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

to achieve better than second-order accuracy without overstepping the storage limitations.
However, the third-order Runge-Kutta scheme has not yet been implemented in our model.
It will be used in a future version.
3.5 Dimensional splitting
In our model, equation q t + f(q) x + g(q) z = s(q) must be solved. The straightforward
approach is to evaluate the three terms f, g, s at the same time and thus updating q with the
three contributions at once. However an other possibility is to split the problem into three
sub-problems and solve them in turn
x-sweeps: q t + f(q) x = 0, (46)
z-sweeps: q t + g(q) z = 0, (47)
source-addition: q t = s(q). (48)
The application of a splitting technique generally gives more accurate results as the values
q are updated between the computation of each term. The successive steps will be detailed
using a forward-in-time integration.
In the x-sweeps we start with cell averages Q n ij at time t n and solve the one-dimensional
problem (46) along each row of cells C ij with j fixed, computing an intermediate solution Q ∗ ij :
Q ∗ ij = Q n ij + ∆t RHS x (Q n ). (49)
In the z-sweeps the values Q ∗ ij are then used as data and (47) is solved along each column of
cells with i fixed. This results in Q ∗∗
ij :
Finally the values Q ∗∗
ij
Q ∗∗
ij = Q ∗ ij + ∆t RHS z (Q ∗ ). (50)
are used to solve (48) and the new value Qn+1
ij
at t n+1 is obtained
Q n+1
ij
= Q ∗∗
ij + ∆t S(Q ∗∗ ). (51)
Without flux splitting, the information that is contained in the adjacent cells C i−1j and
C ij−1 is transmitted to cell C ij at the same time. However this does not take into account the
information that is contained in the diagonally adjacent cell C i−1j−1 , which is transported by
diagonal winds. If a flux splitting technique is used and x-sweeps are preceeding z-sweeps,
then information contained in the diagonally adjacent cell C i−1j−1 is first transmitted to cell
C ij−1 during the x-sweep and it is then further conveyed to cell C ij during the z-sweep.
3.6 Initial conditions
To determine the quantities q(x, z, t) for all times t > t 0 , initial conditions at time t 0 must be
specified. An initial ”background” state is computed for the whole domain and cell averages
are determined for every grid cell. This background state is stored in a variable q 0 , as it
might be further needed for boundary conditions computation (section 3.7) and filtering
(section 3.8). Then, a perturbation is often initiated in one of the variables q depending on
the problem to be solved. Different possibilities of initial states are given is this section.
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