242
5. Finite-Volume Methods
(a)
"\0.5
x
,
I
I""""" ,,"' " "',, ' " "" "",,",,"""" "'" ",I
0.5
x
FIGURE 5.1. Exact (dash-dotted), upstream advective-form (solid) and upstream f1ux-form
(dashed) solutions to the generalized Burgers's equation at I = 2.4 on the subdomain
0.5
x
1. (a) ilx = 0.02, ill = 0.01; (b) ilx = 0.005, ill = 0.0025.
As will be explained in Section 5.1, if the initial conditions are specified by the
step function
I
if x SO,
1f!(x,O)= { 0'
.
otherwise,
(5.3)
the correct solution consists of a unit-amplitude step propagating to the right at
speed
An upstream finite-difference approximation to the advective form (5.1)
was calculated using
4J,!+1 - 4J' !
)
) + (4J'! ) + 4J' !
) -1 )2 (4J'! - 4J' !
)
) -1 ) = 0,
(5.4)
!:lt
2
!:lx
and an upstream approximation to the flux form (5.2) was obtained using
4J,!+1 - 4J' ! (4Jn)3 - (4J'!_ )3
)
) + )
) I
3!:lx
= O.
. (5.5)
!:lt
reet solution in the limit !:lx
0 and !:lt
O. The difficulties that can be