INAUGURAL–DISSERTATION zur Erlangung der Doktorwürde der ...

48 3. Numerical Methods
M(0, 2) = w 1 v 2 1 + w 2 v 2 2 + w 3 v 2 3, (3.10)
M(1, 2) = w 1 r 1 v 2 1 + w 2 r 2 v 2 2 + w 3 r 3 v 2 3, (3.11)
M(2, 2) = w 1 r 2 1v 2 1 + w 2 r 2 2v 2 2 + w 3 r 2 3v 2 3, (3.12)
M(3, 2) = w 1 r 3 1v 2 1 + w 2 r 3 2v 2 2 + w 3 r 3 3v 2 3. (3.13)
These weights and abscissas are computed using the Wheeler algorithm (see Subsection
3.2.3). Similarly, if any of the terms on the right hand side of these Eqs. (3.4) – (3.9)
contain unknown moments, they will be closed in the analogous manner.
To solve Eqs. (3.4) – (3.9) a numerical scheme based on a kinetic transport scheme
to evaluate the spatial fluxes [96, 189] can be employed. A first-order, explicit, finite
volume scheme for these equations can be written for the set of moments
as
M =
M n+1
[
M(0, 0), M(1, 0), M(0, 1), M(1, 1), M(2, 1), M(3, 1)] T
(3.14)
i = Mi n − ∆t [
]
G(Mi n , M n
∆x
i+1) − G(Mi−1, n Mi n ) + ∆tSi n (3.15)
where n is the time step, i is the grid node, S is the right hand side estimate of
Eqs. (3.4)– (3.9), and G is the flux function. Using the velocity abscissas, the movement
of the quadrature node from left to right or right to left is determined. The flux function
at any time step is expressed as [28]
G(M i , M i+1 ) = H + (M i ) + H − (M i+1 ) (3.16)
where
⎛ ⎞
⎛ ⎞
⎛ ⎞
1
1
1
r 1
r 2
r 3
H − (M) = w 1 min(v 1 , 0)
v 1
r 1 v + w 2 min(v 2 , 0)
v 2
1
r 2 v + w 3 min(v 3 , 0)
v 3
2
r 3 v ,
3
⎜
⎝r1v 2 ⎟
⎜
1 ⎠
⎝r2v 2 ⎟
⎜
2 ⎠
⎝r3v 2 ⎟
3 ⎠
r1v 3 1 r2v 3 2 r3v 3 3
(3.17)
and