e u
nþ
1
2
j
¼ e u
nÀ
1
2
j
À
Δτ λ
n
j
2
γΔχ 0
j e ρ
0
j
e p
n
jþ
1
2
À e p
n
jÀ
1
2
þ e q
nÀ
1
2
jþ
1
2
À e q
nÀ
1
2
jÀ
1
2
h
i
,
ð6:29Þ
which is Eq. (10) in the article by Brode. The equation for the normalized particle
velocity, namely, Eq. (6.14), in discrete form gives,
e u
nþ
1
2
j
¼
λ
nþ1
j
À λ
n
j
Δτ
,
hence,
λ
nþ1
j
¼ λ
n
j þ e u
nþ
1
2
j Δτ,
ð6:30Þ
and this is Brode’s Eq. (11). The mass conservation equation, namely, Eq. (6.19), in
discrete form becomes,
e ρ
nþ1
jÀ
1
2
À e ρ
n
jÀ
1
2
Δτ
¼ Àe ρ
nþ
1
2
jÀ
1
2
2e u
nþ
1
2
jÀ
1
2
λ
nþ
1
2
jÀ
1
2
þ
e u
nþ
1
2
j
À e u
nþ
1
2
jÀ1
λ
nþ
1
2
j
À λ
nþ
1
2
jÀ1
2
6
4
3
7
5,
and, by using the expansions for the intermediate points, one obtains,
e ρ
nþ1
jÀ
1
2
À e ρ
n
jÀ
1
2
Δτ
¼ À
1
2
e ρ
nþ1
jÀ
1
2
þ e ρ
n
jÀ
1
2
2 e u
nþ
1
2
j
þ e u
nþ
1
2
jÀ1
λ
nþ
1
2
j
þ λ
nþ
1
2
jÀ1
þ
e u
nþ
1
2
j
À e u
nþ
1
2
jÀ1
λ
nþ
1
2
j
À λ
nþ
1
2
jÀ1
2
4
3
5
¼ À e ρ
nþ1
jÀ
1
2
þ e ρ
n
jÀ
1
2
2 e u
nþ
1
2
j
þ e u
nþ
1
2
jÀ1
λ
nþ1
j
þ λ
n
j þ λ
nþ1
jÀ1 þ λ
n
jÀ1
þ
e u
nþ
1
2
j
À e u
nþ
1
2
jÀ1
λ
nþ1
j
þ λ
n
j À λ
nþ1
jÀ1 À λ
n
jÀ1
2
4
3
5 :
In relation to the quantity within the square brackets in this latter equation, Brode
defines,
W ¼ Δτ
2 e u
nþ
1
2
j
þ e u
nþ
1
2
jÀ1
λ
nþ1
j
þ λ
n
j þ λ
nþ1
jÀ1 þ λ
n
jÀ1
þ
e u
nþ
1
2
j
À e u
nþ
1
2
jÀ1
λ
nþ1
j
þ λ
n
j À λ
nþ1
jÀ1 À λ
n
jÀ1
2
4
3
5 ,
which is his Eq. (13), hence, with this substitution the mass conservation equation
becomes,
290
6 Numerical Treatment of Spherical Shock Waves