5.7 Two Spatial Dimensions
277
5.7 Two Spatial Dimensions
The preceding discussion has focused almost exclusively on problems in one spatial dimension . The most straightforward way to extend these methods to multiple dimensions is through fractional steps, and fractional steps have been used
successfully in problems whose solutions contain discontinuities or poorly resolved gradients. A theoretical basis for the success of these methods was provided by Crandall and Majda (1980a), who showed that convergent approximations to the entropy-consistent solution of two-dimensional scalar conservation
laws can be achieved using the method of fractional steps, provided that consistent conservation-form monotone schemes are used in each individual step.
Nevertheless, as discussed in Section 3.3.1, operator splitting generates only an
o [ßt]-accurate approximation unless the finite-difference operators associated
with each split step commute (and are individually at least second-order accurate). The finite-difference operators cannot be expected to commute un1ess the
corresponding operators in the unapproximated problem commute , and in many
practical cases the unapproximated operators do not commute. For example, the
advective operators
8
u8x
and
8
v8y
do not commute unless
8v
8u
u -=v-=O.
8x
8y
Thus, one drawback to the fractional-step procedure is the likelihood that it will
lead to increased time-truncation errors. Strang splitting (3.57) can be used to
retain 0 [(ßt)2] accuracy when the finite-difference operators don't commute ,
but if the value of the prognostic variable is required at every time step,8 Strang
splitting requires 50% more work per time step than does conventional splitting.
More accurate results, and a more isotropic finite-difference solution, can be
obtained using unsplit algorithms. In the following, we will consider two representative methods : flux-corrected transport (Zalesak 1979) and a flux-limiter
algorithm for two-dimensional nondivergent flow proposed by LeVeque (1996).
Several other schemes with varying degrees of similarity have also appeared in
the literature, including those by Smolarkiewicz (1984), Colella (1990), Saltzman (1994), Leonard et al. (1993), Thubum (1996), and Stevens and Bretherton
(1996).
5.7.1 FCT in Two Dimensions
The extension of the flux-correction algorithm described in Section 5.4.2 to multidimensional problems is straightforward and is discussed in detail by Zalesak
8The values of the prognostic variables are required at every time step during the integration of
systems of equations in which a chemical or physical proce ss (such as cloud condensation and precipitation) is parametrized as a function of the prognostic variables.
277
5.7 Two Spatial Dimensions
The preceding discussion has focused almost exclusively on problems in one spatial dimension . The most straightforward way to extend these methods to multiple dimensions is through fractional steps, and fractional steps have been used
successfully in problems whose solutions contain discontinuities or poorly resolved gradients. A theoretical basis for the success of these methods was provided by Crandall and Majda (1980a), who showed that convergent approximations to the entropy-consistent solution of two-dimensional scalar conservation
laws can be achieved using the method of fractional steps, provided that consistent conservation-form monotone schemes are used in each individual step.
Nevertheless, as discussed in Section 3.3.1, operator splitting generates only an
o [ßt]-accurate approximation unless the finite-difference operators associated
with each split step commute (and are individually at least second-order accurate). The finite-difference operators cannot be expected to commute un1ess the
corresponding operators in the unapproximated problem commute , and in many
practical cases the unapproximated operators do not commute. For example, the
advective operators
8
u8x
and
8
v8y
do not commute unless
8v
8u
u -=v-=O.
8x
8y
Thus, one drawback to the fractional-step procedure is the likelihood that it will
lead to increased time-truncation errors. Strang splitting (3.57) can be used to
retain 0 [(ßt)2] accuracy when the finite-difference operators don't commute ,
but if the value of the prognostic variable is required at every time step,8 Strang
splitting requires 50% more work per time step than does conventional splitting.
More accurate results, and a more isotropic finite-difference solution, can be
obtained using unsplit algorithms. In the following, we will consider two representative methods : flux-corrected transport (Zalesak 1979) and a flux-limiter
algorithm for two-dimensional nondivergent flow proposed by LeVeque (1996).
Several other schemes with varying degrees of similarity have also appeared in
the literature, including those by Smolarkiewicz (1984), Colella (1990), Saltzman (1994), Leonard et al. (1993), Thubum (1996), and Stevens and Bretherton
(1996).
5.7.1 FCT in Two Dimensions
The extension of the flux-correction algorithm described in Section 5.4.2 to multidimensional problems is straightforward and is discussed in detail by Zalesak
8The values of the prognostic variables are required at every time step during the integration of
systems of equations in which a chemical or physical proce ss (such as cloud condensation and precipitation) is parametrized as a function of the prognostic variables.