100
2. Basic Finite-DitTerence Methods
Although the larger-scale features may be approximated with considerable quantitative accuracy , generally one must either be content with a qualitatively correct
representation of the shortest-scale features or must remove these features with
some type of numerical smoothing. Since it is not realistic to expect convergence
to the correct solution in such problems , it is not particularly important to use
high-order methods. Instead, one generally employs the finest possible numerical
grid, selects a method that captures the behavior of moderately resolved waves
with reasonable fidelity, and ensures that any spurious poorly resolved waves
are eliminated by either explicit or implicit numerical dissipation . The numerical
dissipation associated with all the schemes considered in this chapter is applied
throughout the entire numerical domain. An alternative approach will be considered in Chapter 5, in which the implicit dissipation is primarily limited to those
regions where the approximate solution is discontinuous or very poorly resolved.
Given that some degree of dissipation must generally be included to generalize the methods described in this chapter to practical problems involving lowviscosity flow, the neutral amplification factors associated with leapfrog time
differencing and centered spatial differences are less advantageous than they may
first appear. The difficulties associated with time splitting that can arise in nonlinear problems make the leapfrog scheme relatively unattractive in comparison with
the third-order Adams-Bashforth or third- or fourth-order Runge-Kutta methods.
The advantages of these relatively high-order methods are not primarily associated with their small truncation error (since some features will be poorly resolved) but arise from their stability and relative efficiency. The second-order
Magazenkov and leapfrog-trapezoidal methods are also possible alternatives to
the leapfrog scheme. Even forward differencing is a possibility, provided that it
is used in a Lax-Wendroff method and that the implicit diffusion in the LaxWendroff scheme is limited by using a sufficiently small time step.
Now consider the choice of spatial difference approximations. Approximations
based on centered spatial differences typically require the use of an explicit fourthor sixth-derivative dissipative filter and are therefore less efficient than a third -
order upstream approximation. This lack of efficiency is compensated by two
practical advantages. First, it is not necessary to determine the upstream direction
at each grid point when formulating the computer algorithm to evaluate a centered
spatial difference. The determination of the upstream direction is not particularly
difficult in advection problems where all signal propagation is directed along a
clearly defined flow, but it can be far more difficult in problems admitting wave
solutions that propagate both to the right and to the left , The second advantage of
a centered difference used in conjunction with a spatial filter is that one can explicitly control the magnitude of the artificial dissipation, whereas the magnitude
of the numerical dissipation associated with an upstream difference is implicitly
determined by the local wind speed. The compact schemes appear to provide particularly good formulae for the evaluation of centered spatial differences because
they remain accurate at relatively short wavelengths (3D.x or 4D.x) and use information at a minimum number of spatial grid points, which reduces the amount of
special coding required near the boundaries of the spatial domain.
2. Basic Finite-DitTerence Methods
Although the larger-scale features may be approximated with considerable quantitative accuracy , generally one must either be content with a qualitatively correct
representation of the shortest-scale features or must remove these features with
some type of numerical smoothing. Since it is not realistic to expect convergence
to the correct solution in such problems , it is not particularly important to use
high-order methods. Instead, one generally employs the finest possible numerical
grid, selects a method that captures the behavior of moderately resolved waves
with reasonable fidelity, and ensures that any spurious poorly resolved waves
are eliminated by either explicit or implicit numerical dissipation . The numerical
dissipation associated with all the schemes considered in this chapter is applied
throughout the entire numerical domain. An alternative approach will be considered in Chapter 5, in which the implicit dissipation is primarily limited to those
regions where the approximate solution is discontinuous or very poorly resolved.
Given that some degree of dissipation must generally be included to generalize the methods described in this chapter to practical problems involving lowviscosity flow, the neutral amplification factors associated with leapfrog time
differencing and centered spatial differences are less advantageous than they may
first appear. The difficulties associated with time splitting that can arise in nonlinear problems make the leapfrog scheme relatively unattractive in comparison with
the third-order Adams-Bashforth or third- or fourth-order Runge-Kutta methods.
The advantages of these relatively high-order methods are not primarily associated with their small truncation error (since some features will be poorly resolved) but arise from their stability and relative efficiency. The second-order
Magazenkov and leapfrog-trapezoidal methods are also possible alternatives to
the leapfrog scheme. Even forward differencing is a possibility, provided that it
is used in a Lax-Wendroff method and that the implicit diffusion in the LaxWendroff scheme is limited by using a sufficiently small time step.
Now consider the choice of spatial difference approximations. Approximations
based on centered spatial differences typically require the use of an explicit fourthor sixth-derivative dissipative filter and are therefore less efficient than a third -
order upstream approximation. This lack of efficiency is compensated by two
practical advantages. First, it is not necessary to determine the upstream direction
at each grid point when formulating the computer algorithm to evaluate a centered
spatial difference. The determination of the upstream direction is not particularly
difficult in advection problems where all signal propagation is directed along a
clearly defined flow, but it can be far more difficult in problems admitting wave
solutions that propagate both to the right and to the left , The second advantage of
a centered difference used in conjunction with a spatial filter is that one can explicitly control the magnitude of the artificial dissipation, whereas the magnitude
of the numerical dissipation associated with an upstream difference is implicitly
determined by the local wind speed. The compact schemes appear to provide particularly good formulae for the evaluation of centered spatial differences because
they remain accurate at relatively short wavelengths (3D.x or 4D.x) and use information at a minimum number of spatial grid points, which reduces the amount of
special coding required near the boundaries of the spatial domain.