3.2 Threeor More Independent Variables
117
and is identical to that which would be obtained if leapfrog time-differencing was
used to integrate every term. The benefits of forward -backward time-differencing
have been lost. Once again, the numerical scheme includes computational modes,
and for c » IVI the maximum stable time step is one-half that allowed in the
spatially unstaggered scherne . The benefits of spatial staggering are retained, but
apply only to that portion of the total velocity of propagation that is produced by
the pressure gradient and divergence terms (i,e., by the mechanisms that remain
active in the limit U
0). In situations where c » IVI, spatial staggering yields
substantial improvement, but in those cases where IVI » c, the errors in the 2ßx
waves introduced by the advection terms dominate the total solution and mask
the benefits of spatial staggering. One way to improve accuracy when lVI :::
c is to use fourth-order centered differencing for the advection terms. Fourthorder differencing is not used to obtain high accuracy in the well-resolved waves,
but rather to reduce the phase-speed error in the moderately resolved waves to a
value comparable to that generated by second-order staggered differencing (see
Fig.3.2).
3.2 Three or More Independent Variables
In most time-dependent problems of practical interest, the unknowns are functions
of three or four independent variables (i.e., time and two or three spatial coordinates) . The accuracy, consistency, and stability of finite-difference approxirnations to higher-dimensional equations are determined using essentially the same
procedures described in Chapter 2. Two specific examples will be considered in
the following section: scalar advection in two dimensions and the Boussinesq
equations.
3.2.1 Scalar Advection in Two Dimensions
The advection equation for two-dimensional flow can be approximated using
leapfrog-time centered-space schemes that are obvious generalizations of the
finite-difference approximations employed in the one-dimensional problem . New
considerations involving the incorporation of mixed spatial derivatives do, however, arise in designing accurate and efficient forward-in-time approximations.
These considerations will be explored after first examining schemes that are centered in space and time.
Centered-in-TIme Schemes
When explicit finite-difference schemes for the integration of one-dimensional
problems are extended to two or more spatial dimensions, the stability criteria for
the multidimensional problems are often more stringent than those for the onedimensional formulation. As an example, consider the two-dimensional advection
117
and is identical to that which would be obtained if leapfrog time-differencing was
used to integrate every term. The benefits of forward -backward time-differencing
have been lost. Once again, the numerical scheme includes computational modes,
and for c » IVI the maximum stable time step is one-half that allowed in the
spatially unstaggered scherne . The benefits of spatial staggering are retained, but
apply only to that portion of the total velocity of propagation that is produced by
the pressure gradient and divergence terms (i,e., by the mechanisms that remain
active in the limit U
0). In situations where c » IVI, spatial staggering yields
substantial improvement, but in those cases where IVI » c, the errors in the 2ßx
waves introduced by the advection terms dominate the total solution and mask
the benefits of spatial staggering. One way to improve accuracy when lVI :::
c is to use fourth-order centered differencing for the advection terms. Fourthorder differencing is not used to obtain high accuracy in the well-resolved waves,
but rather to reduce the phase-speed error in the moderately resolved waves to a
value comparable to that generated by second-order staggered differencing (see
Fig.3.2).
3.2 Three or More Independent Variables
In most time-dependent problems of practical interest, the unknowns are functions
of three or four independent variables (i.e., time and two or three spatial coordinates) . The accuracy, consistency, and stability of finite-difference approxirnations to higher-dimensional equations are determined using essentially the same
procedures described in Chapter 2. Two specific examples will be considered in
the following section: scalar advection in two dimensions and the Boussinesq
equations.
3.2.1 Scalar Advection in Two Dimensions
The advection equation for two-dimensional flow can be approximated using
leapfrog-time centered-space schemes that are obvious generalizations of the
finite-difference approximations employed in the one-dimensional problem . New
considerations involving the incorporation of mixed spatial derivatives do, however, arise in designing accurate and efficient forward-in-time approximations.
These considerations will be explored after first examining schemes that are centered in space and time.
Centered-in-TIme Schemes
When explicit finite-difference schemes for the integration of one-dimensional
problems are extended to two or more spatial dimensions, the stability criteria for
the multidimensional problems are often more stringent than those for the onedimensional formulation. As an example, consider the two-dimensional advection