During the convective step, all source terms in the compaction
model are set to zero, so that Eq. (1) reduces to
∂w
∂t
+ ∂
∂ξ
f(w) = 0. (32)
This step is important because solutions of this hyperbolic
system can include the evolution and propagation of discontinuitieswhich
are challenging features for numerical methods
to resolve. To numerically solve this problem, it is necessary
to express it in the semi-discrete, conservative form
dWj
dt
+ Fj+1/2 − Fj−1/2
_ξ
= 0, (33)
where Fj}1/2 are numerical fluxes through the computational
cell boundaries located at ξ j}_ξ/2. This conservative
formulation guarantees that all wave speeds are correctly
predicted and that mass, momentum, and total energy are
properly conserved within the physical domain.
To compute the numerical flux function F, it is first necessary
to project Eq. (32) into individual characteristic fields.
To this end, an average state is locally defined at each cell
interface by ˜W j+1/2 = (Wnj
+ Wnj
+1)/2, and ˜λi , ˜ri, and
˜l