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

3.3.4 Richtmeyer Two-Step Lax-Wendroff Method
The Richtmeyer Two-Step Lax-Wendroff method is of second order. This is achieved by using
a better approximation to the average flux (27). In this method a half step in time is first
made using the Lax-Friedrichs method at the cell interfaces (∆x, ∆z and ∆t are replaced
by 1/2∆x, 1/2∆z and 1/2∆t). The numerical approximation of q at the midpoint in time
t n+1/2 = t n + 1 2∆t is given by
and the flux is evaluated at this point
Q n+1/2
i−1/2j = 1 2 (Qn i−1j + Q n ij) − ∆t
2∆x [f(Qn ij) − f(Q n i−1j)], (37)
F n
i−1/2,j = f(Qn+1/2 i−1/2j ) (38)
Although more accurate than the Lax-Friedrichs method, this method often leads to oscillations
in solutions.
3.4 Time integration
In section 3.1, a system of ordinary differential equations (23) for the evolution of cell averages
Q ij (t) in time was derived. Our model solves the partial differential equation (16) that
contains source and correction terms in addition to the flux terms. This is done using a
splitting technique, first solving
and then making the post update
q t + ˜f(q) x + ˜g(q) z = s(q), (39)
q t = r(q). (40)
The flux terms in (39) are tilded as, in addition to the regular fluxes, they contain the flux
correction terms due to the potential energy.
Performing a finite volume discretization for (39) yields the following system of ODEs
dQ
dt = RHS x(Q) + RHS z (Q) + S(Q), (41)
where RHS x (Q) and RHS z (Q) consist in the x- and z-flux terms and S(Q) is the source term.
For simplicity, these three terms can be replaced by F(Q) giving
dQ
dt
= F(Q). (42)
The discretization in time can then be done by using any standard method for systems of
ODEs. Some time-discretization methods will be presented in this section.
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