7.3 Fractional-Step Methods
359
ferenced perturbations. Since the speed of sound is relatively uniform throughout
the atmosphere, it is easy to ensure that the terms evaluated implicitly dominate
those computed explicitly and thereby guarantee that the scheme is stable .
7.3 Fractional-Step Methods
The semi-implicit method requires the solution of an elliptic equation for the pressure during each step of the integration. This can be avoided by splitting the complete problem into fractional steps and using a very small time step to integrate
the subproblern containing the terms responsible for the propagation of the fastmoving wave. Consider a general partial differential equation of the form
-
81/1
+ L(1/I) = 0,
(7.63)
8t
and suppose that
As discussed in Section 3.3, if L does not depend on time, (7.63) can be formally
integrated over an interval 6.t to obtain
1/I(t + 6.t) = exp(6.tL)1/I(t).
7.3.1 Complete Operator Splitting
Let FI (6.t) and F2(6.t) be numerical approximations to the exact operators
exp(6.tL» and exp(6.t.L2) . In the standard fractional-step approach, the approximate solution is stepped forward over a time interval 6.t using
r/l S = FI (6.t)r/ln,
r/ln+J = F2 (6.t)r/l s ,
(7.64)
(7.65)
but it is not necessary to use the same time step in each subproblem. If L2 contains those terms responsible for the propagation of fast-rnoving waves and the
maximum stable time step with which (7.64) can be integrated is M times that
with which (7.65) can be integrated, the numerical solution could be evaluated
using the formula
(7.66)
This approach can be applied to the Iinearized one-dimensional shallow-water
system by writing (3.1) and (3.2) in the form
8r
- + LI (r) + .L2(r) = 0,
8t
(7.67)
359
ferenced perturbations. Since the speed of sound is relatively uniform throughout
the atmosphere, it is easy to ensure that the terms evaluated implicitly dominate
those computed explicitly and thereby guarantee that the scheme is stable .
7.3 Fractional-Step Methods
The semi-implicit method requires the solution of an elliptic equation for the pressure during each step of the integration. This can be avoided by splitting the complete problem into fractional steps and using a very small time step to integrate
the subproblern containing the terms responsible for the propagation of the fastmoving wave. Consider a general partial differential equation of the form
-
81/1
+ L(1/I) = 0,
(7.63)
8t
and suppose that
As discussed in Section 3.3, if L does not depend on time, (7.63) can be formally
integrated over an interval 6.t to obtain
1/I(t + 6.t) = exp(6.tL)1/I(t).
7.3.1 Complete Operator Splitting
Let FI (6.t) and F2(6.t) be numerical approximations to the exact operators
exp(6.tL» and exp(6.t.L2) . In the standard fractional-step approach, the approximate solution is stepped forward over a time interval 6.t using
r/l S = FI (6.t)r/ln,
r/ln+J = F2 (6.t)r/l s ,
(7.64)
(7.65)
but it is not necessary to use the same time step in each subproblem. If L2 contains those terms responsible for the propagation of fast-rnoving waves and the
maximum stable time step with which (7.64) can be integrated is M times that
with which (7.65) can be integrated, the numerical solution could be evaluated
using the formula
(7.66)
This approach can be applied to the Iinearized one-dimensional shallow-water
system by writing (3.1) and (3.2) in the form
8r
- + LI (r) + .L2(r) = 0,
8t
(7.67)