256
5. Finite-Volume Methods
generate spurious negative values from nonnegative initial data. More generally,
monotone schemes yield numerical solutions to either (5.22) or (5.23) that are
free from spurious ripples in the vicinity of discontinuities and poorly resolved
gradients. This is perhaps the most useful property of monotone approximations
to the tracer transport equation, since unlike (5.22), (5.23) is a linear partial differential equation whose weak solutions are uniquely determined by the initial and
boundary data, There is therefore no need to employ monotone schemes (or to
demand satisfaction of some entropy condition) in order to ensure that consistent,
conservation-form approximations to (5.23) converge to the correct solution as
Sx, lly, and llt approach zero.
As discussed previously, monotone methods are not actually used in most practical applications because they are only first-order accurate and highly diffusive.
In regions where the solution is smooth, more accurate approximations to the onedimensional conservation law (5.6) can be obtained using TVD methods . One
might hope to pursue the same strategy in designing approximations to the twodimens ional tracer transport equation, but there are difficulties . The first problem
is that except for special cases of no practical importance, all TVD approximations
to the two-dimensional nonlinear conservation law (5.22) are at most first-order
accurate (Goodman and LeVeque 1985). Thus, in contrast to the one-dimensional
case, there are no second-order accurate TVD approximations to (5.22).
The second and more fundamental problem is that although the entropy-consistent solution to the nonlinear conservation law (5.22) is TVD (or, more precisely,
total variation nonincreasing), the total variation in the true solution to the tracer
transport equation can increase with time--even when the velocity field is nondivergent! The non-TVD nature of the solutions to (5.23) follows from the circumstance that the total variation of 1/J(x , y) is not invariant under coordinate
rotarions.' The total variation of a two-dimensional function is conventionally
defined as
11
00
1
00
TV (1/J) = lim sup -
11/J(x + E, y) -1/J(X, y)1 dx dy
F-+ O f'
+limsup - 11
-00 -00
1
00
00
11/J(x, y + E) -1/J(x, y)1 dx dy.
f-+O E -00 -00
Suppose that the initial conditions for (5.23) are
1/J(x , y, 0) = {
if ]x] 1 and Iyl s I,
otherwise,
and that the flow is in solid-body rotation with u = -y and v = x. After the
distribution of 1/J rotates through an angle of 45°, its total variation will increase by
3The nonconservation of the total variation under coordinate rotations, which was pointed out
to this author by Joe Tenerelli, appears to be a weakness in the mathematical definition of the total
variation of a two-dimensional function. A second weakness appears in the physical units that are associated with total variation. If 1/J and sp are functions in degrees and x and y are spatial coordinates in
rneters, then TV (1/J(x» has units of degrees, whereas TV (
5. Finite-Volume Methods
generate spurious negative values from nonnegative initial data. More generally,
monotone schemes yield numerical solutions to either (5.22) or (5.23) that are
free from spurious ripples in the vicinity of discontinuities and poorly resolved
gradients. This is perhaps the most useful property of monotone approximations
to the tracer transport equation, since unlike (5.22), (5.23) is a linear partial differential equation whose weak solutions are uniquely determined by the initial and
boundary data, There is therefore no need to employ monotone schemes (or to
demand satisfaction of some entropy condition) in order to ensure that consistent,
conservation-form approximations to (5.23) converge to the correct solution as
Sx, lly, and llt approach zero.
As discussed previously, monotone methods are not actually used in most practical applications because they are only first-order accurate and highly diffusive.
In regions where the solution is smooth, more accurate approximations to the onedimensional conservation law (5.6) can be obtained using TVD methods . One
might hope to pursue the same strategy in designing approximations to the twodimens ional tracer transport equation, but there are difficulties . The first problem
is that except for special cases of no practical importance, all TVD approximations
to the two-dimensional nonlinear conservation law (5.22) are at most first-order
accurate (Goodman and LeVeque 1985). Thus, in contrast to the one-dimensional
case, there are no second-order accurate TVD approximations to (5.22).
The second and more fundamental problem is that although the entropy-consistent solution to the nonlinear conservation law (5.22) is TVD (or, more precisely,
total variation nonincreasing), the total variation in the true solution to the tracer
transport equation can increase with time--even when the velocity field is nondivergent! The non-TVD nature of the solutions to (5.23) follows from the circumstance that the total variation of 1/J(x , y) is not invariant under coordinate
rotarions.' The total variation of a two-dimensional function is conventionally
defined as
11
00
1
00
TV (1/J) = lim sup -
11/J(x + E, y) -1/J(X, y)1 dx dy
F-+ O f'
+limsup - 11
-00 -00
1
00
00
11/J(x, y + E) -1/J(x, y)1 dx dy.
f-+O E -00 -00
Suppose that the initial conditions for (5.23) are
1/J(x , y, 0) = {
if ]x] 1 and Iyl s I,
otherwise,
and that the flow is in solid-body rotation with u = -y and v = x. After the
distribution of 1/J rotates through an angle of 45°, its total variation will increase by
3The nonconservation of the total variation under coordinate rotations, which was pointed out
to this author by Joe Tenerelli, appears to be a weakness in the mathematical definition of the total
variation of a two-dimensional function. A second weakness appears in the physical units that are associated with total variation. If 1/J and sp are functions in degrees and x and y are spatial coordinates in
rneters, then TV (1/J(x» has units of degrees, whereas TV (