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Van Leer Flux Splitting Scheme
The Van Leer (1979) flux vector splitting is one of a large body
of similar techniques. Since a general fluid flow contains wave
speeds
that are both positive and negative (so that eigenvalue
information can pass both upstream and downstream), the basic idea
behind all ofthese techniques is that the flux can be split into
two components so that each may be properly discretized using
relatively upwind stencils to
maintain stability and accuracy. There are many possible ways to
split the flux term by basing the splitting on the eigenvalue
structure or
some similar representation of the flow.
From Tannehill, Anderson, and Pletcher (1997), the particular
Van Leer flux splitting sought to correct some problems found at
sonic
and stagnation points for an earlier splitting called
Steger-Warming. The Van Leer splitting offers both zero and first
order continuity
through sonic and stangation points, thus correcting this
problem. This particular variation of flux splitting is based on
Mach number splittin
(Laney 1998). The Mach number for supersonic flow is simply the
full scalar Mach number in the downwind direction, and zero in
the
upwind direction. For subsonic flow, the Mach number is slightly
more complex, and is given in eqn. (21).
(21)
Here, the Mach number is approximated using a second order
polynomial. This insures zero and first order continuity at the
sonic points,M = +1 and M = -1. This can be found by adding the
"plus" and "minus" contributions together at each of these two
points. From this, the
flux can be defined as a result. In a supersonic flow case, the
flux is likewise equal to the full flux from the upwind side (with
0 contribution
from the downwide side). Although notation can be somewhat
confusing here, a simple convention is given which will be used
from this
point on. For a flow moving from left to right, a positive sense
flux will also move from left to right. A negative sense flux will
move from
right to left. Thus, to preserve proper upwind stenciling, a
variable or flux term from the left should be stenciled from the
left, and a variable
or flux from the right should be evaluated using points from the
right. Thus, the term "left" and "+" will be used interchangeably,
and the
term "right" and "-" will be used interchageably.
From this, for positive supersonic flow from left to right (M
> 1.0), the full flux will be a function of variables from the
left or plus. For a
right to left supersonic flow (M < -1.0), the full flux will
be a function of variables solely from the right, or minus. For a
subsonic flow, the
flux is a summation of both plus and minus flux contributions,
with each component appropriately stenciled. This arrangement is
given in
eqn. (22).
(22)
The flux components are defined as given in eqn. (23), which
just result from the Mach number splitting above. Van Leer put
these
components in terms of Mach number to make the analysis more
consistent.
(23)
It is worthwile to note for implementation purposes that there
is a lot of repitition in the split flux definition. It is a simple
substitution to
define some parameters that can be used to avoid redundant
calculations and increase calculation efficiency. This abbreviated
form is given
in eqn. (24).
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8/12/2019 Van Leer Flux Splitting Details
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with...
(24)
Likewise, the full flux, used in the supersonic conditions, can
be defined in terms of Mach number to match the split fluxes. This
is given in
eqn. (25).
(25)
Explicit time integration methods may simply utilize eqns. (24)
and (25) to calculate the flux and corresponding residual dependent
on th
condition given in eqn. (21). The remaining topic to consider is
that for implicit methods, a Jacobian matrix must be calculated for
the left
hand side matrix. This Jacobian requires the derivatives of the
flux values with respect to each conservative variable at a given
cell center
(solution) point. This Jacobian is required for all the solution
points that exist, forming, in general, a penta-diagonal system of
matrices. Th
is somewhat difficult directly since the flux is calculated in
terms of a reconstructed conservative variable set at each
individual face. Hence
the chain rule is applied to get the final derivative for the
flux with respect to the appropriate solution point conservative
variable. This
construction is shown in eqn. (26).
(26)
First, the Jacobian of the full flux term is given since it is
relatively straight forward and simple. The full flux is given in
eqn. (27)
completely in terms of conservative variables since these are
the derivatives that must be known.
(27)
The Jacobian matrix follows by taking the relatively
straight-forward derivative of each term with respect to each
variable. This is given in
eqn. (28).
(28
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Taking a derivative in the split flux case is somewhat more
complicated. It is easiest to take derivatives offaandfb, and then
use thechain rule on eqn. (24) to get the face Jacobian. It happens
that a relatively common form appears in all the derivative terms,
and thus is
defined purely to make the problem more manageable. Calculating
it once in the implementation of the Jacobian programming also
offers a
small increase in efficiency. This commonly seen term is given
in eqn. (29) before providing the derivative terms.
(29)
The derivatives offawith respect to the conservative variables
are given in eqn. (30).
(30)
Similarly, the derivatives offbwith respect to the conservative
variables are given in eqn. (31).
(31)
This can be put all together by using the chain rule on eqn.
(24) to get the Jacobian matrix with respect to the face variable
for either the
plus or minus split flux in terms of derivatives of fa and fb.
The needed Jacobian for the implicit schemes can be found by using
eqn. (32)
coupled with eqns. (30) and (31) as well as eqn. (28).
(click image to enlarge view)
(32)
http://chimeracfd.com/programming/gryphon/flux_vl_eqn16.gifhttp://chimeracfd.com/programming/gryphon/flux_vl_eqn16.gif
