168
3. Beyond the One-Way WaveEquation
(PU
(PU
- - c 2 _
CJt 2
CJx 2
= 0
(a) Write down appropriate modifications for the discretized vertical momentum and buoyancy equations (3.44) and (3.45), and derive the discrete
dispersion relation for this system.
with the expression that arises when h is eliminated from the forwardbackward approximation on the staggered mesh (3.17) and (3.18). What
does this comparison suggest about the number of computational modes
admitted by each numerical approximation?
3. Suppose that numerical solutions to the two-dimensional Boussinesq system (3.39)-(3.42) are obtained using the staggered grid shown in Fig. 3.6
except that the distribution of the variables is modified so that b is colocated with the w rather than the P points.
(b) Assurne that all resolved modes are hydrostatic, so that k; can be neglected with respect to li in the denominator of (3.47) and in the result
derived in (a). Compare the horizontal and vertical group velocities for the
numerical solutions on each staggered grid with the exact expression from
the nondiscretized hydrostatic system.
4. Derive the amplification factor and stability condition given in the text for
the CTU method (3.36) . Show that including thc cross-derivative tenn in
the CTU method always decreases the amplification factor relative to that
obtained with the standard two-dimensional upstream scheme (3.31).
5. Show that the false 2-0 Lax-Wendroff scheme (3.37) is unstable for all !!.t.
6. Derermine the range of !!.t (if any) over which the backward, forward, and
leapfrog schemes give a stable approximation to
d1/l
-
= r1/l.
dt
Consider both the cases r > 0 and r < O.The true solution to this equation
preserves the sign of 1/I(t = 0) . What, if any, additional restrictions must be
placed on !!.t to ensure that the numerical solution for each method is both
stable and sign -preserving.
7. Suppose the Lax-Wendroffmethod is used to obtain an 0 [(!!.t)2]-accurate
approximation to the advection-diffusion equation (3.72). Show that before
discretizing the spatial derivatives, the scheme has the form
_
- - - - + c - - M - -
!!.t
CJx
CJx 2
= -
!!.t ( 2
2
c - - -2cM-- +M - - .
2
CJx 2
CJx 3
CJx 3
3. Beyond the One-Way WaveEquation
(PU
(PU
- - c 2 _
CJt 2
CJx 2
= 0
(a) Write down appropriate modifications for the discretized vertical momentum and buoyancy equations (3.44) and (3.45), and derive the discrete
dispersion relation for this system.
with the expression that arises when h is eliminated from the forwardbackward approximation on the staggered mesh (3.17) and (3.18). What
does this comparison suggest about the number of computational modes
admitted by each numerical approximation?
3. Suppose that numerical solutions to the two-dimensional Boussinesq system (3.39)-(3.42) are obtained using the staggered grid shown in Fig. 3.6
except that the distribution of the variables is modified so that b is colocated with the w rather than the P points.
(b) Assurne that all resolved modes are hydrostatic, so that k; can be neglected with respect to li in the denominator of (3.47) and in the result
derived in (a). Compare the horizontal and vertical group velocities for the
numerical solutions on each staggered grid with the exact expression from
the nondiscretized hydrostatic system.
4. Derive the amplification factor and stability condition given in the text for
the CTU method (3.36) . Show that including thc cross-derivative tenn in
the CTU method always decreases the amplification factor relative to that
obtained with the standard two-dimensional upstream scheme (3.31).
5. Show that the false 2-0 Lax-Wendroff scheme (3.37) is unstable for all !!.t.
6. Derermine the range of !!.t (if any) over which the backward, forward, and
leapfrog schemes give a stable approximation to
d1/l
-
= r1/l.
dt
Consider both the cases r > 0 and r < O.The true solution to this equation
preserves the sign of 1/I(t = 0) . What, if any, additional restrictions must be
placed on !!.t to ensure that the numerical solution for each method is both
stable and sign -preserving.
7. Suppose the Lax-Wendroffmethod is used to obtain an 0 [(!!.t)2]-accurate
approximation to the advection-diffusion equation (3.72). Show that before
discretizing the spatial derivatives, the scheme has the form
_
- - - - + c - - M - -
!!.t
CJx
CJx 2
= -
!!.t ( 2
2
c - - -2cM-- +M - - .
2
CJx 2
CJx 3
CJx 3