7.4 Summary of Schemes for Nonhydrostatic Models
371
One might inquire whether divergenee damping ean also be used to stabilize the
eompletely split approximation to the test ease with vcrtieal shear in the horizontal wind. The norm of the amplifieation matrix for the large-time-step third-order
Runge-Kutta integration (7.68)-(7.70) is strietly less than unity for all sufficiently
small !:!..t . Divergenee damping makes the norm of the amplifieation matrix for the
small time step strietly less than unity for all sufficiently small !:!.. r and thereby
stabilizes the eompletely split seheme by guaranteeing that the norm of the amplifieation matrix for the overall seheme will be less than unity. Nevertheless, divergenee damping only modestly improves the solution obtained with the eompletely
split seheme; the pressure field remains very noisy and eompletely unacceptable.?
The fundamental problem with the eompletely split method appears to be one of
inaeeuraey, not instability. This will be diseussed further in the next seetion.
7.4 Summary of Schemes for Nonhydrostatic Models
One way to eompare the preeeding methods for inereasing the efficieney of numerieal models for the simulation of fluids that support physically insignifieant
sound waves is to eompare the way eaeh approximation treats the velocity divergenee . As before , the mathematies of this diseussion will be streamlined by using
the eompressible Boussinesq equations (7.54)-(7.56) as a simple model for the
Euler equations. The pressure and the divergenee in the eompressible Boussinesq
system satisfy
ap
2
-
at
+c s 8 = F p ,
a8
2
-+'\1 P=G,
at
(7.89)
(7.90)
where 8 = '\I . v, F p = -v · '\I P, and G = -'\I . (v· Vv) + ab/az. The semiimplieit method approximates the left sides of the prceeding equations with a
stable trapezoidal time differenee. Sound waves are artificially slowed when large
time steps are used in this trapezoidal differenee, but the gravity-wave modes are
aeeurately approximated. The implicit eoupling in the trapezoidal differenee leads
to a Helmholtz equation for the pressure that must be solved at every time step.
The prognostic pressure equation (7.89) is disearded in the ineompressible
Boussinesq approximation, and the loeal time derivative of the divergenee is set
to zero in (7.90). This leads to a Poisson equation for pressure that must be solved
at every time step. The eomputational effort required to evaluate the pressure is
similar to that required by the semi-implicit method. The Boussinesq system does,
70ne way to appreciate the difference in the effectiveness of divergcnce damping in the completely
and partially split schemes is to note thc difference in wavelength at which spurious pressure perturbations appear in each solution. The partially split scheme generates errors at much shorter wavelengths
than those produced by the completely split method (compare Figs. 7.3a and 7.5b). and the shortwavelength features are removed more rapidly by the divergence damper.
371
One might inquire whether divergenee damping ean also be used to stabilize the
eompletely split approximation to the test ease with vcrtieal shear in the horizontal wind. The norm of the amplifieation matrix for the large-time-step third-order
Runge-Kutta integration (7.68)-(7.70) is strietly less than unity for all sufficiently
small !:!..t . Divergenee damping makes the norm of the amplifieation matrix for the
small time step strietly less than unity for all sufficiently small !:!.. r and thereby
stabilizes the eompletely split seheme by guaranteeing that the norm of the amplifieation matrix for the overall seheme will be less than unity. Nevertheless, divergenee damping only modestly improves the solution obtained with the eompletely
split seheme; the pressure field remains very noisy and eompletely unacceptable.?
The fundamental problem with the eompletely split method appears to be one of
inaeeuraey, not instability. This will be diseussed further in the next seetion.
7.4 Summary of Schemes for Nonhydrostatic Models
One way to eompare the preeeding methods for inereasing the efficieney of numerieal models for the simulation of fluids that support physically insignifieant
sound waves is to eompare the way eaeh approximation treats the velocity divergenee . As before , the mathematies of this diseussion will be streamlined by using
the eompressible Boussinesq equations (7.54)-(7.56) as a simple model for the
Euler equations. The pressure and the divergenee in the eompressible Boussinesq
system satisfy
ap
2
-
at
+c s 8 = F p ,
a8
2
-+'\1 P=G,
at
(7.89)
(7.90)
where 8 = '\I . v, F p = -v · '\I P, and G = -'\I . (v· Vv) + ab/az. The semiimplieit method approximates the left sides of the prceeding equations with a
stable trapezoidal time differenee. Sound waves are artificially slowed when large
time steps are used in this trapezoidal differenee, but the gravity-wave modes are
aeeurately approximated. The implicit eoupling in the trapezoidal differenee leads
to a Helmholtz equation for the pressure that must be solved at every time step.
The prognostic pressure equation (7.89) is disearded in the ineompressible
Boussinesq approximation, and the loeal time derivative of the divergenee is set
to zero in (7.90). This leads to a Poisson equation for pressure that must be solved
at every time step. The eomputational effort required to evaluate the pressure is
similar to that required by the semi-implicit method. The Boussinesq system does,
70ne way to appreciate the difference in the effectiveness of divergcnce damping in the completely
and partially split schemes is to note thc difference in wavelength at which spurious pressure perturbations appear in each solution. The partially split scheme generates errors at much shorter wavelengths
than those produced by the completely split method (compare Figs. 7.3a and 7.5b). and the shortwavelength features are removed more rapidly by the divergence damper.