AD-0199 392 EFFICIENT IMPLEMENTATION OF ESSENTIALLY NON-OSCILLATORY 1/1

SHOCK CAPTURING S..(U) INSTITUTE FOR COMPUTERAPPLICATIONS IN SCIENCE AND ENGINEERIN. C SNU NAY 6?

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NASA Conrco Report 1783041F

04 ICASE REPORT NO. 87-330r)

00 ICASEEFFICIENT IMPLEMENTATION OF ESSENTIALLY

NON-OSCILLATORY SHOCK CAPTURING SCHEMES

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87 DT159

Efficient Implementation of Essentially 4

Non-Oscillatory Shock Capturing Schemes

Chi-Wang ShuInstitute for Mathematics and Its Applications Accession For

University of Minnesota NTSGAl

Minneapolis, Minnesota 554155 FDTIC TAB CUnannounced 0

and Justifioatio

Stanley Osher'Mathematics Department Ditiuin 0

U"niversity of California AvilbliyvoeLos Angeles, California 90024 Avi ado

Dist Special0

Abstract W I

In the computation of discontinuous solutions of hyperbolic conservation laws, '

TNT (tot al-variat ion-dimin ish Ing), TVI3 (total- variation- bounded) and the recently

developed ENO (essentially non-oscillatory) schemes have proven to be vcry useful.

In this paper two improvements are discussed: a simple TVD Runge-lKutta type

time discretization, and an ENO construction procedure based on fluxes rather0

than on cell averages. These improvements simplify the schemes considerably--p%

especially for multi-dimenisional problemrs or problems with forcing terms. IPrelind~-

nary numerical results are also given.

1'Research for the secondl author was supported by the National Aeronaut ics aid'Space Administration undtr NASA Contract No. NAS]-18107 whiile het wis IIIresidence at the Inst it ute for Computer Applications in Science and llhginerilng(ICASE), NASA Langley Research Center, Hampton. VA 23665. Ak.o. hit W"I"suipported by NSF Grant'No. 1)MS85-03294, AR() Grant No, l)AV.29-85-lK-0I90.DARPA Grant in the ACMIf" Program, ONR Grant Nf(X)0Il-86;-K-U69L . and NASALangley Grant N AG 1-270)

i'os P" 019t da a"wu INdthib~USbNMOUIimW

%

4 . -. f .40

I. Introduction. In this paper we are interested in solving the system of

hyperbolic conservation laws •

d

(1.1a) ut + fi(u)i =0 (or = g(u,z,t), a forcing term)

(l.b) u(X,O) = Uo(Z)

Here u= (ul,.. , um.) ", z = (X , z2 ,... ) zd), and any real combination of the Jaco-

bian matrices L has m real eigenvalues and a complete set of eigenvectors.= 1 0

On a computational grid zi = j Ax, t. = nAt, we use u! to denote the'.P 4.4

computed approximation to the exact solution u(x,, t.) of (1.1).

We also use the abstract form%%

(1.2) Ut = (U)

in place of (I.la). Here C is a spatial operator.

As is well known, the solution to (1.1) may develop discontinuities (shocks, con-

tact discontinuities, etc.) even if the initial condition uo(z) in (1.lb) is a smooth

function. Traditional finite difference methods, even if linearly stable, often give ".-

poor results in the presence of shocks and other discontinuities. Recently there '

has been a lot of activity geared towards constructing efficient finite difference ,

approximations to (1.1). These include TVD (total-variation-diminishing), TVB

(total-variation-bounded) and ENO (essentially non-oscillatory) methods. See, e.g. •

[21, [31, (41, [51, (6], [91, [101, [12], [131, (141, and the references listed therein. Many

of the ideas can be traced back to Van Leer's work in [151, [16).

Usually, rigoious analysis (e.g. total-variation stability, convergence) is only ,

done for the scalar, one-dimensional nonlinear case (i.e. d m = 1 in (1.1)). Some

'.--..'2,.

VO .% -r.

* N4..~~~~~~~~~ ..417N!C1 .~ \ ~ '. '... ., .

-2-

S

partial theory (e.g. convergence for first order monotone schemes, and maximum

norm stability for higher order TVD schemes) exists for scalar multi-dimensional

problems (d > 1 in (1.1)), but a full convergence theory for multidimensional non-

linear systems appears to be extremely difficult. However, numerical experiments

for multi-dimensional problems and/or for systems of equations, using direct gener-

alizations of TVD, TVB and ENO schemes give very good results. Again, see, e.g.,

[21, 14], [61, [101, [I]l. We now shall confine our discussion at first to this one space

dimension, scalar case. Systems and multi-dimensional problems are discussed at . %

the end of Section 3.

We shall always use conservative schemes of the form

a~t(1.3) 1+1= 1-- (fj+, - fj-_), A=--"

2S

with a consistent numerical flux

-e %,

in order to guarantee that any convergent bounded a.e. subsequence has as its limit

a weak solution of (1.1), (Lax-Wendroff Theorem [81), i.e. we construct so-called

"shock capturing methods".

The total variation of a discrete scalar solution is usually defined by

(1.5) TV(u) = I u-

We say the scheme is TVD if -. . ,

% ..- ..(1.6)

-3-

- " ... 4

nAt < T._

S

A nice theoretical advantage of all TVD or TVB schemes is that they have con-

vergent subsequences as Ax -+ 0, and, if a further "entropy condition" is satisfied, 0

then they are convergent. See, e.g. [3].

The formal "order of accuracy" in this paper is in the sense of local truncation 0

errors, i.e. if local truncation error is O(Ax " + I) in smooth regions, we say the

scheme is (formally) r-th order accurate. See, e.g. [2]. k "z,.

There are many TVD schemes constructed in the literature (e.g. [2], [3], [9],1".* ,.

[10], [141). In [10], TVD schemes of very high spatial order (up to 15th order)

were constructed. These schemes can be used for steady state calculations (e.g.

implemented with the TVD Runge-Kutta type time discretizations with large CFL.. ., .

numbers in f131) or for time dependent problems, equipped with a multi-level TVD

high order time discretization in [13] or with a Runge-Kutta type TVD high order 0

time discretization in Section 2 of this paper. These are perhaps the highest order

TVD schemes existing at present. However the definition of total variation (1.5)

implies that these methods must degenerate to first order accuracy at extremacy. A -

TVB modification of such schemes which recovers global high order accuracy even. . '.at critical points is obtained in [12].

, .The above mentioned TVD and TVB schemes use a fixed, wide stencil (for the %

15th order scheme, the stencil is 17 points wide), thus restricting the- advantage of

a9 %%/. *'a a - - ii l'. i l" **4cc'- . ' i -. -. ' -'

-4-

going to higher order through smearing of discontinuities and resulting degradation

of the accuracy. Numerically we observed that third order schemes work quite well .

[12], but we lost accuracy in a fairly large region near discontinuities by using a fifth

order method. Recently Harten, Osher, Engquist and Chakravarthy constructed ' =

ENO schemes which are of globally high order accuracy in smooth regions and .

which use adaptive stencils, thus obtaining information from regions of smoothness

if discontinuities are present. These methods achieve high order accuracy right up

to discontinuities. Analysis and numerical experiments are found in [5], [6], [4]. -.. J.

At present, a convergence theory (e.g. TV boundedness) for ENO schemes is still

unavailable. S

There are two natural directions in which to simplify the ENO or TVD, TVB ......

schemes, especially for multi-dimensional problems or problems with forcing terms:

(1) Time discretization. Usually, semi-discrete (method of lines) versions of -

ENO or TVD, TVB schemes are much simpler than the fully discrete ones. There

are then mainly two ways to discretize in time. One is of Lax-Wendroff type, i.e., : y.

by using ut = ..... 0f- = u At(ut)! +fft2 . .tu' +

and then by discretizing the spatial derivatives. Many second order TVD schemes

(e.g. Harten's in [2]), and the ENO schemes in [5], (6], [4], used this type of

time discretization. The main disadvantages to the procedure is that it is com-

plicated to program, especially for multi-dimensional problems with forcing terms.

One can see this by writing out a third order approximation to the equation

Ut + fl(u) , + fM(u)S = g(u, X1,x 2 ,t). Moreover it is not easy to prove that

this results in a TVD or TVB method, even if the original method-of-lines ODE

and its Euler forward version are both TVD or TVB. The numerical results have

proven satisfactory, but, speaking theoretically, only second order in time TVD or .

-- , - .,,1 •~

-5-

TVB schemes exist of this type. Another way to discretize in time is to use a multi-

level or Runge-Kutta type ODE solver. This is much simpler to program than the

Lax-Wendroff type of discretization for multi-dimensional problems or for problems

with forcing terms, so it is widely used for numerically implementing a method of

lines approximation. However, usually only linear stability analysis is available in

the literature, which is certainly not enough for our purpose since linear stabil-

ity does not imply convergence if shocks or other discontinuities are present. This

is particularly true for ENO schemes which use moving stencils. Linear stability

analysis is based on the fact that the stencil is fixed and the error accumulates in

a predictable pattern, hence it does not apply to ENO schemes at all. For these

reasons we consider TVD time discretizations . In [13], a class of multi-level TVD

time discretizations were constructed and analysed, (numerical results can be found

in [121). However, for easy starting and for storage considerations, one step Runge-

Kutta type schemes are preferable to multi-level methods. In Section II of this ""..* ¢

paper we present a class of high order TVD Runge-Kutta type time discretizations.J.

(2). Avoiding the using of cell-averages. The ENO schemes constructed in [5],

[61, [41 are for cell-averages but involve point values as well. Hence a reconstruction

procedure is needed to recover point values from cell averages to the correct order,

which can be rather complicated, especially in multi-dimensional problems. It is

desirable to use the moving-stencil idea directly on fluxes to get ENO schemes

without using cell-averages. In Section III of this paper a class of such ENO schemes

is constructed.

Some encouraging numerical results obtained by using schemes constructed in

this paper are included in Section IV.

S ."

- 6-.1 P

We conclude these introductory remarks by noting that R. Sanders [17] has

recently devised third order accurate TVD methods which degenerate to second K2 I!

order at extrema. He defines the variation of the numerical solution as the variation

of an appropriately chosen piecewise parabolic interpolant. The numerical results

are very good. However this technique has no method of lines analogue, so we omit

it from our present discussion.

II. High Order Runge-Kutta Type TVD Time Discretizations. Define S

(2.1) W = T(u) = (I + tL)(u)

where T and L are nonlinear discrete operators, L is a r-th order discrete approxi- •

mate to the spatial operator C in (1.2):

(2.2) L(u) = £(u) + O(Axz) %-%

if u is smooth.

Our goal is to get a fully r-th order approximation to the differential equation

(1.2) of the form

(2.3) u = ,S(u")

(The operator S depends on T). This means that if u(x, t) is an exact smooth V0. k % UN

solution of (1.2), then.

(2.4) u(zX, t + ') - S(u') • = O(Axz + ).

We also want the scheme to be TVD:

(2.5) TV(S(u)) ! TV(T(u))

under suitable restrictions on At (or, equivalently, on the CFL number A).VS

-7- 0 .

We call a time discretization (2.3) r-th order TVD if it satisfies (2.4) and (2.5).

If the spatial operator T in (2.1) is TVD or TVB: •

(2.6a) IrV(T(u)) < TV(u) %%%

or

(2.6b) TV(T(u)) < TV(u) + M-At

for 0 < M uniformly bounded as At - 0, then the fully discrete high order scheme

(2.3) is TVD or TVB, owing to (2.5).

In [13], a class of multi-level type high order TVD time discretizations was

constructed. Numerical experiments in [121 were very promising. But there are two

disadvantages of multi-level type methods: (i) for an m-th level method the first m -

1 levels have to be calculated by other methods to the same order of accuracy (e.g. %

by using Taylor series expansions); (ii) we have to store all m level datas, creating a%- .. j. .

rather large storage requirement, stretching up to and beyond the limits of present

day computers for physical problems arising e.g. in computational aeronautics. At

present, Runge-Kutta type methods are more often used in discretizing the method- -

of-lines than are multi-level methods. Since the former consists of one-level methods,

they are self-starting and reduce storage requirements significantly. In the following .we will analyze the nonlinear stability (TVD) of a class of such methods.

Assume (2.1) is TVD (or TVB) under a suitable CFL restriction

(2.7) A < Ao e_

We may also need an approximation to -L to the spatial operator -, which

we take to satisfy

(2.8) i = T(u) = (I - AtL)(u)

(2.9) L(u) = C(u) + O(Ax') ,.. "

"A 0,

5,- - - - -'p ': '-5* - -.. ; ... ~y 5.5 *..5,.%,. .%5.*"

0

-8-

where (2.8) is TVD (or TVB under the same CL rsrcto 2.)

As an example, a very simple first order (r = 1) TVD approximation to

Ut = U,= 'C(U)

is obtained via simple upwind differencing:

L Uu) -tx+ AX) - u(X)

This scheme (2.1) is TVD if A~ < A0 1

The approximation L(u) is defined as6

U(X)~. -- U( -A

L(u)-ux)- xA)AX

and (2.8) is TVD again for A- A <A 0 <1.

This procedure (2.8), (2.9) easily generalizes to any conventional TVD, TVB,

or ENO-approximation (2.1) satisfying (2.2).

The general explicit Runge-Kutta method (we use explicit methods to avoid

4 ~solving nonlinear equations) for (2. 1) is -..

(2.l10a) UM =UO + At cik L(Uk) 1, 2,.. m

(2. 10b) U(o) -n U(") -n

If the operator L also depends explicitly on t, as is the case when the forcing

term g in (1.1a) depends explicitly on t, or when we have time-dependent boundary

conditions, the general explicit Runge-Kutta method takes a more complicated form -

(2. 11a) UM' U(0 ) + 'At E cjkL(uHk, t(0) + dhAt)k=0

where

P.1

k-i * -,. ,€'(2.11b) dk =ZCk

f=0

For details, see any numerical ODE text, e.g [11, [71 (any such method is usu-

ally written in a slightly different form, using kj, k2,.., but that form is clearly

equivalent to (2.10) or (2.11)).

We shall restrict our attention to (2.10); the generalization to (2.11) is clearly

straightforward, using (2.1lb).

In order to get conditions for TVD, we rewrite (2.10) as follows: For ak >.i-1

0, Jaik = 1, we havek=O ii ~"'~

u (i) = aiku 1° + At ciL(u(.))

k=0 k=0

i-1 k-i0) + M Qk )- At M c, L(u( 1 ))+

k=1 t=O

+ At E cjkL(u(k))

k=O

E + (cika- u rtkcki)AtL(U~k)k= --I '.. . ,'-

So if we let jk = cik - ctkcr, then (2.10) may be written in the followingt=k+i I

equivalent form

(2.12) u O [a u(k) + 3jkAtL(u(k))-k=O * .- "e

, .5- , *, .5

It is well-known that we can get (m + 1)-th order accurate methods in the form

(2.10) or (2.11) for m < 3 m-th order methods for m = 4,5,6: or (rn - 1)-th order

methods for m 7, 8 (see, e.g. [1]).

., . .- .. . ,. . , .. . .- . . ... - , - , . :.- .. . . -. . -,. .-',,.. ~ * ,' ' , .. ,,'. .. '. ,- ".- ",.', ' ", " ., ,, . .,.. ' 5,$

0-10- ,

For the classical 4-th order Runge-Kutta methods, the constants cik in (2.10) %

are all non-negative. However, /ik in (2.12) may well be negative. In order to obtain or

TVD we apply a trick used in [131, i e. we replace L in (2.12) by L in (2.8)-(2.9)

whenever 3ik is negative. Now (2.12) becomes a convex combination of TVD (or

TVB) operators under the CFL restriction ..' *. %

'. .'-5 -.

(2.13) A < Aominj~

and we easily get the following

PROPOSITION 2.1. Scheme (2.12) is TVD under the UFL restriction (2.13), :%

if L is replaced by L when 3 jk is negative.

REMARK 2.1. The previous proposition may be put into a more general

framework as follows. The TV in (2.5) and (2.6)(a,b) may be replaced by G(u),

any convex mapping into the non-negative real line, where u, S(u), T(u) belong %.-

to a Banach space of functions B"2. Also if u is a smooth solution of 1.2, then

u(x,t ' +') - S(u(x,t')) = 0((Ax)"+). The statement in the proposition can then NO

be replaced by: G(um) < G(u°); moreover the formal order of accuracy is still r.

Now our goal is to choose the crik and 0k such that (2.12) is of the highest 0

possible order and such that the CFL restriction (2.13) is optimal. We would also

like to minimize the number of negative 3ik's in order to reduce the computational

work involving L. S

One easy way to do this is to use standard Runge-Kutta method and then

rewrite it in the form (2.12) to get cik and ,3i. Unfortunately, most classical

Runge-Kutta methods lead to small CFL numbers in (2.13) as well as negative

- 3ik's. ttence the best way is to consider (2.12) directly. Straightforward but te(lin"', " '

.• ..

%" %",, 0

A .,.'. .

Taylor expansions and an analysis of possible parameters (which we omit) leads us

to the following results:

i) Second order case, m = 1.

For accuracy

a 0 20 21-

(2.14) 0 -!L- k""2.31211 106L",,"O"

01o, a2 i are free parameters.

It can be verified that the "optimal" scheme (considering CFL restriction (2.1'3)

and whether Lappears) is

0)= u o) + AtL(u(°))

(2.15) iu(o) + 1 AtL(u('))2 2

CFL#=1

Notice that L does not appear in (2.15). This is equivalent to

(2.16) u(-) = u(0) + •tL(u()).

(2.16u(2) =u + AtL(u(O)) + L.tL(u( 1)

which is the classical Heun's method or modified Euler method [11.

ii) Third order case, m =2.

For accuracy

1332 = P(/OL-P)

21 1

(2.17) _0 10 628-P 2 -0.3~31 010 *j .

330 1 - 31 30 - a3 2 P - /331- 332

320 = P - 021 /310 - 321

a2 1 (30, a31 , /31o and P = o20 + a21 31o + '32, are free parameters. The solution

(2.17) is written in convenient inductive form.

L.,,. . e-€, .j.'.. .

U., - 2: : ..-::.-o., -' -" - .'- .-,,..:., ,:. ,,"."". ... ,,. "'-=" ","..".'. ..,",." .. ".." '

-12-

Extensive searching leads to the following preferred scheme:

j(I = u(0) + AtL(u (° ) )

U(2) =-u ( °) + lu( + AtL(u('))

CFL#=1

Notice that Ldoes not appear in (2.18). Also in computing u( ') we only need

L(u(i-1)), so there is no need to store the previous L(u(k)), lowering the storage

requirement significantly.

(2.18) is equivalent to

u(i) = u(o) + AtL(u ( °) )

(2.19) u(2) = u(O) + 1AtL(u(0 )) + 1AtL(u('))

+ ) + ) + ,,AtL(u(2))

We have been unable to identify (2.19) with any of the "classical" third order

Runge-Kutta methods. On the other hand, the classical third order Runge-Kutta,. .% .\.

methods in [7], when written in equivalent forms (2.12), lead to negative /3i and .

small CFL numbers (2.13), and are hence inferior to (2.19).

(iii) Fourth order case, m 3

For accuracy we get a system of 7 equations with 16 unknowns, so there are

9 free parameters. Unfortunately this time the solution is not easily obtained in a

convenient form. In [1] a general solution with two parameters is given for the form

(2.10). We can certainly rewrite it in the form (2.12). Extensive searching seems to ..

indicate that we cannot avoid negative constants 3 ik this time. The classical fourth .- "

ILS

5 '.Z.,,

; 'S '* + " ] '' ''" " ' " i ' " "

'** '" ' "' ' ; '-*1" "• | - ".. =: _:

order Runge-Kutta method can be written in the form (2.12) as

u(1) = u(O) + 1ALU()

U (2) = W)- 1Ati(u(0 )) + WO+ -L~tL(u('))

(2.20) U(3 = -. U () jtL(U(0)) + 2U(1) _ Ljit(U(1)) + +2U(2) U(2 ))

U(4) = LuM(1 + -LAtL(L(1 )) + ! A+ tLu()

CFL#=

Notice that we have to compute L(u(0 )) and L(u(')). If we use the more

awkward definition of %

(2.21)0

8000 1600 800 400 200

then the CFL# can be raised slightly to L0000.-~.

Another fourth order scheme with a slightly larger CFL is:

u()- u(O) + 1iAitL(u(0 ))

U()= au(o) - a,~tiL(u(o)) + !u(1) + !AtL(u( 1 ))

U~~~ (3 31UO _J§-tL(s( 0 )) + 460%9 U(1 - __ U()(2.22) 200 +00 00 0

= -1(O) + -l(' + l~tL(u(') + -L + L3 3 3 3 6

CFL# = 0.87

We still need to compute t(u(0 )) and L(u(')). :

(iv) Fifth order case, m =5.

We simply write out the form (2.12) corresponding to a fifth order method oR

-14-

given on page 143 of [7]:

(2.23) ....*

(1)l u(O + 1-AtL(u (° )"

U (2) lu(°) + u( 0) + AttL(u(')) +%+ ,

U(3) :uO) _, tL(u(O)) + !U(1) - tL(u(1)) + !U(2) + !AtL(,U( 2 ))

U(4) : *u(+) _ _ tL(u( ° )) + 40702 L) + U(2) +1AtLlu(2))+1 U () + -L tL ( U ( )). , ...

U (5) 8953 .,, ) + 2276219 t L u O ) + 407023 U (1) + 407023 A t ~ ( ) +, .... .280000 + 4 0 3 200 0 0 ztL(u(0 )) 2880000 672 0 0 0 AtL(u(')-

1611 u(2 ) + AtLlu(2)) + Lo U(3 ) -261AtL(u(3)) + Au( 4 ) + jAtL(u(4 ))

_+ ) - AtLu(1 )) + 8u(3 ) + 2AtL(+("))(+ ) +. .tL(u(5))

CFL# 7

Notice that we need to compute L(U(°)), L(u( 1 )), L(u( 3 )).

III. A Simplified Version of ENO Schemes. To use the Runge-Kutta -'

type TVD time discretizations in Section II, we must have a spatial discrete opera-

tor (2.1) to start with. Theoretically one would like to use a TVD or TVB operator

T satisfying (2.6), because then the full scheme (2.3) would be TVD or TVB. But

as indicated in section I, the existing high order TVD or TVB schemes may smear

discontinuities and pollute the solution (i.e. we may not get high order accuracy in

a fairly large region near discontinuities), due to the fixed, wide stencil. The ENO

schemes constructed in [5], [6], [4] are very promising experimentally and appealing

conceptually, but the fact that they use cell-averages as well as point values via a re-

construction procedure, and that they were implemented using a Lax- Wend roff type...- .

time discretization, makes them rather complicated to program, especially in multi-

dimensional problems, or problems with forcing terms. The Runge-Kutta type TVD

time discretizations in Section II equipped with semi-discrete ENO schemes will sim- .

plify them in many cases (although there is no rigorous theory concerning TVB of

semi-discrete ENO schemes or their Euler forward version, analysis in many cases

and numerical experiments strongly support that the total variation increse at each.

% ,• %

.,,._ . .,.,.,.-.-..,...-, ,-, .....-...-.: . . ,,?.,.. ...?.-,.. ....'. -'-,-'..'.'',..?.. , ....'.-, S.

~%J

step is O(Ax") for a r-th order ENO scheme, see [61. Hence the full scheme (2.3)

in this case should also be TVB). In this section we further simplify non-oscillatory

1"methods by deriving a version of ENO schemes using only fluxes, not cell-averages. ::

We start with a simple first order monotone Lax-Friedrichs type of scheme. If P? J-

we define 5

(3.) f+( ) - (f (U) + Cn), f-( )= -:(5U ) .fu) 2 2a, ~ 1 -u

where a > max If'(u)l is a constant, then clearly "

(3.2) f+t(u) > 0, f'(u) < 0

(3.3) f+(u) +f- (U) = (u) ,

The Lax-Friedrichs scheme is simply (1.3) with'the numerical flux defined by

(3.4) ,+; =f +f7+-

Taylor expansion reveals the existence of constants a2 , a4 ,..., a2m , such

that if .

(3.5) a'k+z "fi+ + E a2k L + OlAx2-+1)

k=i

then the scheme (1.3) will be 2m-th order accurate in space in the sense of (2.2).

For example, a 2 - a4 724 ' a "57601'. .

In light of (3.4), it is natural to require

and to define the positive flux and the negative flux f (in the meaning of '--

(3.2)) separately.

Z: •

r 0"

:,t .t:,:,,:,:,.¢_,,-_.,:..-.,-: 2 -. y ./. > -. ,. -.,..,:.,, ....:,-..-... ,.-,- -, .-....- ,.. .. . . . . ....... -.. :.::. ..-,'.

SW ''q, J'- ." ,32, ', ' .%i, "% % % % "% -% %' '..." " , "- ." " ' . 4 .-. "."--- "- "- .

-16- 6

For accuracy, we require ft+ and f7-+ to satisfy (3.5) separately:+2

rn-i a2k

(3.7) := + 2 a,aIhz(0 *)3 +, + 0(T_2m+i)

We achieve (3.7) by using polynomial interpolants p* of f± to the correct

order:,... d.~..:.

(3.8) p3*(X) -- f(u(x)) + OIAX21 + 1)'","

near x -xj+ L, then define

rn- I 2k(3.9) /± X) 0 €,'=.

(3) ,) + E a2k (k=1

Clearly if (3.8) is true, then the fluxes , defined by (3.9) will satisfy (3.7).-.J, %,

It is in constructing the interpolating polynomials p (x) that we use the ENO a..-

moving stencil idea: in order to achieve (3.8) p:(x) can be polynomials of degree

2m interpolating f+ (u(x)) at any 2m + 1 points near xj+,. We use the ENO ideas

in [5], [6] and [4] to choose the 2m + 1 points automatically from the smoothest

possible region, but start with the correct one according to (3.4).

The algorithm can be written as follows: For constructing pt (x):

(3.10) () k(0 ) -k( 0 ) (0) Wz = f +(uj)Je(31) (1) min --,max= ,Q

n-1) (n-1) and Q(+"-1), -(2) Inductively, assume we have km~i , kman (x), then we com-

pute the n-th divided differences of f+(u(x)): .-.

(3.11 a) an f + [,(,,). (-k:.1,+,)1-(3.1 Ib) b ) = f+[ulzk ) )... u -)]. ".

min mk +

. .

-17-

We proceed to add a point to the stencil according to the smaller n-th divided

difference:

(i) If Ia(~l b(n)I, then

(3. 12a) -n bn

(3.12b) k -1,k)'

O %Ti

(ii) If Ja(*) < Ib(n)j, then0

(3.13a) -~n (n) %%

and finally

(3.14) ~Q~n Q~(z) =Q I'l) (X) + c(n)Hk~,.)( k

min

(3) p() + +~)z

For constructing p- (x):

(1) k0 ) = k( = j +1, Q~2() fu+)max u 1

(3.15) (2) same as (2) above with f+ replaced by f- and Q+ replaced by Q-1p'

(3) p;1(x) Q(-=X%**

REMARK (3.1).

(a) For the first order scheme we just get back the Lax- Fricxdrichs flux (3.4):

%

I~~~~~ W 0 W I I%

0-18-

%

(b) The second order scheme here is similar to the usual minmod second order

TVD scheme, [9] except that the minmod function is replaced by choosing the value

closer to zero (we omit the details of the derivation here); this is still a TVD method.

(c) For the linear equation ut + au. = 0 (the scheme is still nonlinear!), the

schemes here are equivalent to the ENO schemes in [6] using the primitive function

reconstruction, except for a possible difference in the choice of stencil. The ENO

schemes in [61 choose the stencil according to the divided difference table of cell

averages of u, rather than that of fl (u) here. We again omit the details of derivation

here. Since the ENO schemes in [6] worked so well numerically, and our simplified

schemes here are equivalent to those in [6] for linear equations, we expect ours to %* . -. 2

work as well. For preliminary numerical results see section IV.

As mentioned before there is at present no rigorous theory about TVB of this •

type of ENO schemes. Here we make two observations for our schemes along these .. K.. V

lines:

(1) For smooth solutions all the divided differences (3. 11) should be bounded .%%

(by the maximum norm of the n-th derivative of f±. times some cnstant) So if %

we use •

(3.16a) = min(Ic")I. M,1") sign (c(n ) %

or(3.16b) -1= min(lc("I, .\ "," - ) sign (cn)

in the place of r(") in (3.1.-) for n > 2, where .%In i are 'n)tant sh ich are related

to the maximum norm of the n-th derivative of fi in initial sni o,)th regin , it,

not affect the accuracy in regions f sn,,thness We then szet a T'! ,h,.,,eine We

-- , - - -% , - .--" . ."- % % ,"% ,•.% :- . .

-19- -

can easily see that our flux with (3.16) satisfies :4.V "

TD2(3.17a) jj+ = fjv= + + t

where

(3.17b) ja j+'2 1:5 MAX 2

with the constant M depending on the M(n),and jTVD2 is the second order TVD .

flux mentioned in Remark (3.1b) above. Now (3.17) clearly implies TVB of the

scheme. Numerically we do not see any essential difference by using or not using %

(3.16), hence we strongly believe that ENO schemes without (3.16) are also TVB,

at least for most practical problems. •

.*- *.,,,./,

(2) The approach in (3.1) - (3.15) is not the only possible one which can be.,.'A .~.. ,

used to construct an ENO scheme based on interpolating fluxes. At an early stage 0

of our current work we used another approach: starting from f+ and f in (3.1),

then for constructing p+ (z):%

(1) k21 , - j-Ikm2'= j, Q)( ) -f(u) + f+[u(z_,j),u(xj)j( - z); .

(2) & (3), same as the procedures (2) and (3) above in (3.10), "

0*.;.% % "-

Similarly, for constructing p"(z):

1 .(l) =j l- l ) + ,Q(')lx) f-lu))+ f-[uli). (x , )]lxr - , "x.))..-' .2Y

• J % ..

(2) and (3), same as the procedures (2) and (3) above in (3.1.7).

%,, %

Then we take p,(z) = p+ z) + p7(z), and write our scheme as

(3.18) d =u'-A(x x)=, . .'.:,

.4.~~~€ S % ,,,*d.* .4 ..--

"% " " i " " " % •

• " % % ","." . ."h" " " " ,..".". - .r..d"

-20- '..': -?*.° .m

Notice that the scheme (3.18) is simpler than (3.9) - (1.3). The only trouble is that

it is not in conservation form (1.3). However, if we use (3.16), then it is easily seen S' *' .d* .,

that (3.18) can be written as %

(3.19) U.!+l = Ur! A(. -L 2_ A

where kcI ! M, and FLF is the first order Lax-Friedrichs flux (3.4). We can call

such schemes "essentially conservative" because the most important property of a

conservative scheme-the conclusion of the Lax-Wendroff theorem in [8] - is still .. ,-, '4'

valid. Since the scheme deviates from a first order monotone scheme (not only a

TVD scheme) by MAx 2 , we have even a stronger theory than before - we have

the entropy condition, hence full convergence (not just of a subsequence), and also " "."..:',....?

convergence in multi-dimensional scalar problems i.e. we have every convergence

property first order monotone schemes have. Unfortunately numerical experiments

indicate that in some cases, (3.18) is inferior to the fully conservative (3.9) - (1.3). P-J.An illustrative example is to compute the Riemann problem for Burgers' equation '.

ut + uu, = 0 with a moving shock (e.g. u1,f t , Uright -l.) using the fifth

order versions of (3.18) and (3.9) - (3.15), (1.3), equipped with a fifth order multi-

level TVD time discretization in [13]. The procedure (3.9) - (3.15), (1.3) gives good .. t.'4

results, with or without (3.16): while without (3.16), the non conservative (3.18)

gives the wrong shock location. With (3.16) the shock location becomes correct, but r.

the mechanism that enforces this causes a rather severe smearing of the shock. For

these reasons we abandoned the simple and theoretically pleasing version (3.1S).

Finally let us po(int on t that the [,ax- Friedrich s huilding tl,wk is ,)nlv . ,',n-

%enient one: we may also use ,)ther monotone ,or E-fltix% (- e.g [1()]) as (,ur

building blocks. Of course it is n,,t always possible to assoicato f' ani f- as in

13 1) with each E-flux such that (32). (3.3). (3 1) is xalid, but careful in,,pecti)n%'!

P-'

0- 21- x , ,

reveals that we do not need to use the values of f+ and f - only their divided

differences. For each E-flux hi+, we can define

(3.20) drj++ -i+-h+ , df 7 L h + fl... ,

where df+ and df#7 replace the first (undivided) differences fjt+ - anddJ+2+ .:..2f7 I - f7. Hence we can just use the divided difference tables of df+ and df- in

.7+ +place of the divided difference tables of f+ and f in constructing pt and pi and

define ft, f7 in any consistent way such that f+ + f;= fy, e.g. fI = fy, f = 0.

By (3.20)- 5,. ' %

(3.21) df+t + df-+ = fi+I- f4. 'S_%". ,.

hence if p+ and p-" use the same stencil then pt(x) + p- (x) is a polynomial inter-.'.

polating f(u(z)), thus accuracy is guaranteed with (3.9) - (3.6); if the stencils are

different, say -t] has the same stencil as p- but pt does not, then it is easy to show

that + - pt is a sum of r-th order undivided differences of df +} (r is the order of

the polynomials pt, p-) hence as long as these are O(Azx+ ') (valid if the E-flux •

hi+ , is smooth up to order r) we still have the correct accuracy.

The reason one might consider general E-fluxes as building blocks is that the

Lax-Friedrichs flux is considered to be too dissipative. While the first order Lax-Friedrichs scheme is much inferior to upwind schemes (e.g. to Godunov's or the

Engquist-Osher schemes). our numerical experiments show that higher order ENO .

schemes using Lax-Friedrichs building blocks work quite well (although they are still

slightly inferior to the same order ENO schemes based on upwind building blocks.

the difference is much smaller than that in the first order case). The advantage of

the Lax-Friedrichs flux is that it is (', hence the ENO schemes based on it have

full high order accuracy. On the other hand, most other E-fluxes (;odunv's, ,.

". 5o5.5...' 5.* 5.

-- *- m . m- II- I .-- -'---' ' ' -")

-22-

Engquist-Osher's, Entropy condition satisfying version of Roe's, etc. - are not

smooth at sonic points (points at which f'(u) 0), hence ENO schemes based

on them using the methodology of this paper will lose accuracy at sonic points. -.--.

Although we may overcome this by smoothing those E-fluxes at sonic points, in

most cases the simple Lax-Friedrichs building block should be good enough.

Problems in multi-dimensions are approximated by applying the procedure

described in (3.1) - (3.15) or its generalization (3.20), (3.21) to each of the terms 0% %- ' ,,

21, in (1l.a). The Range-Kutta methods devised in section 2 are then used, with

CFL numbers shrunk by a factor (d) - 1.

Systems of equations are approximated using by now familiar field-by-field

decompositions ideas. In (3.1) we replace the scalar constant a by a constant

matrix al, a = {aij)= 1, where the eigenvalues of ML are non-negative and of

are non-positive. . .

Obviously a might be taken to be a sufficiently large positive scalar multiple of

the identity, but this might also lead to some smearing of discontinuities associated

with slower waves. Other more practical choices might involve freezing M at some

constant state a, diagonalizing:

,(T() . T 1 (

and letting .. "

a T(u) ". T- (a) -%-%

a=T~)( ' Ad)

where each A, > A,(u)l throughout the region. To be safe, the margin of difference

between each A, and the maximum value or IA,(u)I has to be sufficiently large.

e':'

-23- .

...r . .

In any case, the corresponding p*(z) are obtained with the help of the left

and right eigenvectors of -- (u), which we denote by f(.i) and i ...d. We

interpolate t(9) f(u) obtaining t.'). p*(x) exactly as in (3.10) - (3.15). We then

define

The fluxes f 1 are defined through (3.9).+2

Generalizations of the type described in (3.20), (3.21) using approximate Rie-

mann solvers for Aj,k , appropriately smoothed at sonic points, may also be ob- .

tamed.

Work is currently under way with various colleagues applying these methodse

to Euler's equations of compressible gas dynamics in multi-space dimensions. J>"'S

IV. Preliminary Numerical Results. The numbers in this section are often

10-3. % -% % 1

written in exponential form, e.g. 4.2-3 means 4.2 i0.

EXAMPLE 1. The ENO schemes (3.1) - (3.15) in section III combined with,.

the Runge-Kutta type TVD time discretizations (2.19) - (2.20) in section i are used

to solve the nonlinear Burgers equation with periodic initial conditions:

: .';.f

0

(4-1) -< X< 1(4.1), 0) ++sinirx

The exact solution is smooth up to t u cothen it develops a moving shock

which interacts with the rarefaction waves. We get the exact solution by using a

Newton iteration. For details, see o6. -

Since there is a sonic point, we use the smooth LF (Lax-Friedrichs) building

block in our ENO schemes. Both 3-3-LF-ENO (third order in time and space wi)

. " .-. .-

-24-,s...

schemes with Lax-Friedrichs building blocks) and 4-4-LF-ENO are used.

We use a CFL number of 0.8 for 3-3-LF-ENO and 0.6 for 4-4-LF-ENO.

The errors of the numerical solutions at t -- 0.3 are listed in Table 1. Since the

exact solution is still smooth, we get the full order of accuracy in both L, and L.--

norms.

At t =,2,the shock begins to form. We use Ax = and print out the errors40

at 10 points near the shock:% ," "'*.

3-3-LF-ENO: -8.7-4, -2.6-3, -7.1-3, -1.3-2, -1.2-1, * >. '-

8.2-2, 8.0-3, 1.1-3, 2.6-4, 1.9-4

4-4-LF-ENO: -1.5-4, -3.2-4, -1.6-3, -1.5-2, -1.2-1, *

7.8-2, 7.9-3, 9.2-4, 1.8-4, 9.5-5 0

where the * is the position of the shock.

We see that there is a very good shock transition. (No oscillations are observed).

Figures 1-6 show the shock transitions. -- %

In smooth regions the numerical solutions are very accurate. We compute the -

L, and Lo, norms in the region a distance of 0.1 from the shock (i.e. Ix- shock locationi >

0.1) and list them in Table 2. From the table we can see that the errors are of the

same magnitude as in the smooth case when t = 0.3.

At t = 1.1, the reaction between the Shock and the rarefaction waves is over.

The solution becomes monotone between the shocks. We again print out the errors

at 10 points near the shock for Ax =

3-3-LF-ENO:-1.0-4, 1.6-4, 4.2-4, -3.3-2. -8.3-3,.

'.'.-%'S

'- %j p -%#

-25-

3.9-2, 1.7-3, -1.6-4, -2,5-5, -7.0-6.%

4-4-LF-ENO: 1.1-5, 6.6-4, -1.6-3, -5.7-2, -1.3-3, %

5.7-2, 2.7-3, -2.4-4, -5.8-5, -1.2-6 ,.,>

Figures 7-12 show the shock transitions.

The errors where the solution is smooth are again listed in Table 2.

We can see the excellent behavior of ENO schemes in this example.

EXAMPLE 2. A two-dimensional version of Example IA '. *°",+ (qL+q =0

(4.2) U ))+ W 2 Y - 2 < x, y < 2u (X, y,0)= + sin '+ ,..""""..

is tested'using the same schemes as in Example I in a dimension by dimension

fashion (i.e. T(u) = (I + AtL, + AtLy)(u) in (2.1), together with the R - K time

discretization: (2.19)-(2.20)). The exact solution is one-dimensional depending only

on = : + y, however our grid points are rectangular in (x, y) coordinates, andA-A'%.%

thus this example is a truly 2-dimensional test problem. .43-".":"-d"

A<..-/.|The CFL number is always taken to be half of the one dimensional analog, i.e.

0.4 for the 3-3-LF-ENO and 0.3 for the 4-4-LF-ENO.

As in Example 1, we collect the L, and L, errors at t = 0.3 (smooth solution) .

in Table 3 and the L, and L, errors in regions at a distance of 0.1 from the shock

at times t = and t = 1.1 in Table 4. We also print out 10 points near the shockif

when x=0, t= 2 and t= 1.1 for Ax=ANy=

Ir 200

t -0:if

3-3-LF-ENO: -9.7-4. -2.3-3, -7.6-3, -4.5-3, -1.2-1,

_7A'p -

4-4-LF-ENO:- -1.-4 -3.-4 -163 152,-.-,N

3.9-, 17-3,-1.-4,-2.55, 7.0- -2e6-

4-4-LF-ENO: -1.15, 6.-4, -1.6-3, -.- 2, -1.-, ..

57-2, 27-3, 9.2-4, -. 8-, -.- 6

The hoc trnsiiongrahs ae vry imiar o Fgure 1-2, enc weomi

3-LFat-rENO: sch0-4, 4.6-4 4.2-4 -3.-2 mu8t3-3, enioa prbes

3.9-2, 1.7-3, +1.-4, -2.5-5,<-7.0-

(4.3)F-NO 1.1-5 6.,-.-, -5.7-, -1.33,Y * S

4 10

The shoc t68prns in graphs anrderyt smila toe Ftiigure an12 thec weun oi

Wmearn obsrv disnutillyf the se resthos.a nte1dmEape1 hsm

cTe hatmoriEal schemeos wor well(ftr ero in mut-dmnina d roblems. yand~~~~~~~~. y~ -04 r ipae nFgrs1-0

-27- -, , , #, -

Observations: (1) the 2 dimensional schemes are stable under CFL numbers

one half of those used for I dimension; (further experiments using 1-dimensional

CFL numbers led to instability-overflow),

(2) The 4-th order scheme resolves the discontinuities better than the 3rd order 'Aw

method.

(3) Overshoots and undershoots, if any, are negligible. .* 0

,, 4-,..-. ,

EXAMPLE 4. We did not prove for these ENO methods that limit solutions"4 , .d*

4 , '

satisfy the entropy condition. However, numerical experiments in [6], including

some tests using nonconvex fluxes, indicated the convergence of ENO schemes to I S

the correct entropy solution. We test our schemes 3-3-LF-ENO and 4-4-LF-ENO

for Riemann problems for two such fluxes. One is - %.

((4.4)) f u) - (u2 - 4).. . -4

with UL = 2, UR = -2 (the exact solution is a shock followed by a rarefaction

wave followed by another shock) and with UL = -3, UR = 3 (a stationary shock at

X = 0). See [61 for details. Our schemes converge to the correct solutions in both

cases with good resolution. The results are displayed in Figures 21-32.

Another nonconvex flux we test is the well known Buckley-Leverett example: .1* .P

(4.5) f(u) = 4u2,+(1-.u42+ (I - U)2.,.._,

with initial data u = 1 in [-1,01, and u = 0 elsewhere.

The exact solution is a shock-rarefaction-contact discontinuity mixture. Our

schemes resolve the correct solution well. However, the 3-3-LF-ENO (using the Lax-

Friedrichs building block) smears more than the 3-3-EO-ENO (using the Engquist-

Osher building block) (Figures 33-38). Although in this example f'(u) > 0 so

* S

01' %.,..

.... .. . .. .~ ~* J* *** ~ -. .,,. , , - - ,. % 4 . 4 **,4 44, ,%,4 W . . .. ... -- -

0-28-

%do ..

there was no need to smooth the flux, the observed improvement using the upwind

building block indicate that sometimes it is worthwhile spending the effort to smooth e .- 1

the EO or other upwind flux rather than to use the simple LF building block.

* .2. .- ?

'a '. '

-. J'..k

%00

IN.

tA. -A%

'1 S

"or

-

m - °' a-%

, "-"r ,. " ' r " "

,'k . -,..-'.--'.t' '. _,.' .,''.. '. . ' S - - " , :' ' ' " "

" " "

' " = " : " , " ""-* " " "' " " - " ' "a *

, , - . ' P m " ." " ." " ,, ..- ,% -%- '.• "°' '.' .. •°% -. a. *% .

,

.P %.~-29-

0

Table 1. (Example 1).

t = 0.3; E type of error, r• numerical order

t:33-ENO r 4-4-ENO r 33ENO r_ 4-4-ENO r"i

-L 2.6-3 2.1-3 7.2-4 5.3-410

A. 2.3-4 3.50 1.1-4 4.25 5.7-5 3.66 2.5-5 4.41 e

3.1- 2. 5.- 4.35 6.5-6 3.13 9.2-7 4.76-"- '

• ~~., .' '

5-.40b

__3-3-ENOr 4-4-ENO r-N 3-3-ENO 4--N -rN 4-4-ENO

2.- .- 4 31.-4 .5 55 3.66 .7-6 4.4-6

**. , ,,%

Tak .1-5pl .8 5.-,435 6.--313 92-"47,,.-.,.'

o .-

-5.Erros- i sm ot regio 1.6- s k 0. 1; 3.5 5 8.-1.- . - , .

51" . *%"A

-..1=21.",w--;.,'.- '-.;'j ...'- , .-. -.- , . ,/....r. • . . . . . -"--'-.1

-"-,,uI 3, a , .''-3EN- 4,-4 ,-ENO, - --.. -ENOL. 4,', -4-ENO a- 3- ENO"."' -4.-ENO" .". 3-3-N ." 4 ". -4 -•--.."

. . 7 4... - 1.6 - 1' " .6 L,-5 . 5 ''.,N"-5 8,.6. - . 7-6 1. , ..4. '-6 "" """" -

- 30- 0

Table 3. (Example 2). '"A

t = 0.3; E: type of error, r: numerical order

AX =Ay\x Lew LI

3-3-ENO r 4-4-ENO r 3-3-ENO r 4-4-ENO r

L 2.7-3 2.1-3 1.4-3 2.7-4

2.3-4 3.55 1.1-4 4.21 1.1-4 3.67 1.3-5 4.40... %. _-.

3.2-5 2.85 5.6-6 4.36 1.3-5 3.08 4.5-7 4.8020 "%pj

o" ., .

Table 4. (Example 2). k', "

Errors in smooth region [(x,y)- shockj > 0.1; Ax A= ;

200

L ,, L I ' '

t -- 2/ t=1.1 t 2/v t =1.1 .

3-3-ENO 4-4-ENO 3-3-ENO 4-4-ENO 3-3-ENO 4-4-ENO 3-3-ENO 4-4-ENO ,, .

9.9-4 1.5-4 1.6-4 1.7-5 7.9-5 4.8-6 1.5-5 7.7-7 .% -.

q p. . .

8-6 . -5 •.,

lap.

- 31-..,.., ,

In the following figures, the solid lines are for exact solutions, and the circles

are for numerical solutions. S

Figures 1-12 are for Burgers' equation (4.1);, %. -

Figures 13-20 are for the linear two dimensional equation with discontinuous % %

initial data (4.3), with Ax = #;",0

Figures 21-32 are for Riemann problems for the nonconvex flux (4.4). Figures S

21-26 correspond to Uleft = 2, Uright = -2; Figures 27-32 correspond to ulft .

- 3 , U rijg = 3 ;

Figures 33-38 are for Riemann problems for the nonconvex flux (4.5). **5"*.

- , % .

- -'. -*

... ....- 5.

- '. 5..

'.-,S J' q

.. ',,5,,

\ % 5 * V\, .

6'.2 , .'', .''. . ,',''2 j . ','-,',',"., -. "..-., .,-.:. .. ,.. ..5* 5. ,-

[ '.,' , ,"" .,,.. ' ... "' ' "--:'- h.a" €' a,;, :..."'" ,. "' 4 " \,." ." - " ", , '.'..,",.," ".""..o ..... 55." ,.'

%0%

I- 'i r '-,-"' Z'

[I] (ear, Numerical Init al \alue l drlems, in ()rlir r. l))ifI.rait l lrni;- ,. %

tions: Prentice-l1all. 1971. .. ,

[2] .A. liarten, High Resoltion Schem,.s fi~r 1lyl-rlho ic ( ',, '. :ll . ..

(omputi. lhys. N-19, 19 3. pp. 357-393.

[3] A. Hlarten. On a ('lass of High Resolution Ttal-Variation-St.,,l, Iiite I ),tf.r-

ence Schemes: SI A M J. Numer. Anal., V21 i .9, pp. 1-23 '

[4] A. Ilarten. Preliminary R( sults on the Extensi(m of EN() Schemes to

Dimensional Problems: Proceedings of the International ('n fernce )ni 1I1 pr-

bolic Problems, Saint-Ftienne, January 1986..V %

[51 A. Ilarten and S. Osher, U niformlv High Order Accurate N ,n-()scillat,,r%

Schemes. 1: MR( Technical Summary Report #2823. May 1985, to ;lpe;Ir -* . 1-.S

in SIAM .1. Numer. Anal.

[6] A. Harten, 13. Engquist, S. Osher and S. ('hakravarthy. I iifor milv Olli )r- %

der Accurate Essentiallv Non-Oscillatory >cherfes. Ill .. (r1,iplt ri hs.,

appear.

[71 J.D. Lambert, (omputational Methods in Ordinary l)iff'rer,, i]l ldiiltI i,, 77,K-,

.John "ilev & S(ons Ltd.. 1973..- ;. _d,. \"

[ >.I). Lax and 13. \Vendroff. Svsteiis of (, ,rx ait Wion L.; (',,llilll 'ir .A p.

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Standard Bibliographic Page .. ~%-

.Report No- NASA cR-178 304 2.co tnn ceso o fspalt" C atailog No

ICASE Report No. 87-33 -

4Title and Subtitle R~Iepor Dat Je 11.

EFFICIENT IMPLEMENTATION OF ESSENTIALLY May 1987

NON-OSCILLATORY SHOCK CAPTURING SCHEM4ES 6. l'erforruing ( )rgAoiZlti~ (od) 'MI

7.~~~~ Auhrs)Perforinig Organ~ization Report No

Chi-Wang Shu and Stanley Osher8-3

____________________________________________W. _______ - . L k ititN

9. U 5ications in Science 505-9-10

and Engineering 11 '1 ( t rt or G;rant NcMail Stop 132C, NASA Langley Research Center NASI-18107

Hampton, VA 23665-5225 __ __________~.. - VIIoIeorad'rod(ord

12.Sponsoring Agency Nanie anrI Address

Contractor ReportNational Aeronautics and Space Administration I4 io. oring Agency ('ode ? -.

Washington, D.C. 20546 ___ ,%.

I Supplemecntary Notes J.'

Langley Technical Monitor: Submitted to J. Comput. Phys.% %

J. C. South

Final Report

16). AhsItra(t

In the computation of discontinuous solutions of hyperbolic conservation

laws, TVD (total,&variatiorv-diminishing), TVB (total-variation- bounded) and therecently developed ENO (essentially non-oscillatory) schemes have proven to bevery useful. In Lhis paper two improvements are discussed: a simple TVD Ruinge-

Kutta type time discretization, and an ENO construction procedure based on flux-es rather than on cell averages. These improvements simplify the schemes con-siderably--especially for multi -dimensional problems or problems with forcingterms. Preliminary numerical results are also given.-

7 Key %%ords, (Suggested hv Authrs~l ) IS. D~istrbution Stitpem]eit

~construction laws, essentially 64 -'Numerical Analysis

non-oscillatory, TVD,' Runge-ktitta

Unclassified - unlimit ed

1) cujov(Ilitmeif (oj this report) 20. Sec urity (1,si (f this page 21. No. of Plages 22. Ptr e(

Tnc1 siie Unclassife 73 AL ~

For sale fly the N'ational Technical Informtat ion Service, Springfield. Virginlia 22161

NASA Laniqley 11,487

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