5.71\\'0 Spatial Dimensions
291
5.7.5 When Is a Flux Limiter Necessary?
Use of a flux limiter can be essential to ensure the convergence of numerical approximations to problems with shocks or discontinuous solutions. On the other
hand, in an advection problem such as that considered in the preceding section,
the initially smooth concentration field never develops a discontinuity in a finite
time; there are no spurious weak solutions, and the flux limiter is not required
to guarantee convergence. The use of a flux limiter in numerical approximations
to the advection equation is optional and can be considered as a device for converting one type of error, namely undershoots and overshoots, into a less easily
quantifiable but more acceptable form.
Consider the advantages and disadvantages of using flux-limited or flux-corrected methods to obtain approximate solutions of advection problems. As revealed by the example shown in Fig. 5.25, flux limiters are not always helpful
and can actually degrade the result if the solution remains well-resolved on the
numerical mesh. Nevertheless, in many practical applications the solution is not
well-resolved, either due to discontinuities in the initial data or to deformation in
the velocity field that stretches a well-resolved initial field until one of the scales
characterizing the field contracts to the scale of the numerical grid. In these situations flux limiters can eliminate the spurious overshoots and undershoots that
develop as a consequence of poor numerical resolution. Except for the absence
of undershoots and overshoots, the overall character of the flux-limited solution
may, nevertheless, be rather similar to that obtained using an accurate linear finitedifference scheme (compare Figs. 5.20 and 5.21). The need for the flux limiter
is therefore most pronounced in problems where undershoots and overshoots in
the tracer concentration field can couple with other physical processes to trigger spurious behaviors. An example of such coupling can occur in simulating the
evolution of atmospheric clouds. A spurious cloud can be generated where an
error in the advective transport of water vapor produces an overshoot in which
the water-vapor mixing ratio exceeds the saturation mixing ratio. Latent heat is
released as the water vapor condenses to form the spurious cloud, and this heat
generates buoyancy perturbations that feed back on the flow field, thereby altering
the subsequent evolution of the system.
A second example of coupling between advectively generated undershoots and
other physical processes involves the generation of negative chemical concentrations in simulations of chemically reacting flows. The mixing ratio of a chemical
species should never drop below zero, but numerically generated undershoots may
produce false negative concentrations that destabilize the integration by triggering nonphysical chemical reactions. As an example, cons ider the following pair
of equations describing the advection and interaction of two chemical species :
(5.56)
(5.57)
291
5.7.5 When Is a Flux Limiter Necessary?
Use of a flux limiter can be essential to ensure the convergence of numerical approximations to problems with shocks or discontinuous solutions. On the other
hand, in an advection problem such as that considered in the preceding section,
the initially smooth concentration field never develops a discontinuity in a finite
time; there are no spurious weak solutions, and the flux limiter is not required
to guarantee convergence. The use of a flux limiter in numerical approximations
to the advection equation is optional and can be considered as a device for converting one type of error, namely undershoots and overshoots, into a less easily
quantifiable but more acceptable form.
Consider the advantages and disadvantages of using flux-limited or flux-corrected methods to obtain approximate solutions of advection problems. As revealed by the example shown in Fig. 5.25, flux limiters are not always helpful
and can actually degrade the result if the solution remains well-resolved on the
numerical mesh. Nevertheless, in many practical applications the solution is not
well-resolved, either due to discontinuities in the initial data or to deformation in
the velocity field that stretches a well-resolved initial field until one of the scales
characterizing the field contracts to the scale of the numerical grid. In these situations flux limiters can eliminate the spurious overshoots and undershoots that
develop as a consequence of poor numerical resolution. Except for the absence
of undershoots and overshoots, the overall character of the flux-limited solution
may, nevertheless, be rather similar to that obtained using an accurate linear finitedifference scheme (compare Figs. 5.20 and 5.21). The need for the flux limiter
is therefore most pronounced in problems where undershoots and overshoots in
the tracer concentration field can couple with other physical processes to trigger spurious behaviors. An example of such coupling can occur in simulating the
evolution of atmospheric clouds. A spurious cloud can be generated where an
error in the advective transport of water vapor produces an overshoot in which
the water-vapor mixing ratio exceeds the saturation mixing ratio. Latent heat is
released as the water vapor condenses to form the spurious cloud, and this heat
generates buoyancy perturbations that feed back on the flow field, thereby altering
the subsequent evolution of the system.
A second example of coupling between advectively generated undershoots and
other physical processes involves the generation of negative chemical concentrations in simulations of chemically reacting flows. The mixing ratio of a chemical
species should never drop below zero, but numerically generated undershoots may
produce false negative concentrations that destabilize the integration by triggering nonphysical chemical reactions. As an example, cons ider the following pair
of equations describing the advection and interaction of two chemical species :
(5.56)
(5.57)