vertical
pressure gradient and the divergence of the vertical velocity could be integrated
361
pressible two-dimensional Boussinesq equations, which could be split into the
form (7.67) by defining
x
a
.c.=
.
)0
0
v - V
0
0
0
0
0
v ·V
-1
(
N 2 0
Oz )
o .
c:ox c:o z 0 0
0
v·V
0
0
0
0
0
0
r = ( u w b P {,
(7.73)
Suppose that N and C s are constant and that the full nonlinear system is linearized
about a reference state with a mean horizontal wind U (z) , The operators associated with this linearized system will not commute unless dU [dz is zero.
As in the one-dimensional shalIow-water system, the advection operator .cl can
be approximated using the third-order Runge-Kutta method (7.68)-(7.70). The
second fractional step may be integrated using trapezoidal differencing for the
terms governing the vertical propagation of sound waves and fotward-backward
differencing for the terms governing horizontal sound-wave propagation and buoyancy oscillations. The resulting scheme is
u m+ 1 - um opm
- - - - + - - = 0 ,
OX
m+1 _ u/"
0 (pm+1 + pm)
w
- - - - + -
-bm=O,
oz
2
pm+1 _ pm
b m + 1 - b"
- - - - + N
2 w
m
+
1 = 0,
2 oum+ 1
2 0 (w m+ 1 + wm)
- - - - - + C s - - + C s -
=0,
OX
oz
2
(7.74)
(7.75)
(7.76)
(7.77)
This approximation to exp( r .c2) is stable and nondamping, provided that the
number max(csK,
r is less than 2. The trapezoidal approximation of the
terms involving vertical derivatives does not significantly increase the computations required on each small time step because it generates a simple tridiago -
nal system of algebraic equations for the w m +I throughout each vertical column
within the domain. If the horizontal resolution is very coarse, so that K « N / cs,
further efficiency can be also obtained by treating the terms involving buoyancy
oscillations with trapezoidal differencing. Since these terms do not involve derivatives, the resulting implicit algebraic system remains tridiagonal.
As an alternative to the trapezoidal method, the terms involving the vertical
pressure gradient and the divergence of the vertical velocity could be integrated