5.4 F1ux-Corrected Transport
257
a factor of ../2. Aceurate finite-volume approximations to (5.23) cannot therefore
be TVD . Useful schemes for the simulation of tracer transport can nevertheless
be obtained by borrowing techniques used to generate TVD approximations to the
one-dimensional conservation law (5.6).
Instead of demanding that the scheme be TVD, it is possible to control the development of spurious oscillations by regulating the behavior of the local maxima
and minima in the solution. Smooth solutions to the nonlinear conservation law
(5.22) also satisfy the advective-form equation
df 81/1
dg 81/1
81/1
-+--+--=0.
(5.24)
8t
d1/l 8x
d1/l 8y
!fthe velo city field is nondivergent, (5.23) may be written in an analogous advective form,"
81/1 + u 81/1 + }1/1 = O.
(5.25)
8t
8x
8y
Solutions to both (5.24) and (5.25) conserve the amplitude of all local maxima
and minima in the initial data . Flux-corrected transport algorithms, which will
be the considered in the next section, exploit this property of the true solution to
control the development of ripples near a discontinuity.
The remainder of this chapter will be primarily devoted to the examination of
methods for the simulation of discontinuities or poorly resolved gradients in nondivergent flow. The first topic considered is one-dirnensional nondivergent flow,
which can occur only if the velocity is constant. The one-dimensional constantwind-speed advection equation is also a member of the family of autonomous
conservation laws of the form (5.6). As a consequence, the study of the constantwind-speed advection equation serves as an introduction to both fluid transport
problems of the fonn (5.23) and nonlinear hyperbolic conservation laws of the
form (5.22) . The extension of these results to nonuniform two-dimensional flow
is discussed in Section 5.7. The extension of the one-dimensional constant-windspeed problem to nonlinear systems of conservation laws, which is beyond the
scope of this text, is discussed in LeVeque (1992) and Godlewski and Raviart
(1996).
5.4 Flux-Corrected Transport
Flux-corrected transport, or FCT, was proposed by Boris and Book (1973) as a
way of approximating a conservation law with a high-order scheme in regions
where the solution is smooth while using a low-order monotone scheme where
4Equation 5.25 states that the tracer concentration (typically expressed as a dimensionless ratio,
such as grams per kilogram or parts per billion) is conserved following the motion of each fluid parceI.
In contrast, (5.23) states that the local rate of change of the mass of the tracer at a fixed point in space
is determ ined by the divergence of the tracer mass-f1uxat that point.
257
a factor of ../2. Aceurate finite-volume approximations to (5.23) cannot therefore
be TVD . Useful schemes for the simulation of tracer transport can nevertheless
be obtained by borrowing techniques used to generate TVD approximations to the
one-dimensional conservation law (5.6).
Instead of demanding that the scheme be TVD, it is possible to control the development of spurious oscillations by regulating the behavior of the local maxima
and minima in the solution. Smooth solutions to the nonlinear conservation law
(5.22) also satisfy the advective-form equation
df 81/1
dg 81/1
81/1
-+--+--=0.
(5.24)
8t
d1/l 8x
d1/l 8y
!fthe velo city field is nondivergent, (5.23) may be written in an analogous advective form,"
81/1 + u 81/1 + }1/1 = O.
(5.25)
8t
8x
8y
Solutions to both (5.24) and (5.25) conserve the amplitude of all local maxima
and minima in the initial data . Flux-corrected transport algorithms, which will
be the considered in the next section, exploit this property of the true solution to
control the development of ripples near a discontinuity.
The remainder of this chapter will be primarily devoted to the examination of
methods for the simulation of discontinuities or poorly resolved gradients in nondivergent flow. The first topic considered is one-dirnensional nondivergent flow,
which can occur only if the velocity is constant. The one-dimensional constantwind-speed advection equation is also a member of the family of autonomous
conservation laws of the form (5.6). As a consequence, the study of the constantwind-speed advection equation serves as an introduction to both fluid transport
problems of the fonn (5.23) and nonlinear hyperbolic conservation laws of the
form (5.22) . The extension of these results to nonuniform two-dimensional flow
is discussed in Section 5.7. The extension of the one-dimensional constant-windspeed problem to nonlinear systems of conservation laws, which is beyond the
scope of this text, is discussed in LeVeque (1992) and Godlewski and Raviart
(1996).
5.4 Flux-Corrected Transport
Flux-corrected transport, or FCT, was proposed by Boris and Book (1973) as a
way of approximating a conservation law with a high-order scheme in regions
where the solution is smooth while using a low-order monotone scheme where
4Equation 5.25 states that the tracer concentration (typically expressed as a dimensionless ratio,
such as grams per kilogram or parts per billion) is conserved following the motion of each fluid parceI.
In contrast, (5.23) states that the local rate of change of the mass of the tracer at a fixed point in space
is determ ined by the divergence of the tracer mass-f1uxat that point.