van leer flux splitting
TRANSCRIPT
FLUX-VECTOR SPLITTING FOR THE EULER EQUATIONS
Bram van Leer
Institute for Computer Applications in Science and Engineering
and
Leiden State University, Leiden, The Netherlands
Abstract
The flux vector in the Euler equations of compressible flow, with the
ideal-gas law inserted, is split, in the simplest possible way, in a
continuously differentiable forward-flux vector and backward~flux vector. The
first-order upwind scheme based on these fluxes produces steady shock profiles
wi th at mos t two zones. A comparison with thé split fluxes of Steger and
Warming, which are not differentiable in aonlc and stagnation points, shows
the advantage of the present fluxes.
Research was supported under NASA Contract No. NASl-15810 while the author was
in residence at IeASE, NASA Langley Research Center, Hampton, VA 23665.
Paper presented at the 8th International Conference on Numerical Methods inFluid Dynamics, Aachen, Germany, June 28 - July 2, 1982.
Introduction
To approximate a hyperbolic system of conservation laws wt + {f(w)}x = OI
with so-called upwind differences, we must first establish which way the wind
blows. More precisely, we must determine in which direction each of a variety
of signals moves through the computational grid. For this purpose, a physlcal
model of the interaction between computational cells is needed; at present two
such models are in use.
In one model neighboring cells interact through discrete, finite-
amplitude waves. The nature, propagation speed and amplitude oE these waves
are found by solving, exactly or approximately, Riemann's initial-value
problem for the discontinuity at the cell interface. We may call this th.e
Riemann approach (Fig. 1 (a)). The numerical technique of distinguishing
between the influence of the forward- and the backward-moving waves ls called
flux-difference splitting; examples are the methods of Roe [l] and of Osher
[2 J •
In the other model, the interaction of neighboring cells is accomplished
through mixing of pseudo-particles that move in and out oE each cel1 according
to a given velocity distribution. l-/emay call this the Boltzmann approach
(Fig. 1 (b)). The numerical technique of distinguishing between the inEluence
of the forward- and the backward-moving particles is called flux-vector
splitting or simply flux-splitting; an example is the "beam scheme" of
Prendergast [3], rediscovered by Steger and Warming [4].
80th kinds of splitting are discussed by Harten, Lax snd Van Leer [5J.
The present paper is restricted to flux-vector splitting· for the Euler
equations of compressible flow, with the ideal-gas law used as equation of
state.
(a)
o x
u -c o u U +c q
Figure 1. Two physical models for upwind differencing. (a) Space-time
diagram showing waves in the solution of a Riemann problem: forward shocks and contact discontinuity c, backward rarefaction r. (b) Velocitydistribution function used in the Boltzmann modela The densities of the
forward-moving and backward-moving par tic les are represented by the
differently shaded areas.
Goal
Wewishto spli tthe fluxf(w)
flux
C(w),that is,
(1)
f(w)=f+(w)+ f-(w),
(2)
dE+/dwmusthavealleigenvalues
dE-/dw
musthavealleigenvalues
under the following restrictions:
(3) f±(w) must be continuous, with
f+(w) - f(w) for Hach-numbers
f- (w) - f(w) for
in a fotward flux f+(w) and a backward
;> O,
<: O,
M ;> 1,
H <: -1,
(4) the componen ts oE together must mimic the symmetry of f
with respect to tI (all other state quantities held constant), that is
± fk (-M),
2
(5) df±/dw must be continuous,
1// (6) df±/dw mutilt hay€! one eigenvalue vaniah for 'MI <: 1,/
(7) f± no t like f (M), must be a polynomial in M, and of the lowes t possible
degree.
Restriction (3) ensures that in supersonic regions flux-vector splitting
leads to standard upwind differendng. . Restrittions (4) and (5) are self-
evident, although (5) was not satisfied in [4], with negative consequences for
the smoothness of numerical results. Restriction (6) is crudal, greatly
narrowing down the choice of functions. The degeneracy of f±(w) in subsonic
cells makes it possible to build stationary shock structures with no more than
two interior zones, just as does the flux-difference splitting technique in
[2]. Final1Yt requirement (7) makes the splitting unique.
Derivation
Consider the' one-dimensional Euler equations. We shall regard the full,
forward, and backward fluxes as functions of density p, sound speed c and
Mach number M. The full flux reads
(8)
( pcM )
f(p,c,M) = pc2(M2 +~)
3(1 2 1)pc Md2M + y-1
where y is the specific-heat ratio.
From coriditions (3) and (5) it follows that+
f (p,c,M) as well as
must vanish for M t 1. Condition (7) then leads to the restriction that
f+ includes a factor (M+1)2 and f a factor (_H+1)2, for IMI < 1.
Without introducing further factors depending on M we can now achieve the
splitting óf the mass flux:
3
(9 )
satisEying (1), (3), (4), (5) and (7). In order to split the momentum flux in
agreement with (3) and (5) we need eubie polynomials In M:
2 1 2 y-1.. 2 . 2 1 2 y-1. 2pe { /2 (M+1)} (-----r1 + -) + pc { /2 (-H+1)} (- -'--11 + -), 1M I < 1;y y y y
again, (1), (3), (4), (5) and (7) are satisHed.
The splitting of the energy flux can now be aehieved by eombining the
split mass- and momentum-fluxes:
(11 ) f±energy
2Y2
2(y-l)IMI < 1.
The acale factor ls needed to satisfy (3). The relation (11) between the
components of f± makes these fluxes degenerate: df±(w)/dw has a zero,
eigenvalue for IMI < 1. Thus condition (6) i9 fulfilled. l1oreover, since
the vanishing elgenvalue i8 eontinuou8 for M = ± 1, we have sufEieient
informatlon about the smoothness of f± to conc1ude that condition (5) i8
fully sati8fied.
trivial.
Testing, the fulfillment of (7), (4) and (1) by (11) 18
We still must determine if condiUon (Z) ls satisfied by the splitting
(9), (10), (11). There ls a good reason to believe thls ls lndeed the case:
the eigenvalue±
111of thát 18 most likely to violate eondition (2),
i.e. to have the wrong sign, has been forced te vanish fer IMI < 1. We find
that the non-zero eigenvalues
quadratic equation:
+IlZ 3,
4
dE df+/dw are the roots of the
(12) (1'+) 2 _ + 3 () y-1 ( ) {( ) 2 () }~ cll • 2 1+1'1 [1 - 12y(y+1) 1'1-1 Y M-1 + 2y M-1 - 2(y+3) ]
1'1-1
8y(y+l) {4y(y-1 )(1'1-1) + (y+1) (3-y)} Jo, 11'11 < 1.
In the relevant range 1 ( Y •• 3, the roots of (12) are positive. The
negativity of 112 3 follows by symmetry.,This completes the derivation of the split fluxes for the one-dimensional
•Euler equations. The formulas for the three-dimens10nal equat10ns are given
in Table 1 (full eguat1ons) and TabIe 11 (constant enthaIpy assumed).
Stability .
The stability anaIysis for the first-order upwind scheme based on the
above split fluxes 1s complicated' by the fact that and dC/dw
commute neither with each other nor with df/dw, for 11'11 < l. This leads to a
reduction of the eFL limit; in the worst case,
shortest waves:
(13) 1.~ < 2y!(y+3).
A practical local stability criterion is
M = O, we find for the
(14 )MM(lul+c) •• {2y + IHI(3-y)}/(y+3), 11'11 < l.
shock structures that satisfy the steady upwind-dlfference equations using
only two interior zones.
5
Table 1. Flux-splitting for the Euler equations. M = u/c.
conserved
x-fluxf+
IMI< 1forward x-flux f ,quantity
mass
pu pc{ 1/2(M+1)}2
2 2
+x-momentum p(u +c /y) f '{(y-1)u+2c}/ymassy-momentum
puv f+'vmass
z-momentum
puw f+'wmass
1 2 2 2
2+ . 2 2 1 2' 2total energy
pu { 12 (u +v +w )+ c /(y-l) }f o[{(y-l)u+2c} /{2(y -l)} + /2 (v +w)]mass .
Table 11. Flux-splitting for the isenthalpic Euler equations.
H :::'enthalpy,c* = 
M
oIII L p Q R (R) )(
(15 )
Figure 2. Steady shock profile from a fiux-split upwind scheme.
Consider the one-dimensional Euler equations. '.J'edenote the supersonic
pre-shock state by Lt the subsonic post-shock state by R and the interior
states by P and Qt as in Fig. 2. To require stationarity means to require
constancy of net flux:
fR = fL
f+ + fL P
f+ + fP Q+
fQ + fR'
Assume that the first equalltYt 1.e't the jump condition across the full shock
structure, holds. The second equality ts automatically satisfied H zone Pt
- +like Lt ls supersonict with fp = O, fL = fL• The fourth equality boils down
to
7
(16 )
which, if zone Q is subsonic, implies on1y 2 independent equations. The third
equality stands for 3 equations, so that we end up with 5 equations for 6
unknowns, the components of and The solutions form a one-parameter
family of steady shock profi1es, the parameter re1ating to the sub-grid shock
position. There exists one profile with on1y one interior zone that is
precisely sonic. In constrast, Goc;iunov~s and Roe's schemes yield steady
shocks with one or no interior zone.
Another scheme that yields steady shock structures with two interior
states is Osher' s flux-difference scheme [2]. For a acalar conservation law
it boils down to a f1ux-splitting method.
Unlike Osher's scheme, the present flux-vector splitting can not preserve
a stationary contact discontinuity; in fact, no flux-vector splitting scheme
can (see [5]). This is readi1y understood from the underlying pbysical
model: cell-interactions are achieved through mixing, a diffusive process.
Present research is aimed at the development of a computationally simple
"collision term" that could prevent the diffusion across a steady contact
discontinuity.
A comparison
For one-dimensional isothermal flow (y=l, constant e) the flux-vector
splitting of Steger and Warming, alias the beam scheme, can be derived from
the beams.
the assumption that there are two kinds of gas particles, equally abundant but
Per unit volume, 1/2p moves withmoving at different r~tes:
velocity u-e and V2 p with velocity u+c, yielding the correct average flow
speed u. For IMI < 1 there is a forward and a backward beam, with
associated fluxes satisfying (1), (3) and (4):
8
(17)±fB/SW (ljzp(U±C) )
1 2, lul < c.12 p(u±c)
In Fig. 3 (a) the dependence of the component8 of fB/SW on M 18 shown.
The momentum flux la continuously dtfferéntiable at M = tI; the mass flux ls
noto The splitting given by Eqs. (9) and (10) for y = 1 reads
(l8 )
2
(± 1/4 p(u±c) / c)112 p(u±c), lul < c,
with the same momentum-flux splitting but improved mass-flux splitting. The
dependence on M is shown in Fig. 3 (b). The Figs. 4 (a) and 4 (b) are graphs
1 M
Y=1VL
'.'." .
-1
. 1
-1
Y=1B/SW
...... ......
f f
2
-1
..'.'.........',.'
.......f;..'.'.'.'.'
fi .
'..'..'.
..... '.
-1
Figure 3. Flux-splitting for the isothermal Euler equations (y=l).Platted against Hach number are mass fluxes (subscript 1) and momentum fluxes
(subseript 2), normalized by pe and pe2, respectively. (a) Beam seheme/Steger andWarming; (b) Van Leer.
9
of the eigenvalues of ± /'dfB/SW dw
and Their values are given by
= 1/4 c( S-M) (1 +M) ,
(19)= O,
+ I(S+4H)},
IMI < 1,
..'.......¡.t;
,.'.'
2
.'
-1-1
11+ ,/t-'"2 /
/ 1/////
II+~JI t-'"1\ -------
_._~~ ::::._ •• _.. '; _ •• _ •• _:.:,..... •• •• oo.
(-1 , g. 1 M~¡.t1
..'
-2 y=1B/SW
-2 y=1VL
Figure 4. Eigenvalues of split-flu~ derivatives_ (y=1). Plotted againstHach number are the eigenvalues of df /dw and df /dw, normalized by c.(a) Beam scheme/Steger and Warming; (b) Van Leer.
10
••• <. •••• __ ••• -.-- ----,..-,-.~'Pr..•."""'~.:,
and the sYlllmetry condition (4). The e1genvalues shown in Fig. 4 (a) whlle
satisfying (2), have a discontinuity at M = +1 or -l. In a stationary
numerical solution this causes a diseontinuity in the gradient at the sonie
point, as seen in Fig. S, curve BIs,,,. Correcting the mass-flux splitting
performs a filmal1 mirada. as aeen from curva VL. The gradient-discontinuity
disappears, and . the numerical diffusion in the subsonic region i9
substantially reduced. This reduction corresponds to the reduction of the
eigenvalues±
I-ll to zero.
20
u
15
10
e
5180" 2700 ~ 3600
Figure 5. Numerical solutions of the periodic, nozzle-type, cosmic flowproblem (y=l) from [7] by flux-split upwind schemes, on a l28-zone grid.Plotted is velocity against coordinate angle. B/sw = steady solution by Beamscheme/Steger and lvarming; VL = steady solution by Van Leer.
11
Conclusions
For the full or isenthalpic Euler equations combinedwith the ideal-gas
law, the flux-vector splitting presented here is, by a great margin, the
simplest means to implement upwind differencing. For a polytropic gas law,
with y > 1, closed formulas have not yét been derived.
The scheme produces steady. shock proEiles with two interior zones. 1t
has been conjectured by Engquist and Oshér [10] that, among implicit versions
of upwind methods, those with a two-zone steady-shock representation may give
faster convergence to a steady solution than those with a one-zone
representation. This has not yet been demonstrated in practice.
A disadvantage in using any flux-vector splitting is that it leads to
numerical diffusion oE a contact discontinuity at resto This diffusion can be
removed; present research i8 aimed at achieving this with minimal
computational effort. Numerical solutions obtained with first- and ~econd-
order schemes including the above split fluxes can be found in Refs. [6], [7]
(one-dimensional) and [8], [9] (two-dimensional).
12
·~·~W~77J,.:Ro¡.~W¡;:'¡&(;'
y
References
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hyperbolic systems of conservation laws", North-Holland Hathematical
Studies!!L (1981), 179.
3. R. H. Sanders snd K. H. Prendergast, Astrophys. J. 188 (1974), 489.
4. J. L. Steger and R. F. Warming, J. ComRutational Phys. 40 (1981), 263.
5. A. Harten, P. D. Lax and B. van Leer, '~pstream differencing and Godunov
type schemes," TCASE Report No. 82-5; to appear in SIAM Review.
6. J. L. Steger, "A preliminary study of relaxation methods for the inviscid
conservative gasdynamics equations using flux-vector splitting,"
Repart No. 80-4, August 1980, Flow Simulations, lne.
7. G. D. van Albada, '8. van Leer and W. W. Roberts, Jr., "A comparative
study of computational methods in cosmic gas dynamics," Astron.
AstroRhys. 108 (1982), 76.
8. G. D. van Alhada and W. W. Roberts, Jr., AR' J. 246 (1981), 740.
9. M. D. Salas, "Recent developments in transonic flow over a circular
cylinder," NASA Technical Memorandum 83282 (April, 1982).
la. B. Engquist and S. Osher, Math. Comp• .1i (1980), 45.
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