NASA/CR-2000-21 0632 leASE Report No. 2000-49 NASA-CR-2000-21 0632

20010029126

A Technique of Treating Negative Weights in WENO Schemes

ling Shl, Changqing Hu, and Chi-Wang Shu Brown University, Providence, Rhode Island

December 2000 1111111111111 1111 1111111111111111111111111111

NF01367

,

I I I

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NASA/CR -2000-210632 ICASE Report No. 2000-49

11111111111 1111 1~~~~I[i[fl\jrli\1 ~11{il~111 1111 III 111111 3 1176 01452 7502

A Technique of Treating Negative Weights in WENO Schemes

ling STu, Changqing Hu, and Chi-Wang Shu Brown University, Providence, Rhode Island

leASE NASA Langley Research Center Hampton, Virginza

Operated by Universities Space Research Association

National Aeronautics and Space Administration

Langley Research Center Hampton, Virginia 23681-2199

Prepared for Langley Research Center under Contract NAS 1-97046

I I I I I Ii I I

December 2000

AVaIlable from the follOWIng

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A TECHNIQUE OF TREATING NEGATIVE WEIGHTS IN WENO SCHEMES·

JING SHIt, CHANGQING HUt, AND CHI-WANG SHU§

Abstract. HIgh order accurate weIghted essentIally non-oscIllatory C\vENO) schemes have recently been

developed for fimte dIfference and fimte volume methods both m structured and m unstructured meshes

A key Idea m 'VENO scheme IS a lmear combmatIOn of lower order fluxes or reconstructIOns to obtam a

lllgher order approxImatIOn The combmatIOn coeffiCIents, also called lmear \velghts, are determmed by

local geometry of the mesh and order of accuracy and may become negatIve 'VENO procedures cannot

be applIed dIrectly to obtam a stable scheme If negatIve lInear weIghts are present PrevIOus strategy for

handlmg tillS dIfficulty IS by eIther regroupmg of stencIls or reducmg the order of accuracy to get nd of the

negatIve lmear weIghts In tIus paper we present a sImple and effectIve techmque for handlIng negatIve lmear

weIghts wIthout a need to get nd of them Test cases are shown to Illustrate the stabIlIty and accuracy of

tIus approach

Key words. weIghted essentIally non-oscIllatory, negatIve weIghts, stablhty, lugh order accuracy, shock

calculatIOn

Subject classification. Apphed and Numencall\lathematIcs

1. Introduction. HIgh order accurate weIghted essentIally non-oscIllatory (\vENO) schemes have re­

cently been developed to solve a hyperbohc conservatIOn law

Ut + V' f(u) = 0 (1 1)

The first 'VENO scheme was constructed m [18] for a thIrd order fimte \olume verSIOn m one space dImensIOn

In [10], thIrd and fifth order fimte dIfference 'VENO schemes m multI space dImensIOns are constructed, wIth

a general framework for the desIgn of the smoothness mdlcators and nonlmear wClghts Later, second, tlurd

and fourth order fimte volume WENO schemes for 2D general tnangulatIOn have been developed m [4] and

[8] Very hIgh order fimte dIfference WENO schemes (for orders between 7 and 13) have been developed m

[1] Central WENO schemes have been developed m [12], [13] and [14]

WENO schemes are desIgned based on the successful ENO schemes m [7, 23, 24] Both ENO and

'VENO use the Idea of adaptIve stencIls m the reconstructIOn procedure based on the local smoothness

of the numencal solutIOn to automatIcally aclueve hIgh order accuracy and non-oscIllatory property near

dlscontmmtles ENO uses Just one (optImal m some sense) out of many candIdate stencIls when domg the

reconstructIOn, wlule 'VENO uses a convex combmatIOn of all the candIdate stencIls, each bemg assIgned

a nonhnear weIght WhICh depends on the local smoothness of the numencal solutIOn based on that stencIl

'VENO Improves upon ENO m robustness, better smoothness of fluxes, better steady state convergence,

better provable convergence propertIes, and more efficIency For a detaIled reVIew of ENO and 'VENO

schemes, we refer to the lecture notes [21, 22]

'Research supported by ARO grants DAAG55-97-1-0318 and DAAD19-00-1-0405, NSF grants DMS-9804985 and ECS-

9906606, NASA Langley grant NAG-1-2070 and Contract NASl-97046 whIle the thIrd author was In resIdence at ICASE,

NASA Langley Research Center, Hampton, VA 23681-2199, and AFOSR grant F49620-99-1-0077 tDlvlslOn of ApplIed MathematICs, Brown UnIversIty, ProvIdence, RI 02912 E-maIl shl@cfm brown edu tDlvlslOn of ApplIed MathematICs, Brown UnIversIty, ProvIdence, RI 02912 Current address Oracle, 30P6, 500 Oracle

Parkway, Redwood Shores, CA 94065 E-maIl phu@us oracle com §DlvlslOn of ApplIed MathematICs, Brown UnIversIty, ProvIdence, RI 02912 E-maIl shu@cfm brown edu

1

WENO schemes have already been wIdely used m apphcatlOns Some of the examples mcIude dynamIcal

response of a stellar atmosphere to pressure perturbatlOns [3], shock vortex mteractlOns and other gas

dynamIcs problems [5], [6], mcompresslble flow problems [26], HamIlton-JacobI equatlOns [9], magneto­

hydrodynamIcs [11], underwater blast-wave focusmg [15], the composIte schemes and shallmv water equatlOns

[16], [17], real gas computatlOns [19], wave propagatlOn usmg Fey's method of transport [20], etc

A key Idea m \VENO schemes IS a hnear combmatlOn of lower order fluxes or reconstructlOns to obtam

a hIgher order approximatlOn The combmatlOn coeffiCIents, also called hnear weIghts, are determUled by

local geometry of the mesh and order of accuracy and may become negatIve \VENO procedures cannot

be apphed dIrectly to obtam a stable scheme If negatIve lmear weIghts are present PrevlOus strategy for

handhng tlns dIfficulty IS by either regroupmg of stenCils (e g III [8]) or reducmg the order of accuracy (e g

m [12]) to get nd of the negatIVe hnear Weights In thIs paper we present a sImple and effectn e techmque

for handhng negative hnear weIghts WIthout a need to get nd of them Test cases WIll be shown to Illustrate

the stablhty and accuracy of thIs approach

We first summanze the general \VENO reconstructlOn procedure, consIstmg of the followmg steps \Ve

assume we have a gIVen cell 6 (whIch could be an mterval mID, a rectangle m a 2D tensor product mesh,

or a tnangle m a 2D unstructured mesh) and a fixed pomt xC wlthm or on one edge of the cell

1 \Ve IdentIfy several stenCIls S), J = 1, ,q, such that 6 belongs to each stenCil \Ve denote b} q

T = U S) the larger stencIl winch contallls all the cells from the q stencIls )=1

2 \Ve have a (relatIvely) lower order reconstructlOn or mterpolatlOn functlOn (usually a polynomial),

denoted by p)(x), associated WIth each of the stenCils S), for J = 1, ,q \Ve also have a (relatively)

hIgher order reconstructlOn or lllterpolatlOn functlOn (agam usually a polynomial), denoted by Q(x),

associated WIth the larger stencIl T 3 \Ve find the combmatlOn coeffiCIents, also called hnear weIghts, denoted by /'1, , /'q, such that

q

Q(xc ) = L /'lPJ(XC

) (1 2) )=1

for all pOSSIble gIven data m the stenCIls These hnear weIghts depend on the mesh geometry, the

pomt xc, and the speCific reconstructlOn or mterpolatlOn reqmrements, but not on the gIven solutlOn

data m the stenCils

4 We compute the smoothness llldicator, denoted by (3), for each stenCIl SJ' wincIl measures how

smooth the functlOn p) (x) IS III the target cell 6 The smaller thIS smoothness mdlcator (3), the

smoother the functlOn p) (x) IS m the target cell In all of the current \VEN 0 schemes we are USlllg

the followmg smoothness mdlcator

(1 3)

for J = 1, ,q, where k IS the degree of the polynomial p)(x) and 161 IS the area of the cell 6 m 2D

This factor IS dIfferent for ID or 3D the purpose of It IS to bnng the smoothness mdlcator mvanant

under spatial scahng

5 \Ve compute the nonlmear weIghts based on the smoothness mdicators

(1 4)

2

where "/J are the lmear weIghts determmed m step 3 above, and c: IS a small number to aVOId the

denommator to become 0 We are usmg c: = 10-6 m all the computatIOns m tIllS paper The final

WENO approXImatIOn or reconstructIOn IS then gIven by

q

R(xG) = 2:= w)p) (xG

) (1 5) ;=1

We remark that all the coefficIents m the above steps whIch depend on the mesh but not on the data of

the numencal solutIOn, should be computed and stored at the begmmng of the code after the generatlOn of

the mesh but before the time evolutlOn starts

\Ve now use a sImple example to Illustrate the steps outlIned above \Ve a')sume we are gIven a umform

mesh I, = (X,-1/2,X,+1/2) and cell averages of a functIOn u(x) m these cells, denoted by u, \Ve would

lIke to find a fifth order \VENO reconstructlOn to the pomt value U(X,+1/2), based on a stencIl of five cells

{II - 2,!,-I,!,,!,+l,!,+2}, WIth the target cell contammg the pomt X'+I/2 chosen as 6. = I, In step 1 above we could have the followmg three stenclls

whIch make up a larger stencIl

In step 2 above we would have three polynomials PJ(x) of degree at most 1\,0, wIth theIr cell averages

agreemg WIth that of the functIOn u m the three cells m each stencIl S) The Illgher order functIOn Q(x) IS a

polynomIal of degree at most four, WIth ItS cell averages agreemg WIth that of the functlOn u m the five cells

m the larger stencIl T The three lower order approxImatIOns to U(X'+I/2), associated 'Hth p)(x), m terms

of the gIven cell averages of u, are gIven by

1 7 11 PI (X,+1/2) = 3"U'-2 - GUI - I + (fu"

1 5 1 P2(X,+l/2) = -GU,-I + GU' + 3"U'+I, (1 G)

1 5 1 P3(X,+l/2) = 3"u, + GU,+1 - GU,+2

Each of them IS a tlurd order approximatlOn to U(X'+l/2) The hIgher order approxImatIOn to U(X'+l/2),

assocIated WIth Q(x), IS gIven by

whIch IS a fifth order approxImatIOn to U(X'+I/2)

In step 3 above we would have

1 "/1 = 10'

3 "/2 = 5'

It can be readIly venfied, usmg (1 G) and (1 7), that

for all possIble gIVen data UJ , J = Z - 2, Z - 1, Z, Z + 1, Z + 2

3

3 "/3 = 10

(1 7)

2r

0 0

-2 2

-4 -4

6 -6

-8 -8

-10 -10

-12 -12

-14 14

"t -18

-20

-22

24 --~ 1

01 -"-~_~~ __ I

o 01 02

16

18

20 ([)

B-e--EH: \ I

22 \ I

-24 __ ~_I_~ __ ---'~~~---'---- __ I

-01 0 01 02 X X

FIG 1 1 ReconstructIOns to U(X.+1/2) Solzd lmes exact functIOn, symbols numerzcal approximations Left fifth order

WENO Right fifth order traditional

In step 4 above we could easIly work out from (1 3) the three smoothness mdicators gIven by

13 ? 1 ?

/31 = 12 (U,-2 - 2U,-1 + u,t + 4" (U,-2 - 4U,-1 + 3u,t ,

13 _ _ _ 2 1 _ _ 2 /32 = 12 (U,-l - 2u, + U,+l) + 4" (U,-l - u,+!) ,

13 ., 1 ., /33 = 12 (u, - 2u,+! + U,+2t + 4" (3u, - 4U,+1 + U1+2t

\Ve notice m particular that the lmear weights 1'1,1'2,1'3 m step 3 above are all posItive In such cases, the

\VENO reconstructIOn procedure outhned above and the scheme based on It work very well In Fig 1 1 we

plot the approximatIOn to u(x) for a dlscontmuous functIOn u(x) = 2x for x :S 0 and u(x) = -20 otherWIse,

by the fifth order \vENO reconstructIOn on the left and by the fifth order traditIOnal reconstructIOn (1 7)

on the nght, WIth a mesh x, = (z - 0 4965)~x With ~x = 002 \Ve can clearly see that \vENO a"Olds the

over and undershoots near the dlscontmUIty

\Ve now look at another simple example where some of the hnear weights m step 3 above ",ould become

negative \Ve have exactly the same settmg as above except now we seek the reconstructIOn not at the cell

boundary but at the cell center x, TIns IS needed by the central schemes With staggered gnds [12] Thus,

step 1 would stay the same as above, step 2 would produce

1 1 23 pdx,) = - 24 U,-2 + 12 U,-l + 24 u"

1 13 1 P2(X,) = - 24 U,-l + 12u, - 24 U,+l, (1 8)

23 1 1 P3(X,) = 24 U, + 12 u,+! - 24 U,+2

Each of them IS a third order reconstructIOn to u(x,) The higher order reconstructIOn to u(xl ), associated

With Q(x), IS gIven by

(1 9)

4

~ I \

0

-2

-4

-6

-8

-10

-12

-14

-16

-18

-20

-22 l

-01 0 X

4

2

0

-4

-6

-10

-12

-14

-16

-18

-20

L~~ __ -' -22 01 02

o

L_~~-"_~_--'--"---_~~L_~ __ ' -01 0 01 02

X

FIG 1 2 Reconstructions to u(x,) Solid lmes exact functIOn, symbols numerical apprOXimatIOns Left fifth order

WEND Right fifth order traditIOnal

whIch IS a fifth order reconstructIOn to u(x,) Step 3 would produce the followmg weIghts

49 12 = 40'

9 13 =--

80

NotIce that two of them are negatIve The smoothness mdicators m step 4 WIll remam the same TIns

tIme, the 'VENO approxImatIOn, shm,n at the left of FIg 1 2, IS less satIsfactory (m fact, even worse than

a tradItIOnal fifth order reconstructIOn show on the nght), because of the negatrve lmear weIghts

\Ve remark that negatIve lInear weIghts do not appear m fimte dIfference \VENO schemes III any "patial

dImensIOns for conservatIon laws for any order of accuracy [10], [1], and they do not appear m one dimenslOIldl

as well as some multl-dlmenslonal fimte volume \VENO schemes for conservatIOn laws Unfortunately, tll('\

do appear III some other cases, such as the central \VENO schemes usmg staggered meshes "e ha\e sePlI

above, Illgh order fimte volume schemes for two dImensIOns descnbed III [8] and m tins paper, and filllte

difference WENO approxImatIOns for second denvatrves

\VhJle on approXImatIOn alone the appearance of negatIve IIllear weights might be annoymg but perhap"

not fatal (Fig 1 2), m sohmg a PDE the result mIght be more senous As an example, m Fig 13 we shm\

the results of usmg a fourth order fimte volume \VENO scheme [8] on a non-umform tnangular mesh shown

at the left, WhICh has negatIve lInear weIghts, for solvmg the two dImensIOnal Burgers equatIOn

(1 10)

m the domam [-2,2] x [-2,2] With an mitral conditIOn uo(x,y) = 03+07sm(}-(x+y)) and penodlc

boundary condItIOns \Ve can see that senous oscIllatIOn appears m the numencal solutIOn once the shock

has developed The oscIllatIOn eventually leads to IllstabilIty and blowmg up of the numencal solutIOn for

thIS example

The mam purpose of thiS paper IS to develop a Simple and effectIve techmque for handlmg negative lInear

WeIghts Without a need to get nd of them Test cases Will be shown to Illustrate the stabIlIty and accuracy

of thiS approach

5

-,

-1

-, -1 0

X

FIG. 1.,3. 2D Burgers' equation. Left: non-uniform triangular' mesh used in the computation. Right: fourth order- WRNO

result at. t = 0.47:3, cjl=O.2, without any special treatment for the negative linear- weights.

2. A splitting technique. We now introduce a splitting technique to treat the negative weights. It

is very simple, involves little additional cost, yet is quite effective. Tlw WENO procedure outlined in the

previous section is only modified in step 5 in the following way:

5' If minb, , ... , rq) 2: 0 proceed as before. Otherwise, we split the linear weights into two parts:

positive and negative. Define

i = 1, ... ,q (2.1)

where we take () = 3 all the numerical tests. We then seale them by

q

a± = Lit; i = 1, ... , q. (2.2) j=J

We now have two split polynomials

q

L rtPj(:l/') (2.3) j=1

which satisfy

-Q-( G) a x. (2.4)

We can then define the nonlinear weights (1.4) for the positive and negative groups r:f separately,

denoted by cv;, based on the same Rmoothness indicator (3j. We will then define the WENO

approximation R±(:rP) separately by (1.5), using w.f, and form the final WENO approximation by

- R-( 0) a. x .

We remark that the key idea of this decomposition is to make sure that every stencil has a significant

representation in both the positive and the negative weight groups. Within each group, the WENO idea of

6

-2 I -4 I

I -6 I -8 I

-2

-10

-12 I

-14 I -16 I

I -18 I -20

-22 , ., , , ·-1' ~"~ -0.1 0 0.1 0.2

X

FIC.2.1. WENO approximations with the splitting treatment for negative linear weights_ Left: appr'oximation to v.(x;).

Right: Burgers equation, solution at t 5/rr2 , cfi=O.2_

redistributing the weights subject to a fixed sum according to the smoothness of the approximation is still

followed as before_

For the simple example of fifth order WENO reconstruction to U(:r:i), the split linear weights corrnspond­

ing to (2.1) are, before the scaling,

9

40' _+ _ 49 __ 49 12 - 20' 12 = 40'

_+ _ 9 _ I:) - 80' 1;3

9 40

We notice that, as the most expensive part of the WENO procedure, namely the computation of the

smoothness indicators (1.3), has not changed, the extra cost of this positive/negative weight splitting is very

smalL

However this simple and inexpensive change makes a big difference to the computations_ In Fig. 2.1 we

show the result of the two previous unsatisfactory cases, the fifth order WENO reconstruction to U(:r:i) in

Fig. 1.2 left, and the approximation to the Burgers equation in Fig. 1.3 right, now llsing WENO schemes

with this splitting treatment. We can see dearly that the results are now as good as one would get from

WENO schemes having only positive linear weights.

It is easy to prove that the splitting maintains the accuracy of the approximation in smooth regions_

We will demonstrate this fact in the following sections. We will also demonstrate the effectiveness of this

simple splitting technique through a few selected numerical examples in the next sections. The main WENO

schemes we will consider are fifth order finite volume WENO schemes on Cartesian meshes, and the third

and fourth order finit.e volume \VENO schemes on triangular meshes. In both cases negative linear weights

appear regularly.

The calculations are performed on SUN Ultra workstations and also on the IBM SP parallel eomputer

at TCASCV of Brown University_ The parallel efficiency of the method is excellent (more than 90%)_

3. 2D finite volume WENO schemes on Cartesian meshes.

3.1. The schemes. 'We describe two different ways to construct fifth order finite volume WENO

schemes on Cartesian meshes_ Comparing with finite diffc-m~nce WENO methods [10], finite volume mc"th-

7

ods have the advantage of an applIcabilIty of usmg arbitrary non-umform meshes, at the pnce of mcreased

computatIOnal cost [2]

We define the cell

(3 1)

for z = 1, , Tn, J = 1, , n, \Vhere I,,} needs not be umform or smooth yarymg

The three-pomt Gaussian quadrature rule IS used at each cell edge when evaluatmg the numerical flux

m order to mamtam fifth order accuracy Let (xC, yC) denote one of the Gaussian quadrature pomts at the

cell boundary of I,,) given by r == {x = X,_!,y)_! :S y:S YJ+~} There are two ways to perform a WENO

reconstructIOn at the pomt (xc, yC)

Genuzne 2D The first \VENO reconstructIOn IS genu me 2D fillite volume We can see that there are

totally nme stencils S8,1 (s,t = -1,0,1) Each stencil S8,1 contams 3 x 3 cells centered around I'+8,)+t

On each stencil we can construct a Q2 polynomial (tensor product of second order polynomials III x and y)

satlsfYlllg the cell average conditIOn (I e ItS cell average III each cell IIlslde the stenCIl equal,> to the given 1

yalue) Let T = U S8,t, wIndl contams 5 x 5 cells centered around I,,) On T we can construct a Q4 8,1=-1

polynomial satIsfymg the cell average conditIOn The \VENO reconstructIOn IS then performed accordmg to

the steps outlllled III sectIOns 1 and 2

\Ve \',ould lIke to make the followmg remarks

1 By uSlllg a Lagrange mterpolatlOn basIs, we can eaSily find the ullique hnear "eights

2 Even for a umform mesh, a negative hnear weight appears for the middle Gaussian pomt (xC, yC) = (x,_ 1.., y)) Such appearance of negative lmear weights has also been observed III the central \VENO

2

schemes [12], see the example III sectIOns 1 and 2 before

3 By Taylor expanSIOns, we can prove that the smoothness IIldlcators Yield a umform fifth order

accuracy III smooth regIOns See [10] for the method of proof

Dzmenswn by Dzmenswn The second \VENO reconstructIOn explOits the tensor product nature of the

mterpolatlOn we use This 'VENO procedure IS performed on a dimensIOn by dimensIOn fasluon The \VENO

schemes applIed m [5], [6] belong to tlus class ConSider the pomt (xC,yC) as above First \\e perform a

one dimensIOnal \VENO reconstructIOn m the y directIOn, m order to get the one dimensIOnal cell averages

(m the x directIOn) w(., yC) Then we perform another one dimensIOnal 'VENO reconstructIOn to w III the

x directIOn, to obtam the final reconstructed POlllt value at (xC, yC)

We "ould lIke to make the followmg remarks

1 For a scalar equatIOn, the underlymg lInear reconstructIOns of the above t"o versIOns are eqUivalent

For nonlInear \VENO reconstructIOns they are not eqUivalent Both of them are fifth order accurate

but the actual errors on the same mesh may be different, see Table 3 1 below

2 For systems of conservatIOn laws such as the Euler equatIOns of gas dynamics, both versIOns of the

'VENO reconstructIOn should be performed III local characteristic fields

3 The dimensIOn by dimensIOn versIOn of the \VENO reconstructIOn IS less expensive and reqUires

smaller memory than the genullle two dimensIOnal versIOn The CPU time savmg IS about a factor

of 4 for the Euler equatIOns m our ImplementatIOn The computed results are mostly Similar from

both versIOns

In the followlllg, we "Ill give numencal examples computed by the above WENO schemes Sphttlllg

techmque has been used III all the computatIOns when negatl\e lmear weIghts appear \Ve will show the

8

TABLE 3 1

2D vortex evolutIOn

II II dlm-bv-dlm II genu me FV . N .6.x Loo error order LOO error order

20 G 71E-l 438E-2 52GE-2

40 377E-l 310E-3 459 5 GGE-3 38G

80 2 OlE-l 120E-4 515 39GE-4 422

lGO 1 OOE-l 439E-G 47G 79GE-G 5 G2

320 500E-2 188E-7 453 290E-7 477

results for both smooth and dlscontmuous problems

3.2. 2D vortex evolution. Flrst, '\e check the accuracy of the \VENO schemes constructed above

The two dllnenslOnal vortex evolutlOn problem [21], [8]lS used as a test problem

\Ve solve the Euler equatlOns for compresslble flow m 2D

Ut + J(U)x + g(U)y = 0, (32)

where

U = (p,pu,pv,Ef,

J(U) = (pu,pu 2 +p,puv,u(E+p))T,

g(U) = (pv,puv,pv2 + p,v(E + p))T

Here p IS the denslty, (u, v) IS the veloclty, E IS the total energy, p IS the pressure, related to the total energy

by E = ~ + ~p(u2 + v 2) wlth / = 1 4

The setup of the problem IS the mean flow IS p = 1, p = 1, (u,v) = (1,1) and the computatlOnal

domam IS [0,10] x [0,10] We add, to the mean flow, an Isentroplc vortex (perturbatlOns m (u, v) and the

temperature T = P., no perturbatlOn m the entropy 5 = *) p p

E 2 (ou,ov) = _e05(1-r )(-y,x),

27r 'T = _ h - I)E2 l_r2 u 8 2 e ,

/7r -

where (Y, y) = (x - 5, Y - 5), r2 = £2 + fP, and the vortex strength E = 5

05 = 0

\Ve use non-umform meshes wInch are obtamed by an mdependent random sInftmg of each pomt from a

umform mesh m each dlrectlOn Wlthm 30% of the mesh Slzes The solutlOn IS computed up to t = 2 Table 3 1

shows the L oo errors of p \Ve can see that both the genume t\\O dlmenslOnal fimte volume \VENO scheme

and the dlmenslOn by dlmenSlOn fimte volume \VENO scheme can aclneve the deslred order of accuracy

wInle the genume two dlmenslOnal scheme glves smaller errors for the same mesh

3.3. Oblique shock tubes. The purpose for thls test IS to see the capablhty of the rectangular \VENO

schemes m resolvmg waves that are obhque to the computatlOnal meshes For detmls of the problem, we refer

to [10] The 2D Sod's shock tube problem IS solved where the Imttal Jump makes an angle 0 agamst the x

axlS We take our computatlOnal domam to be [0, G] x [0,1] and the Imttal Jump startmg at (x, y) = (2 25,0)

9

05 c

05

o~~

o

08

06

04

02

0~------~~----~~~2~~~~3-~~~--~4

X

gt..nUltlt. I V dun hy-t.illli

5 6

FIG 3 1 Dbllque Sod's problem Density p Top contour, genume two dlmenswnal WEND, middle contour, dzmenswn

by dlmenswn WEND, bottom cut at the bottom of the computatwnal domam, the solzd lme IS the exact solutwn, the trzangles

are the genume two dlmenswnal WEND results, and the czrcles are the dlmenswn by dlmenswn WEND results

and makmg a () = % angle wIth the x aXIS The solutIOn IS computed up to t = 1 2 on a 9G x 1G umform

mesh In FIg 31 we plot the denSIty contours computed by the above two WENO schemes and the denSIty

cut at the bottom of the computatIOnal domam We can see that both schemes perform equally well m

resolvmg the waves The genu me two dImensIOnal scheme gIves a shghtly better resolutIOn m the contact

dlscontmUIty and the rarefactIOn wave

3.4. A Mach 3 wind tunnel with a step. ThIS model problem IS ongmally from [25] The setup

of the problem IS The wmd tunnel IS 1 length umt wIde and 3 length umts long The step IS 0 2 length

10

, 'I ---------,,-"'''~~''''''--,,-,~.-----~~~==~-, I

°o~--"

FIG 3 2 Forward step problem, flx = fly = to, -to, I!O' 3~O from top to bottom 30 contours from 0 12 to 641, dzmcTlslOTi

by dzmenslOn WEND

umts lugh and IS located 06 length umts from the left-hand end of the tunnel The problem IS InItIalIzed

by a rIght-gOIng Mach 3 flow ReflectIve boundary condItIons are apphed along the wall of the tunnel and

Inflow /outflow boundary condItIOns are applIed at the entrance/exIt The corner of the step IS a sIngular

POInt and we treat It the same ,,,ay as In [25], wInch IS based on the assumptIOn of a nearly steady flow m

the regIOn near the corner \Ve show the denSIty contours at tIme t = 4 In FIg 3 2 Only the results from

the dImensIOn by dImenSIOn \VENO scheme are shown Umform meshes of 6.x = 6.y = 410' lo' I~O' 3~O are

used

3.5. Double Mach reflection. ThIS problem IS also OrIgInally from [25] The computatIOnal domaIn

for thIS problem IS chosen to be [0,4) x [0, 1) The reflectIng wall lIes at the bottom, startIng from x = i ImtIally a rIght-mOVIng Mach 10 shock IS pOSItIOned at x = t, y = 0 and makes a 600 angle WIth the x aXIS

11

075 -

x

FIG 3 3 Double Mach reflectIOn, box = boy = 2!O (top and lower left) and 4~O (middle and lower Tight) Genume two

dimensIOnal WEND Blow-up regIOns at the bottom for details

For the bottom boundary, the exact post-shock conditIOn IS Imposed for the part from x = 0 to x = t and

a reflective boundary conditIOn IS used for the rest At the top boundary, the flow values are set to descnbe

the exact motion of a :t\Iach 10 shock \Ve compute the solutIOn up to t = 0 2 Fig 3 3 and Fig 3 4 show

the equally spaced 30 denSity contours from 1 5 to 22 7 computed by the genu me two dimensIOnal and the

dimenSIOn by dimenSIOn WENO schemes \Ve use ullIform meshes \Hth ~x = ~y = 2!O' 4!O We can see

that the results from both schemes are comparable

4. 2D finite volume WENO schemes on triangular meshes. Both third and fourth order £illite

volume WENO schemes on tnangular meshes have been constructed m [8] The optIOnallmear weights m

such schemes are not umque These are then chosen to aVOid negative weights whenever pOSSible, and If that

fails, a groupmg (of stencIls) techmque IS used m [8], which works fairly well m the third order case With

qUite general tnangulatIOn but can Yield posItive weights for the fourth order case only With faIrly umform

tnangulatIOn In thiS sectIOn, we do not seek posItive hnear wClghts as m [8], but rather use the sphttmg

techmque to treat the negative hnear weights when they appear For scalar equatIOn, the scheme IS stable

12

075 ...

05,-----------------------~

x

" x

" X

FIG 3 4 Double Mach ref/ectlon, Llx = Lly = 2!O (top and lower left) and 4~O (middle and lower rzght) DimensIOn by

dimensIOn WEND Blow-up regions at the bottom for details

m all runs For systems of conservatIOn laws, there are still occasIOnal cases of overshoot and mstablhty, the

reason seems to be related to characteristic decompositIOns and IS still bemg mvestlgated

4.1. Accuracy check for a smooth problem. We solve the 2D Burgers equatIOn (110) wIth the

same mltlal and boundary conditIOns as before usmg the fourth order finIte volume WENO scheme [8) The

solutIOn IS computed up to t = ~ when no shock has appeared The meshes used are 1) unIform meshes

with eqUilateral tnangulatlOn and 2) random tnangulatIOn For the UnIform meshes we do not seek posItive

weights as was done m [8), rather we use the sphttmg technIque to treat the negative lmear weights when

they appear Table 4 1 mdlcates that close to fourth order accuracy can be achieved

4.2. Discontinuous problem 1: Scalar equation in 2D. Havmg shown the stable results \\lth the

sphttmg treatment of negative hnear weIghts for a fourth order finIte volume 'VENO scheme for the Burgers

equatIOn m sectIOn 2, we now test the fourth order \VENO scheme on the Buckley-Leverett problem whose

13

TABLE 4 1

2D Burgers equation accuracy check

III umform mesh III nonumform mesh III ~x LOO error order ~x LOO error order

257E-1 622E-4 267E-1 211E-3

1 29E-1 46IE-5 375 1 26E-1 235E-4 292

643E-2 218E-6 440 632E-2 290E-5 303

32IE-2 138E-7 398 334E-2 26IE-6 378

16IE-2 693E-9 432 166E-2 271E-7 324

808E-3 670E-10 340 744E-3 157E-8 355

FIG 4 1 2D Buckley-Leverett equation the mesh

flux IS non-convex

u2

f(u) = u2 +025(1-u)2' g(u) = 0

WIth the ImtIal data u = 1 when - J ::; x ::; 0 and u = 0 elsewhere The solutlOn IS computed up to

t = 0 4 The exact solutlOn IS a shock-rarefactlOn-contact dlscontmmty mIxture The mesh we use here l~ a

non-umform tnangulatlOn, shown 1Il FIg 4 1 FIg 4 2 shows that the waves have been resolved very well

4.3. Discontinuous problem 2: System of equations in 2D. We consIder the 2D Euler equatlOns

m the domalll [-1,1] x [0,02] The Sod and Lax shock tube mltIal data IS set m the x dlrectlOn and penodlc

boundary condltlOn IS apphed 1Il the y dlrectlOn \Ve use the fourth order fimte volume scheme Oil tnangular

meshes to solve the above problem The mesh we use here IS umform But we do not seek posItive weIghts as

was done m [8], rather we use the sphttmg techmque m sectlOn 2 to treat the negative Imear \\ eIghts when

they appear In fact we set dehberately certam hnear weIghts to be negatIve to test the sphttmg tcchmque

FIg 4 3 shows the numencal results of the Sod and Lax problems

It seems that there are still oSClllatlOns and lIlstablhty for some non-umform tnangular meshes for the

fourth order \VENO schemes apphed to Euler equatlOns As the method works well for the same meshes

wIth a scalar equatlOn, the problem mIght be from the charactenstlc decomposltlOns ThIS IS stIlI under

mvestlgatlOn

5. Concluding remarks. We have devIsed and tested a SImple sphttmg techmque to treat the negatIve

hnear weIghts 1Il WENO schemes ThIS techmque mvolves very httle addltlOnal CPU time and gIves good

results m most numencal tests The only case where It stIlI YIelds osclllatlOns and mstablhty IS when a fourth

order fimte volume \VENO method IS used on some non-umform tnangular meshes for Euler equatlOns, the

reason of whIch, presumably related to charactenstIc decomposltlOns, IS stIll under mvestlgatlOIl

14

08

06

::::l

04

02

01

, '---OS 0 05

X

FIG 42 2D Buckley-Leverett equatIOn at t=o 4, with splzttmg Left the solution surface, Right the cut at y = 0 1 (solid

lme exact solutIOn, symbols numerical solutIOn)

14

12 08

06 c- o.

08

04 06

"'--02 04

, -1 -05

-.

o X

c

0

-"'

,

r;o

0

p

05

FIG 43 Density plot, Left Sad problem, Right Lax problem, with spllttmg Roughly 100 pomts m the x direction

REFERENCES

[1] D BALSARA AND C -\V SHU, Monotomczty preservzng wezghted essentzally non-osczllatory schemes

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[2] J CASPER, C - \V SHU AND H L ATKINS, Comparzson oj two Jormulatzons Jor hzgh-order accurate

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[3] L DEL ZANNA, 1\1 VELLI AND P LONDRILLO, Dynamzcal response oj a stellar atmosphere to pressure

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[4] 0 FRIEDRICHS, Wezghted essentzally non-osczllatory schemes Jor the znterpolatzon oj mean values on

unstructured grzds, Journal of ComputatlOnal PhysIcs, 144 (1998), pp 194-212

[5] F GRASSO AND S PIROZZOLI, Shock-wave-vortex znteractzons Shock and vortex deJormatzons, and

sound productzon, Theor Comp FIUld Dyn , 13 (2000), pp 421-456

[6] F GRASSO AND S PIROZZOLI, Shock wave-thermal mhomogenezty znteractzons Analyszs and numerzcal

szmulatzons oj sound generatzon, Phys FIUlds, 12 (2000), pp 205-219

15

[7] A HARTEN, B ENGQUIST, S OSHER AND S CHAKRAVARTHY, UnzJormly hzgh order essentzally non­

osczllatory schemes, III, Journal of ComputatIOnal PhYSICS, 71 (1987), pp 231-303

[8] C Hu AND C -W SHU, Wezghted Essentzally Non-Osczllatory Schemes on Trzangular Meshes, Journal

of ComputatIOnal PhYSICS, 150 (1999), pp 97-127

[9] G JIANG AND D -P PENG, Wezghted ENO schemes Jor Hamzlton-Jacob! equatlOns, SIAr..I Journal on

SCIentIfic ComputIllg, v21 (2000), pp 2126-2143

[10] G JIANG AND C - \V SlIU, Efficzent zmpiementatzon oj wezghted ENO schemes, Journal of Computa­

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[11] G JIANG AND C -C \Vu, A hzgh order WENO jinzte dzfference scheme Jor the equatzons oj zdeal

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[14] D LEVY, G PUPPO AND G Russo, A thzrd order central WENO scheme Jor 2D conservatzon laws,

Appl Numer r..Iath 33 (2000), pp 415-421

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16

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A techlllque of treatmg negatIve weIghts m 'VENO schemes C NASl-97046 WU 503-90-52-01

6 AUTHOR(S)

Jmg SIll, Changqmg Hu, and CIll-'Vang Shu

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13 ABSTRACT (Maximum 200 words)

HIgh order accurate weIghted essentIally non-OSCIllatory ('VENO) schemes have recently been developed for filllte dIfference and filllte volume methods both m structured and m unstructured meshes A key Idea m WENO scheme IS a llllear combmatlOn of lower order fluxes or reconstructIOns to obtam a hIgher order approxlInatlOn The combmatlOn coeffiCIents, also called lmear weIghts, are deternuned by local geometry of the mesh and order of accuracy and may become negatIve 'VENO procedures cannot be applIed dIrectly to obtam a stable scheme If negatIve llllear weIghts are present PrevIous strategy for handllllg thIs dIfficulty IS by eIther regrouplllg of stenCIls or reducmg the order of accuracy to get rId of the negatIve llllear weIghts In tIllS paper \\e present a SImple and effectIve techlllque for handlmg negatIve lmear \\elghts WIthout a need to get rId of them Test cases are shown to Illustrate the stabIlIty and accuracy of tIllS approach

14 SUBJECT TERMS 15 NUMBER OF PAGES weIghted essentIally non-OSCIllatory, negatIve weIghts, stabIlIty, lllgh order accuracy, 22 shock calculatIOn 16 PRICE CODE

A03 17 SECURITY CLASSIFICATION 18 SECURITY CLASSIFICATION 19 SECURITY CLASSIFICATION 20 LIMITATION

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NSN 7540-01-280-5500 Standard Form 298.(Rev 2-8.9) Prescrtbed by ANSI Std Z39 18 298 102

mcompresszble Navzer-Stokes equatzons, Journal of ComputatIOnal PhYSICS, 146 (1998), pp 464-487

17

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