3.5 LinearEquations with Variable Coefficients
157
properties of the continuous problem. The averaging operator is also a natural
choice on staggered meshes in which the velocity normal to the interface between
each pair of grid cells is located at the center of that interface (as in Fig. 3.6).
Comparison 0/Flux and Advective Form
The transport of a passive scalar in a nondivergent fiow field is also described by
the equation
81ft + V' . (1ftv) = 0,
8t
(3.108)
which can either be derived from physical principles or obtained by combining
(3.102) and (3.103). Equation (3.108) is referred to as being influx form, whereas
(3.102) is in advective form. The fiux form (3.108) is an example of a conservation
law. Conservation laws can be expressed by equations in the general form
81ft
-
8t
+V' ·f=O,
which states that the local rate of change of 1ft is determined by the convergence of
a fiux f. It follows from the divergence theorem that the integral of 1ft over the entire domain, denoted by 1ft, is determined by the net fiux through the boundaries,
and thus, 1ft is conserved if f is periodic over the domain or if the component of
f normal to the boundary vanishes at the boundary. Solutions to arbitrary conservation laws need not, however, conserve
need not conserve 111ft 112 when V' . v 1= O.
For example, solutions to (3.108)
Conservation laws may admit solutions that contain shocks or discontinuities,
and as discussed in Chapter 5, when simulating solutions with shocks or discon -
tinuities it is essential to use a scheme that conserves the discretized equivalent
of 1ft. Even when the solution remains smooth and well-resolved, it is generally
advantageous to choose a scheme that conserves 1ft. A natural way to achieve
this conservation is to evaluate the fiux at each cell interface and then difference
those fiuxes across each cell. Assuming that the fiuxes are originally available on
the same mesh points as the scalar field, the fiuxes at the cell boundaries can be
computed by spatial averaging, in which case the time tendcncy of r/J is given by
dr/J dt + s. (Ix)
x
+ s, ()Y /y
= 0,
(3.109)
where /x and /y are the x and y components of f. Two possible ways to arrive at a
finite-difference approximation to the fiux form of the transport equation (3.108)
are
and
(3.110)
157
properties of the continuous problem. The averaging operator is also a natural
choice on staggered meshes in which the velocity normal to the interface between
each pair of grid cells is located at the center of that interface (as in Fig. 3.6).
Comparison 0/Flux and Advective Form
The transport of a passive scalar in a nondivergent fiow field is also described by
the equation
81ft + V' . (1ftv) = 0,
8t
(3.108)
which can either be derived from physical principles or obtained by combining
(3.102) and (3.103). Equation (3.108) is referred to as being influx form, whereas
(3.102) is in advective form. The fiux form (3.108) is an example of a conservation
law. Conservation laws can be expressed by equations in the general form
81ft
-
8t
+V' ·f=O,
which states that the local rate of change of 1ft is determined by the convergence of
a fiux f. It follows from the divergence theorem that the integral of 1ft over the entire domain, denoted by 1ft, is determined by the net fiux through the boundaries,
and thus, 1ft is conserved if f is periodic over the domain or if the component of
f normal to the boundary vanishes at the boundary. Solutions to arbitrary conservation laws need not, however, conserve
need not conserve 111ft 112 when V' . v 1= O.
For example, solutions to (3.108)
Conservation laws may admit solutions that contain shocks or discontinuities,
and as discussed in Chapter 5, when simulating solutions with shocks or discon -
tinuities it is essential to use a scheme that conserves the discretized equivalent
of 1ft. Even when the solution remains smooth and well-resolved, it is generally
advantageous to choose a scheme that conserves 1ft. A natural way to achieve
this conservation is to evaluate the fiux at each cell interface and then difference
those fiuxes across each cell. Assuming that the fiuxes are originally available on
the same mesh points as the scalar field, the fiuxes at the cell boundaries can be
computed by spatial averaging, in which case the time tendcncy of r/J is given by
dr/J dt + s. (Ix)
x
+ s, ()Y /y
= 0,
(3.109)
where /x and /y are the x and y components of f. Two possible ways to arrive at a
finite-difference approximation to the fiux form of the transport equation (3.108)
are
and
(3.110)