300
5. Finite-Volume Methods
where J.L = e6.t I Sx , Compare the implicit numerical diffusion generated
by this scheme with that produced by upstream differencing. Show that the
ratio of the leading-order numerical diffusion in the upstream scheme to
that in the Lax-Fredrichs method is J.LI(l + J.L).
6. Suppose that the constant-wind-speed advection equation (5.18) is approximated using the scheme
where y is a user-specified parameter determining the amount of numerical
smoothing.
(a) What is the largest value of y for which the scheme can be monotone?
(b) Suppose that y is specified as some value yo for which the scheme can
be monotone. For what values of J.L = e6.t I 6.x will the scheme actually be
monotone?
(c) For what value of y is this scheme equivalent to the Lax-Fredrichs
scheme (5.62)?
7. Explain why a scheme that is monotonicity-preserving need not be TVD
(or more precisely, total variation nonincreasing). Explain why being TVD
does not imply that a scheme is monotone.
8. Show that no new maxima or minima can develop in smooth solutions to
the conservation law (5.6).
9. Show that Harten 's criterion (5.32) insuring that schemes of the form (5.31)
are TVD is not sufficient to guarantee that they are monotone.
0
if ab s 0,
aj = { sgn(a)la + bl/2 otherwise,
where
and
if e ::: 0,
ife < o.
11. Suppose that Lax-Wendroff solutions are sought to a one-dimensional advection equation (5.18) and that the velocity e(t) depends on time but not