7.2 The Semi-implicit Method
Then the resulting semi-implicit system has the form
b n + 1 = b n - I _ 2Öot (v
n . vb" + N 2 w n ) ,
v n + 1 + ÖotVpn+1 = G,
pn+1 + c;ÖotV . yn+1 = h.
357
(7.57)
(7.58)
(7.59)
Here
and
h = pn--I - c;Öot [V . v n - I + 2v n . V pn] .
A single Helmholtz equation for pn+ I can be obtained by substituting the divergenee of (7.58) into (7.59) to yield
2 n+1
V P
-
pn+1
V . G
h
=---------;:;(C s Öot )2
Öot
(C s ö.t )2 ·
(7.60)
The numerieal solution of this Helmholtz equation is trivial if the Fourier speetral
method is employed in a reetangular domain or if spherieal harmonie expansion
funetions are used in aglobaI speetral model. If the spatial derivatives are approximated by finite differenees, (7.60) yields a sparse linear-algebraic system
that can be solved using the techniques described in Section 7.1.3. After solving
(7.60) for pn+l , the momentum equations can be stepped forward, and the buoyancy equation (7.57), which is completely explicit, can be updated to complete
the integration cycle,
This implementation of the serni-implicit method is closely related to the projection method for incompressible Boussinesq ftow. Indeed, in the limit C s
00
the preceding approach will be identical to the leapfrog projection method (described in Seetion 7.1.2) if (pn+1 + pn-I)/2 is replaced by P" in (7.60) . AIthough the leapfrog projection method and the serni-implicit method yield algorithms involving very similar algebraic equations, these methods are derived via
very different approximation strategies. The projection method is an efficient way
to solve a set of continuous equations that is obtained by filtering the exact Euler equations to eliminate sound waves. In contrast, the semi-implicit scheme is
obtained by directly approximating the full compressible equations and using implicit time-differencing to stabilize the sound waves . Neither approach allows one
to eorrectly simulate sound waves, but both approaches allow the accurate and
efficient simulation of the slower-moving gravity waves.
Now consider the version of the semi-implicit approximation in whieh those
terms responsible for gravity-wave propagation are also approximated by trape-
Then the resulting semi-implicit system has the form
b n + 1 = b n - I _ 2Öot (v
n . vb" + N 2 w n ) ,
v n + 1 + ÖotVpn+1 = G,
pn+1 + c;ÖotV . yn+1 = h.
357
(7.57)
(7.58)
(7.59)
Here
and
h = pn--I - c;Öot [V . v n - I + 2v n . V pn] .
A single Helmholtz equation for pn+ I can be obtained by substituting the divergenee of (7.58) into (7.59) to yield
2 n+1
V P
-
pn+1
V . G
h
=---------;:;(C s Öot )2
Öot
(C s ö.t )2 ·
(7.60)
The numerieal solution of this Helmholtz equation is trivial if the Fourier speetral
method is employed in a reetangular domain or if spherieal harmonie expansion
funetions are used in aglobaI speetral model. If the spatial derivatives are approximated by finite differenees, (7.60) yields a sparse linear-algebraic system
that can be solved using the techniques described in Section 7.1.3. After solving
(7.60) for pn+l , the momentum equations can be stepped forward, and the buoyancy equation (7.57), which is completely explicit, can be updated to complete
the integration cycle,
This implementation of the serni-implicit method is closely related to the projection method for incompressible Boussinesq ftow. Indeed, in the limit C s
00
the preceding approach will be identical to the leapfrog projection method (described in Seetion 7.1.2) if (pn+1 + pn-I)/2 is replaced by P" in (7.60) . AIthough the leapfrog projection method and the serni-implicit method yield algorithms involving very similar algebraic equations, these methods are derived via
very different approximation strategies. The projection method is an efficient way
to solve a set of continuous equations that is obtained by filtering the exact Euler equations to eliminate sound waves. In contrast, the semi-implicit scheme is
obtained by directly approximating the full compressible equations and using implicit time-differencing to stabilize the sound waves . Neither approach allows one
to eorrectly simulate sound waves, but both approaches allow the accurate and
efficient simulation of the slower-moving gravity waves.
Now consider the version of the semi-implicit approximation in whieh those
terms responsible for gravity-wave propagation are also approximated by trape-