318
6. Semi-Lagrangian Methods
and is confined to the time coordinate, whereas the phase errors in the solutions
to (6.29) and (6.30) appear in both time and space.
The Courant number is increased to unity and wt:J.t is decreased to -11/1000
in the second test case shown in Fig. 6.3b. At t = 20 the correct solution is again
identical to the initial condition. Since the wind speed is constant and the Courant
number is unity, the semi-Lagrangian advection is exact, and the only source of
error is in the integration of the forcing . The solution obtained using (6.26) is
stable and almost perfect, whereas the solutions produced by (6.29) and (6.30) are
corrupted by growing 2t:J.x disturbances. These unstable 2t:J.x waves completely
dominate the solution computed using (6.29) by t = 27 and that obtained using
(6.30) by t = 34.
In the third case, shown in Fig. 6.3c, the Courant number is 2.5, wt:J.t =
-11/500, and the exact solution at t = 20 is once again identical to the initial condition. Although it has been somewhat diffused by the cubic interpolation in the
serni-Lagrangian advection, the solution produced by (6.26) is free of instability
and noticeable phase-speed error. In contrast, the solutions obtained using (6.29)
and (6.30) are both contaminated by unstab1e long-wavelength disturbances.
Instabilities develop in the second and third cases even though the magnitude of
the forcing is very small (lwt:J.t I < 0.01). These resu1ts,together with the stability
analysis presented in Fig. 6.2, demonstrate the need to compute forcing terms
that depend on the solution, i.e., forcing of the form S(1/!), using data along the
backward trajectory.
6.3 Systems of Equations
One of the most important app1ications of semi-Lagrangian methods in atmospheric science is in global weather prediction. This application was pioneered by
Robert (I 981, 1982), who showed that the equations describing large-scale atmospheric motion cou1d be efficient1y integrated using semi-Lagrangian methods in
conjunction with a semi-implicit approximation of those terms in the governing
equations representing the pressure gradient and velocity divergence. The essential elements ofthe semi-Lagrangian semi-implicit method will be explored in this
section by examining numerical approximations to simple shallow-water equations. The shallow-water system also provides a convenient example in which to
illustrate the difference between the semi-Lagrangian approach and the c1assical
method of characteristics.
6.3.1 Comparison with the Method ofCharacteristics
Semi-Lagrangian approximations to the equat ions describing the advection and
reaction of chemical tracers, such as (6.25), may be regarded as an algorithm for
numerically imp1ementing the c1assica1 method of characteristics (Courant et al.
1952; see also Gustafsson et al. 1995). The advection equation is, however, a
6. Semi-Lagrangian Methods
and is confined to the time coordinate, whereas the phase errors in the solutions
to (6.29) and (6.30) appear in both time and space.
The Courant number is increased to unity and wt:J.t is decreased to -11/1000
in the second test case shown in Fig. 6.3b. At t = 20 the correct solution is again
identical to the initial condition. Since the wind speed is constant and the Courant
number is unity, the semi-Lagrangian advection is exact, and the only source of
error is in the integration of the forcing . The solution obtained using (6.26) is
stable and almost perfect, whereas the solutions produced by (6.29) and (6.30) are
corrupted by growing 2t:J.x disturbances. These unstable 2t:J.x waves completely
dominate the solution computed using (6.29) by t = 27 and that obtained using
(6.30) by t = 34.
In the third case, shown in Fig. 6.3c, the Courant number is 2.5, wt:J.t =
-11/500, and the exact solution at t = 20 is once again identical to the initial condition. Although it has been somewhat diffused by the cubic interpolation in the
serni-Lagrangian advection, the solution produced by (6.26) is free of instability
and noticeable phase-speed error. In contrast, the solutions obtained using (6.29)
and (6.30) are both contaminated by unstab1e long-wavelength disturbances.
Instabilities develop in the second and third cases even though the magnitude of
the forcing is very small (lwt:J.t I < 0.01). These resu1ts,together with the stability
analysis presented in Fig. 6.2, demonstrate the need to compute forcing terms
that depend on the solution, i.e., forcing of the form S(1/!), using data along the
backward trajectory.
6.3 Systems of Equations
One of the most important app1ications of semi-Lagrangian methods in atmospheric science is in global weather prediction. This application was pioneered by
Robert (I 981, 1982), who showed that the equations describing large-scale atmospheric motion cou1d be efficient1y integrated using semi-Lagrangian methods in
conjunction with a semi-implicit approximation of those terms in the governing
equations representing the pressure gradient and velocity divergence. The essential elements ofthe semi-Lagrangian semi-implicit method will be explored in this
section by examining numerical approximations to simple shallow-water equations. The shallow-water system also provides a convenient example in which to
illustrate the difference between the semi-Lagrangian approach and the c1assical
method of characteristics.
6.3.1 Comparison with the Method ofCharacteristics
Semi-Lagrangian approximations to the equat ions describing the advection and
reaction of chemical tracers, such as (6.25), may be regarded as an algorithm for
numerically imp1ementing the c1assica1 method of characteristics (Courant et al.
1952; see also Gustafsson et al. 1995). The advection equation is, however, a