
41Chapter three: DG method for 1D nonconservative equations
+
∂
∂
=()
ˆ
()
0NxqdxNx
C
x
dx
i
x
x
i
x
x
s
e
e
e
s
e
e
e
(3.44)
+−
∂
∂
=() () ()
()
() 0NxNxdx qNxC
Nx
x
Nxdx C
ij
x
x
ji
x
x
i
j
x
x
j
s
e
e
e
s
e
e
e
s
e
e
e
(3.45)
∫
∫
∆
ξξξ
+ξ −
∂ξ
∂ξ
ξξ
=
−
−
−
2
() () ()
()
() 0
1
1
1
1
1
1
x
NNdqNC
N
NdC
ij ji
i
jj
(3.46)
∆
+
−
−
−−
=
2/61/6
1/62/6
1/21/2
1/21/2
0
0
1
2
1
2
1
2
x
q
q
C
C
C
C
(3.47)
=
∆
−−
−
∆
−
−
1
33
33
1
42
24
1
2
1
2
1
2
q
q
x
C
C
x
C
C
(3.48)
A similar procedure as described above for Equation (3.41) is adopted
for Equation (3.42), and the results are given by Equations (3.49) to (3.53). In
Equation (3.54), the numerical uxes are c ...