Diploma report Implementation and verification of a simple ... - LPAS

Figure 1: Illustration of a finite volume method where the average value Q ij is updated by
the fluxes at the cell edges.
which is a discrete approximation to the integral equation (22). The sum of the flux differences
cancels out except for the fluxes at the extreme edges x I1 −1/2, x I2 +1/2, z J1 −1/2 and
z J2 +1/2. Therefore there will be exact conservation of the total quantity within the computational
domain and this quantity will vary only due to the fluxes at the boundary.
In order to obtain a fully discrete numerical method the second step is to discretize (23)
in time. Some techniques of time integration will be discussed in section 3.4. One possibility
of discretization is given here as an example and a fully discrete method is derived.
Let ∆t = t n+1 − t n be the length of a time step, and let Q n ij represent the fully discrete
approximation to Q ij (t n ). Then the time discretization of (23) is done by simply taking the
difference between the values Q n+1
ij
and Q n ij and dividing by ∆t. The numerical method thus
obtained has the form
Q n+1
ij
= Q n ij − ∆t
∆x (F i+1/2,j n − F i−1/2,j n ) − ∆t
∆z (Gn i,j+1/2 − Gn i,j−1/2
), (26)
where the numerical fluxes Fi−1/2,j n and Gn i,j−1/2
are some approximations to the time averages
of the fluxes along the cell interfaces over the time interval [t n , t n+1 ]:
Fi−1/2,j n ≈ 1 ∫ tn+1
∫ zj+1/2
f(q(x
∆t∆z
i−1/2 , z, t))dzdt,
t n z j−1/2
G n i,j−1/2 ≈ 1 ∫ tn+1
∫ xi+1/2
g(q(x, z
∆t∆x
j−1/2 , t))dxdt.
t n x i−1/2
(27)
If these average fluxes can be computed based on the values Q n , then we will have a fully
discrete method.
For hyperbolic problems, information propagates with finite speed, so F n
i−1/2,j and Gn i,j−1/2
can be assumed to be based only on the cell averages on both sides of the interfaces x i−1/2
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