U"0RMLY HIGH-ORDER ACCURATE NON-OSCILLATORY SCHEMES I. 

BY -cc 

AMImm 

Dm. OF MATHEMATICS, UCLA 

(NASA-CR-181173) NUPIERIC'AL SOLUTION 3P THE N87-25546 

UNSTEADY N AVIER-S TO KES EQU AT I: ON Semia n nrld 1 

Report (California Univ.) 61 p Avail: 

NTIS HC AOlr/ClP A01 CSCL 20D unclas 

G3/34 0085370 

_- 

1) Research supported by ARO &tract #DUG -29-80 coo41 while in Mdence at 

IR390-403 and ARO Grant # DAAG 29-82-0090 while at UCLA - 

the MRC, university of WiscaaSin, and by NASA Comortium Agreanmt ACNCA2- 

2) Research supported by NSF Grant H2-00788, ARO Grant #DMG 29-82-0090, NASA-Consortium Agreement #NCA2-IR390-403, and NASA Langley Grmt~ # NAG 

1-270, and NAGf-506. 

-1- 
Abstract. We begin the construction and the analysis of nondatory shock capturing 

methods for the approximation of hyperbolic conservation laws. These schemes 

share many desirable propctties with total variation diminishing schemes, 

but 'IVD schemes have at most first order accuracy, in the sense of 

truncation error, at extrema of the solution. Yn this paper we coqstruct 

a uniformly second order approximation, which is nonosdlatory in the sense 

that the number of extrema of the discrete solution is not increasing in time. 

This is achieved via a nonmtory piecewise hear reconstruction of the 

solution from its cell averages, time evolution through an approximate 

solution of the resulting initial value problem, and averaging of ahis 

approximate solution over each cell. 

AMSMOS Classification: Primary 65 MlO, secondary 65 MOS 

Key Words: Cmservation Laws, Flnte Difference Scheme, Nonoscillatory, TVD. 

-2- 
, 1'1 

1. Introduction. In this paper we consider numerical approximations to weak 

solutions of the scalar initial value problem (IVP) 

(1.la) . ut + f(u)= = ut + a(u) ux = 8 

The initial data u&) arc assumed to be piecmvise-smooth functions that 

are either periodic or of compact support. 

Lct vy = v&, t,,), xj = jit, t,, = n7, denote a numerical approximation in 

conservation form 

which is consistent with (l.la) in the sense that 

(1-2a) 

_... 

f(U,I( ,..., u) = f(u) . 

We consider the numerical appmximation vh(x,t) in (1.2) bo be a 

pi~-amstantfunction 

Accordingly we dcfine its total variation in x to be 

Tv(v") = TV(Vh(',t")) = 1vy+1 - YJl * 

J 

-3- 
If the total variation of the numerical solution is uniformly bounded in h for 

OSZST 

then any refinement sequence h - 0, 7 = O(h) has a subs~umc~ hj - 0 SO 

that 

where u is a wcak solution of (1.1). 

If all limit solutions (1.6) of the numuid solution (1.2) satisfy an entropy 

condition that implies uniqueness of the IVP (l.l), then the numerical scheme is 

convergent (see e.g 131, [12]). 

Recently we have introduced the notion of total variation Aiminisling (TVD) 

schemes (sec [3]), where the approximatesolution operator is required to diminish the total variation (1.4) of the numerical solution at each time-step 

thcse schemes trivially satisfy (1.5) with C = 1. Some early work along thesc 

lines was done by van Leer in [15]. 

TVD schemes are non-adlatory in the sense that the number of local 

extrema ig the numerical solution is diminishing in time (as is customary we use 

"diminishing" loosely as short for %on-increasing", throughout this paper). 

Moreover, the value of an isolated local maximum may only decrease in time, 

while that of a local minimum may only increase. 

We were able to construct 'SVQ schemes that in the sense of local truncation 

-4- 
error are high-order accurate everyarlhexe except at local extrema where they 

necessarily degenerate into iirst-order accuracy (see [4], [13], [IO], ['.I], [14]). 

The perpetual damping of local extrema determines the cumulative global error of 

the"high-orderTVDschemes"tobc O(h) inthe L, nm, O(h9 inthe L2 

norm and O(h9 in the L1 'norm (see ill). 

In this paper we introduce a larger class of non-odlatory schemes. in which 

the solution operator is only required to diminish the number of local extrema in 

the numerical solution. Unlike TVD schemes, which are a 3ubset of this ciass, 

non-osdllatory schemes arc not required to damp she values of each local 

extremum at every single time-~tcp~ but arc allowed to occasionally accentuate a 

local extremum. 

In a se~ucnce of pap, of which the present papa is the first, we show how 

to construct non-oscilhtory schemes that +re uniformly high-order accurate (in 

the sense of global erro~ for smooth soluti~ of (1.1)). In this iirst paper we 

describe a serxmd-order accurate schemeofthis type. 

The fact that the number of load & in the numerid solution may ~nly 

diminish in time is suffidmt by itself to guarantee that the application of the 

scheme to monotone data results in a monotone function. Thus non-oscillatory 

schemes, like m schemes, are monotonicity preserving. In particular, when 

applied to a stepfunction, they do not generate 3gurious oscillations. 

We note that since the number of local extrema in the solution of nonoscillatory schemes is bounded by that of the initial data, uniform bundedness of 

its total variation (1.5) follows immediately if the maximum nom of the solution 

is shown to be uniformly bounded. *. 
2. Design Principle and Overview 

In this section we describe how to lxrnstmct a nan-oscillatory scheme that is 

uniformly second-order accuratewhere 

(2. lb) 

and 

(2. IC) 

I: 

We observe that although (2.la) is a relation &tween the ell-averages iij’ 

and q+’, the evaluation of the fluxes h+m(u) in (2.lb) requires knowledge 

of the solution itself and not its d-averages, 

As in Godunm’s scheme and its second-order extension by van k [16] and 

Colella and Woodward [2], we derive OUT schcmc as a direct apgroximation to 

(2.1). We denote by vj’ the numerical approximation to the cell-averages of 

the exact solution in (2.1c), and set v! to be the cell-averages of the initid data. 

Given vn = {vfl we compute v”+l as foflows: 

First we rcconstruct u(x,t,,) out of its approximate ell-averages {vn to the 

appropriate accuracy and denote the result by L(x; v”). Next we solve the XVP 

(2.2) ’V, + f(v)x = 0, v(x,O) = L(x; v”) 
and denote its solution by v(x,t). Fmally we obtain vY+l by taking cellaverages of Y(X,T) 

The averaging operator in (2.3) is non-osdlatory, therefore the number of 

local extrema in v"+l (interpreted as a mesh-function or the piecewise-constant 

function (1.3)) does not exd that of Y(x,T). &sdg v(x,t) Po be the exact 

solution of (2.2) implies (since the uact solution operator is TM>) that the 

number of local extrema in V(X,T) is less than or equal to that of 

v(x,O) = L(x; v"). Therefore if the number of local extrema in L(x; vn> does 

not exceed that of v", then the resulting sczlcme is non-osdatory. 

We amdude that the design of non-oscillatory high order amate schemes 

essentially boils down to a problem on the!level of approximation of functions: 

Given cell-averages Zj of a piecewk-smooth function u(x>, reconstruct M(X) 

to a desired accuracy. Prior to studying this problem we tackle another related 

question in approximiition of functions, thafof constructing a non-oscillatory 

high-order accurate interpiation of piecewise-smooth functions. 

and satisfies, wherever u(x) is smooth, 
In section 4 we make use of this non-oscillatory piecewise-parabolic interpolaut to design a non-osciUatq reconstruction of a pkewise-smooth €unction 

from its cell-averages. As in 1161, [2], [SI, and [9] we take L(x: ii) to be the following piecewise-linear function 

Unlike the above references that present "second-order accurate" TVD 

schemes, we compute the slopes Sj/h from Q(x; i7j by 

Here m(x,y) is the xu@ mod function 

We show in section 4 that L(x; 9 is a pr+r reconstruction of u(x) in the 

sense that whenever u(x) is smooth 

(2.7a) 

_- 

L(x;u') = u(x) + O(h3 

hn Here Z(x) = I, u(x + y) dy and &; Z) = 

L(x + y; Z) dy; like Q(x; 4, the latter iS also a nm-datory hn 

= h-1 

J-hi2 

piecewise-parabolic interplant of E(%), 

(2.7~) L(xj; 9 = i(xj) 

- 8- 
We remark that the "second-order accuraten WD schemes described in the 

above mentioned references use a slw Sj/h in @.Sa) that approximates 

(&&) u(xj) to O(h), and their loss of seama-order accuracy at id extrema 

points is due to lack of smoothness of the coefficient in the O(h) term at these 

pFts I. This problem is circumvented in the present scheme by taking Sj/h to 

be (2.5b) which is an O(h2) approximation to (dldx) u(xj). Unfortunately there 

is a price to pay for +this extra accuracy, namely the loss of the "FVD propcry. As 

inTvDschcmes 

(2.8) Tv(y"+') Tv(L(*; v")) , 

however here 

Tv(L(-; P)) z W(V") 

and indeed the scheme may occasionally incrcasc the variation af the numerical 

solution. ~lthough we prove that the scfiimc is non-oscillatory we have not been 

able as yet to complete a proof of uniform boundedness of the total variation of 

the numerical soiution; this is due to lack ofjechniques to verify miform hdedness of the maximam norm af the numerical solution. 

I! 

In section 5 we study the praposed scheme in the amstant coefficient case. 

We verify that it is uniformly second-ordcx accurate, examine its behavior at local 

extrema points and get estimates for the possible haease in total variation per 

time-step. 

In this paper where we amsider numerid schemes of the form (1.2) that are 

1. We repeat that the results of [8] and Ell] imply that TVD schemes, no matter how 

they are constructed, must have this loss of accuracy at local extrema 

-9- 
derived from approximating the relation (2.1), it is only natural. to consider truncation error in the sense of cell-averages, Le. we say that the scheme (1.2) is 

second-order accurate if 

where u“ is the cell-average (2.14 of the exact solution. Since 

(2.10) ;(XI = u(x) + O(h2) 

whenever u(x) is smooth, (2.9) holds also for pointwise values of the solution. 

However, in the context of 3rd and higher order accurate schemes, this difference 

in definitions of truncation error wiU be not only amqtual but of practical 

importaace as well. 

Up to this point we have assumed that Y(X,T) in (2.3) is the exact solution 

e: * 

to (2.2). The resulting scheme 

(2. Ila) vy+l = vy - Ayt,+,W - i;-m(v)l 9 

where h+m(v) is (2.1b) applied to v(x,t), 

_- 

1 (2.11b) 5+mM = 7 JT fW4) dt 9 

is certainly second-order accurate in the sense of (2.9). Starting with the exact 

cell-averages vy = iij’ in (2.11) we get h (2.7a) that 

(2.12a) v(x,t) = u(x,t + t,,) + O(h3 for 0 5 t 5 T 

and consequently 

(2.12b) &w2(4 =ftj+lR(u) + w2) 9 

which implies (2.9) due to the sufficient smoothness of the coefficient in the 

.” 10- 
O(h3 term in (2.12b). 

In section 6 we replace the exact solution v(x,t) in (2.3) by an approximate 

one, which we denote by vn(x,t) . This approximate sohtion is consemative. 

TVD, and second-order accurate in the sense of 92.12a). Thus replacing v(x,t> 

in (2.3) by this approximate solution results in a consenrativc: scheme that is nonoscillatory and uniformly seumd order accurate. 

We remark that an alternative approach to the above is to approximate 

&+m(v) in (2.11b) by using a midpoint rule (or trapezoidal de) for the 

integral and by replacing v(x,t) with a non-osdlating second-ordcr accurate 

approximate one v,(x,t) (see [16] and [2]). The resulting scheme 

(2.13a) 

(2.13b) 

- 

is certainly seamd-ordcr accurate, and it ismon-oscillatory in the constant coefficient case. Since we have not used the cell-averaging (2.3) to derive this scheme, 

wecannotasartam * in gend that the resulting scheme is non-oscillatory. 

Nevertheless, our numerical experiments as well as many other experiments in 

the context of TVD schemes (see e.g. 111, [2]) demonstrate that the numerical 

results are non-oscillatory in many (if not all) applications. 

In &on 7 we prcsent some numerical experiments that compare the present 

scheme with a typical "secmd-order amuate" TVD scheme. 

3, Noaosdllatoty interpolation. 

The oscillatory nature of second order amate Lax-Wendroff type schemes 

- 11- 
results from a Gibbs phenomenon associated with E&-order interpoiation across 

discontinuities. In this section, as a preparatory step towards designing a nonoscillatory approximation to (I.I.), we construct a non-oscillatory piecewiseparabolic interpoiant Q(x; u> to a pkcewke-smooth function u(x) such that 

(3.la) 

wherever u(x) issmooth. 

Q(x; u) is non-osdlatory in the sense that the numk of its ld extrema 

docs not exceed that of u(x). 

SinCe 

where 

(3.2b) 

- 12- 
We consider as candidates for qjA1/2 the two quadratic polynomials 6 

and 6+l, interpolating U(X) at (xj-19 xjy xj+l) and (xi, xi+2), raptiveiy, and choose qi+m to be the one that is least oscillatory in [xi, xi+l]. 

Both 5, j = i and j = i + 1, a be written (3.2a) With DiA1/2 u = Dju 

where 

since the least oscillatory of 5 'and g+l can be characterized as the one that 

deviates the least from the line amnecting (xi,ui) with (Xi+l,uj+l) we chouse 

Di+m u in (3.2a) to be 

where m(x,y) is the min mod fuucticm 

If u(x) is smooth in [xj- 1 ,xj 11, then 6 as a quadratic interpoIlant of u 

_- satisfies 

If Diu - DiAlu z 0 then qi+m is either 6 or GA1. Wenvise we set 

Di+m u = 0, but then smoothness of u implies that Dp = 0(h3) and comequently qi+m - 4J = O(h3) for j = i, i + 1. Thus (3.1~) follows from (3.4). 

We turn now to prove that Q(x; u> is a nondatory interpolant of u, 

Le. that the number Of its local extrema does not exceed that of u. We do so by 

showing a one-to-one correspondence between local extrema of Q to those of the 

13- 
mesh €unction (uj), the number of which certainly does not exceed that of u(x). 

Q may have a local atremum in either the interior of some intmal 

(xi,xi+1) or at a mesh point xi. The first case, which will ?x refered to as 

interior-extremum, occufs when there is a pint X* xi < x* < xi+ Ir such that 

From (32) if follows that Q has an interior-extremum, in (Xi, q+l) if and 

only if 

(3.7a) Di u . Di+1-rr > 0 

Tb implies that qi+m has a locail h (xi, xi+l> if and only if both 6 

ad G+1 also have a local extremum in (Xi, Xi+l) and of the same kind. Since 

a parabola has at most one local extremum, it follows then that 6 does not have 

a local extremum in (xi-1, xi) md G+1 dm not have 

Consequently Q is ~llo~loto~lt in both (xi-1, Xi) and (xj+l, XI-L~), but in 

in (xj+l, xj+2). 

opposite sense, i.e. di-m u - di+x u < 0; the latter implies that u has a local 

extremum in [xi, and that either uj or ui+l is a local extremum of the 
mesh iunction {uj) (€or obvious reasons the case uj = uj+l is counted as a 

single-extremum). The above analysis also shows that interior-extrema are isolated, i.e. if has an interior-extremm in (xi, xi+& then it is the miy id 

extremumof Q in (xi-1, xj+2), 

We turn now to examine the case that Q has a local extrmum at a mesb 

point xi; this will be refered to as a mesh-extremum. The above observation 

that interior extrema are isolated excludes the possibility that Q has an interiorextrem~m in &ha (Xi-1, Xi) 011 (xi, Xj+l) and cons~q~~~~tly Q is monotone in 

these intervals. This implies that di-m u dj+a u < 0 and bhercforc= ui is a 

local extremum of the mesh function {y). This concludes the proof that Q(x; u) 

is a non-oscillatory hterpolant of u. 

We next express the nonetory nature of Q in terms of total variation. 

If Pj+a UI s 21d'+m u3 then (2.5) impyes that e is monotone in [xj, xjsl]. - 

nus 

Using (2.6) we get 

We conclude that 

- 15 - 
The sum in the RHS of (3.84 is taken over the set of indices M of intervals 

(xm, %+I) in which Pm+m u1 > 214n+y2 uI, Le. where Q has interiorextrema. 

Next we show that if u(x) is a piecewise-smooth function of bounded variation, then 

lim TV(Q(-; u)) = W(u) . h-0 (3.9) 

We obseme that in this case the number of intervals in M is finite and is uniformly bounded by the number of local qpcma in u(x). Hence (3.9) will follow 

ifweprovethat Dm+~u-O as h-0 forall rn €M. Toaccomplishthiswe 

showthat for h suffidentysmall M doesnotincludeintervals in 

which u(x) is discontinuous. To see that let us examine the case where ~(x) 

ha a discontinuity at x' € (Xi, Xi+ 1) . clearly di+ VL u approaches the size Of 

thejumpin u while dl-1/2rc approachcszeroas h-0. Consequently 

whichimplies i4M. 

- 16- 
4. Non-osdlatory Reconstruction. 

Let u(x) be a piecewise-smooth function and denote by G(x> its mean over 

(x - h/2g + h/2), i.e. 

We denote 5 = .'(.j) and r&er to these values as cell-averages of u(x). Given 

Cui), the task in hand is to reconstruct u(x) to O(h3 in a nonetory way; 

denote the appmxhatdy reconstructed function by L(x; q. To achieve O(h2) 

accluacy it is sufficient to consider L(x; ii) to be a piecewise-linear function. To 

make L(x; ii) a non-tory approximation we use the non-datory piecewise parabolic interpolation Q(x; Z) to compute its slopes as follows: 

(4.2b) 

Here m is the min mod function (3.3); d&q u' and D~+Y;! U are (3.2b) and 

(3.2d), respectively. 

We note that L(x; Z) may be discontinuous at {xj+m) and that 

- (4.3a) L(xj; q = uj . 

To see that wherever u(x) is smooth 

(4.3b) 

we observe that in this case 

(4.4a) 

L(x,~ - U(X) = O(h2) 

Z(x) = u(x) f O(h2) 

- 17- 
and therefore it follows from (3.1~) that 

(4.4b) 

Consequently the RHS of (4.2a) can be expanded as 

d (4.4~) L(x; G) = uj + (X - xj) - * u(xj) + O(h2) 

1 = U(X) + O(h2) for - ~rl < h 

and thus (4.3b) follows. 

¬e by L(x; ii) the mean value L(X; ii) in (x - h/2, x + h/2), i.e. 

(4.5a) L(x; ii) = L Jx+k/2 LO; ii) rJ, . h x-m 

Using (4.2a) to evaluate the integral in (4.5a) we find 

(4.5b) c(x; ii) = iij + dj+m u' I! 

- (4.5c) 

H~CC &; 4, like Q(X; q, is a pi-e-parabolic intcrlpolant Of ;(XI. 

qxj; ii) = uj . 

comparing (43) with (3.2) wc find that fot Xj S x Xj+l 

d 

& From (4.4b) we see that Si = h - i((xi) + 0(h3) (Note that this is true 

- 18- 
also at local extrema points) and therefore 

sjq - sj = h'- d2 u(~jAUg + 0(h3) , 6xz 

On ?he other hand (3.2) shows that 

Dj+m U = h2 & .(xj+& + O(h3) . dx' 

Therefore 

(4.6b) Sj+l - S' - Dj+m u'= O(h3) 

which shows that RHS of (4.6a) is O(h3). Since (2.1~) shows that 

Q(x; iq - iqx) = O(h3) 

we conclude from (4.6a)-(4.6b) that 

that ~(x; 3 is; non-datory approximation to 

that L(x; ii) is a non-oscillatory approximation to _- 

u(x>. We shall do so by showing that W[x,,xj+~(L(.; ii)), the tow-variation of 

L(x; u7) in [Xj, Xj+l], which has the value 

Then it follows immediately form (4.7b), (4.3a) and (3.8) that L is monotone in 

[xj, xj+l] if and only if Q is; consequently L is a non-oscillatory approximation 

- 19- 
to u(x) in exactly the same sense as Q is to the interpolated €unction (see seetion 3). 

Next let us denote 

(4.8a) . 

and observe that (4.2b) implies that 

- 20- 
we conclude from (4.10) that 

It foUows from (4.12) that 

Sj- 2s; =dj+m u>o 24 - - q+m 

-1 (4.13a) 

- rs&, . 9 

- Dj+a u = SG1 2 (4.13b) 0 > dj+m U + 

The relations (4.13) and the definitions (4.8a), (4.2b) imply that 

(4.14a) -1 

2 Sj = S,? = dj+m u - - Dj+m u 

- 21 - 
1 - = dj+m u't - Dj+m u 2 (4.14b) Sj+ 1 = sjq1 

Here M is the set of indim of intervals in the hterior of which L 

(and also Q) has a local extremum. The number of these htemais is €bite and 

is bounded by the number of local extrema of ii(x>. Canparing (4.9) with (34 

we note that 

(4.16) 

,. 

TV(L(*; q) 2 W(Q(-; q> . 

_- 5. The constant soef€ident case. 

In this section we study the constant coefficient case 

(5.1) ut + ou, = 0, a = const. 

The exact solution of the IVP (2.2) is 

(5-2) v(x,t) = L(x - at; Y"> . 

Hence our scheme (2.3) is 

(5.3a) 

'22" 
where f is (4.5a). We have shown in section 3 that the number of local 

extrema in L(x; v”> does not exceed that of Y”. Since vntl in (5.3a) is a cellaverage of L, it follows that the number of Id extrema in v” + does not 

exceed that of v”, and consequently the scheme (5.3a) is son-oscillatory. 

Using (4.33) in (5.3a) we get the following expression for the scheme 

Eh denotes the operator form of the finite difference scheme; p = ha, the 

CFLnumber, is assumed to satisfy 

We turn now to prove that (5.3) is secimd order accurate in the sense of 

(2.9), i.e. if u”(x) denotes the mean value (4.1) of u(x,t,) then ..- 

(5.44 q+l- (E,, - u”>i = 0;h3) . 

To show that we observe that in the amstant coefficient case (5.1) 

q+1 = u”(xi - UT), and by (5.3a) (E,, i”)i = L(xj - UT; 2’). Hence the lLHs 

of (5.4a) is nothing but 

Next we study the time-dependence of the total variation and the maximum 

norm of the numerical solution (5.3). In section 2 we have pointed out that * 

-23- 
(5.5a) TV(V"'1) 

Using (4.15) and (5.5a) we get the fo2lowing upper bound OIL the possible growth 

of the t0ta.I variation of &e numerid solution ger time-step 

(5.5b) TV(vR+l) - m(vn) s C - pm+lR~nl - ]4+m~n[] . m€Ma I' 2 

Here M,, is the set of indices of intervals (+, xmdl) in the interior of 

which L(x; v") (and also Q(x; v")) has a local extremum. The number of thesc 

intmals is finite and remains unifdy bounded in be by the number of local 

extrema in the initial data. 

Clearly the upper bound (5.Sb) is overly pessimistic. It estimates the possible increase in variation in the rcamstructicm step due to replacing the cellaverages vi" by the picccwise-hear function L(x; v"). It dots not take into 

account the possible decrease in variation'& the averaging step (2.3), resulting 

from doing just the opposite, Le. replacing the piecewise-linear function 

L(x - UT; vn) in (5.2) by its cell-averages (5.3a). 

In the following we shall de the temporal behaviout of the local 

extrema of the numerical solution and its total variation by analysing the explicit 

values of the cell-averages 

let us assume that a > 0. 

given by (5.3b). To simplify our 'presentation 

First we note that (4.8b) implies 

1 (5.W wj - +ll s Wji + ls,,,l 5 2-(l+-mvnl, y iq-mv"l) - 

Hence 

- 24- 
where 

We see that the CFL condition 0 < q S 1 and (5.6b) impiy that 

thus we conclude 

Relation (5.8) shows that if vR is monotone for JL 5 j zs JR, Le. 

tone for Jt + 1 s j s JR, and in the same sense. Relation (5.8) also shows 

that mesh-extrema of vn, i.e. those for which Q has its local extremum at a 

mesh point, are being damped at the n-th time-step Namely, 

1 (5.9a) Idj*m vnl z - 2 vnI, yf”-1 S I$’ 2 q max(vf+ll vi”+”) 5 vi” 

We turn now to consider interior local extrema of vn, i.e. those for which 

-25- 
Q has its local extremum in the interior of some (xi, xi+ 1). We recall that such 

an extremum is characterized by idjlb2 Y"! C b2 PJ'+~ vnf and that Sr and 

SrA 1 in this case are given by (4.14); therefore Si", 1 - Sr = DiAm Y". From 

(5.3a) and (4.6a) we see that in general 

Relation (5.10b) amfirms the secwd order accuracy of the scheme at local 

extrema. Although it does not necessitate accentuation of the exaemal values, as 

v?:) in (5.10b) may still be in [vf9 vr+l], it does allow vf:? to deviate from 

this interval by as much as 

Thus (5.10b) is the essential difference ..- between the present scheme and the 

"second order" TVD schemes. 

A similar analysis, which we do not present here, shows that if vp is a 

mesh-extremum then vJ+l, j = i, i + 1, relates to Q(xj - a7; Y") in the fdlowing way: 

' 

(5.11a) vj n+l 2 e(xj - a~; Y"), j = i, i + 1, if vr is a maximum 

(5.11b) vY+l s Q(xj - a?; v"), j = i, i + 1, if vr is a minimum 

From (5.9)-(5.11) we deduce the following relation between the total varia- 

- 26 - 
tim of the numerical solution and that of its piecewise-parabolic interpolant Q: 

(5.12) TV({Q(xj - UT; v"))) W(V"+') TV(Q(*; Vn)) . 

The LHS of (5.12) is the totai variation oi the mesh function 

{Q(xj - UT; Y")}. Relation (5.12) suggests to consider an equivalent definition 

Tv of the total variation of the numerical approximation of the form 

with the hope that the sch~~~~e (5.3) is TVD with RS- to phis PIlOditied dchnition. Unfortunately our numerical CJrPeriments have shown that there are 

instances, although rather rare, that Tv(v") is increasing with ar; the same is 

true for iT(vn> = TV(L(-; vn)). 

As we have mentioned in the introduction, because of the nonoscillatory 

nature of the scheme, uniform total variation boundedness of the numerical solution is implied by uniform "Iboundedness of irs maximum nom. If we follow a 

particular local maximum of the hitid data we see from (5.9)-(5.10a) that it 

actually decreases most of the time, and whever it does increase (5.10~) and 

(3.10) suggest that it does so by a "small amount" that vanishes with h - 0. 

Since the initial data is only giectwise-smooth we have not &en able as yet to 

rigorize these 

We remark that our numerical experiments clearly indicate that in a normal 

ComputatiOnal situation the maximum norm of the numerical solution is indeed 

uniformly bounded. We feel that OUT inability to prove this fact stems only from 

lack of theoretical tools to andyse pointwise regularity of the numerical solution. 

= 27- 
6. The nonlinear case. in this section we describe an approximate solution 

v,(x, t) of [SI for the IVP (2.2) 

This approximate solution is consistent with the conservation form of the equation 

(6.1) in the sense that the cell-averaging (2.3) results in a scheme in conservation 

form Le. 

(6.2) . 

where the numerical flux jf+m is Consistent with f(u) in the sense of (1.2~). 

Furthermore, the approximate solution opcrator is TVD 

and thus by the reasoning presented in section 2, the resulting scheme (6.2) is 

non-oscillatory . 

We turn now to outline the derivation of this approximate solution. To simplify our presentation we ignore entropy coosiderations and refer the reader to 

future papers for details of aBpropriate modifications. We assign to the point 

xj+ 1/2 a &aracrcristic speed that corresponds to the Rankine-Hugoniot speed 

- 

aj+ 1/2 of the two neighbouring cell-averages vj’ and vj’+l 

if vf f Vf+l 

and denote by Z(x) the piecewise-linear interpolant of {Zj+&, i.c. 

-28- 
The approximate solution v,(x,t) is defied by speafying its constancy 

along the approximate characteristics 

(6.5a) x(t) = xo + ii(xo) t 

Le. 

(6-m vn(x(t), t) = Vn(X0,O) = L(x0; y") * 

Using (6.5a) and (6.4b) to express xo in texms of x and t 

where 

. (6.6b) xi+ a(t) = xi + IR +t*G+m. 

Taking cell-averages of the approximate solution (6.6) we get the conservation 

form (6.2) 

(6.7a) q+l= vy - Xct',,, -&-l.d 

- 29 - 
! 

I 

with the numerical flux 

where 

(6.7~) 3j = s~”/[I + ~c;lli+m - . 

Note that (6.7) is identical to (5.30) in the constant cocfficicILt case. 

We turn now to prove that the scheme (6.7) is uniformly seamd-order accuin (6.7) rate in the sense of (2.9). Starting with the exact d-averages YY = 

this amounts to showing that 

with a sufficiently smooth coefficient in tbJ: O(h2) term: here fj+m is the 

numerical flux (6.7b) wmputcd with the exact cell-averages, and j$+,(u) is 

(2.lb). We shall do so in two steps: first we shall show that 

-- 

1 (6-sb) 4+m(~) = yEf(L(xj+ut; u”>> + t(L(xo(xj+,, 7); 011 + 0!h2> 9 

where xo(xj+m,T) is (6.5~1, ad then we shall verify that 

Speciai attention will IX given to the smoothness of the 0(h2) coefficients. 

To show (6.8b) we start by using the trapezoidal rule to approximate the 

integral in (2.lb); we get 
The smoothness of the O(h7 term follows from that of f(u) and u(x,t). Next 

we observe that Z(x) in (6.4b) approximates a(u(x,tn)) to O(h2>, and therefore we can use the approximate characteristic be (63) to trace 

u(xj+M, tn + T) to u(xg(xj+m, T), tJ WMI 0(h3) accuracy; consequently 

Fdy we obtain (6.8b) by approximating u(x,t,) in (6.9a) and (6.9b) to 

O(h2) by L(x; u") (set (4.4)). Thesmoothness of the O(h2) term in this 

approximation is due to (4.44: 

Sf = h %(x, r,) + O(h3) . 

(We recall that the degeneracy to first order accuracy at ld extrema points of 

some "seamd-order accurate" TVD s&cmcs is due to lack of smoothness there of 

the O(h3 term in (2.7a)). 

', i. 

We turn now to verify (6.84. Fmt laus consider the case Zj+m 2 0: 

and expand the LHS of (6.84 around q. We get 

Siarly in the casc 4+m s 0: 

- 31 - 
We see from (6. loa) and (6.10b) that independently of the sign of ij+ m, the 

O(h3 term in (6.84 is the same, namely 

h2 . 

8 (2W - 1) - a‘ ’ (ux)2b+yz. 

This completes the proof that the scheme (6.7) is second-order accurate in the 

sense of (2.9) wherever u(x,t) is smooth, including local extrema and sonic 

-- 

(f’ = 0) points. 

Remarks: (1) The numerical flux (6.7b) can be rewritten as: I 

(6.11) f,+m = + f(Vf+l> - lq+lRl (.i”+l - vi”> 

- 32- 
(2) Our proof that the scheme (6.7) is non-oscillatory is based on the 

representation of (6.7) as the cell-average (6.2) of the non-oscillatory approximate solution v,,(x,t) in (6.6). To ensure that v,(x,t) remains univalud ior 

0 I; t s T we have to restrict the time-step T so that for all j 

(6.12b) T w@y2 - 3+uz) S A. 

On the other hand, to derive the particular form of the numerical flux (6.7b) we 

have made the assum@on 

which implies the condition. 

(6.13b) 7 maxjIiq&l< h . 

The two be-step restrictions (6-12b) and (6.13b) are cplaracteristic to 

Godunm-type schemes. The practice of rec_onciling the mo conditions by 

is completely unnecessary: The scheme (6.7), as the original Godunov scheme, 

remains non-datq (or l"VD in the case analysed in [IO]) €or the full ClXrestriction (6.13b). The reasoning for this observation is as follows: The approximate solution (6.6) under the CFL restriction (6.13b) may fail to be univalued in 

thej-th cell only if > 0 and ii;'+m < 0. In this casc the numerid fluxes 

as defied by cell-averaging in the neighboring cells j 2 1, remain (6.7b). 

Thus the only thing that needs to be changed in this case is the definition of 

- 33 - 
v,(x,T) in the j-th cell. 

(3) We obwme that once va(xct) is defined globally in (6.5) there is no 

intrinsic need to average it on the original mesh. We may average it on different 

intervals and still conclude that the resulting approximation is non-oscillatory and 

conservative. Furthermore, the construction d the interplant Q, the approximation L and the approximate characteristic field Z(x) needed to defiae 

v,(x, t), does not depend on the uniformity of the mesh. Therefore the scheme 

(6.7) generalizes immediately to non-uniform moving meshes. Of particular annputational interest are the self-adjusting moving grids of the type described in 

[U], which make it possible to obtain perfectly resolved shocks and contact 

discontinuities. 

(4) We note that since the I;upproximatc solution v,,(x,t) in (6.6) is comervative, it is possible to consider an associa@ random-choict method obtained by 

replacing the cell-averaging in (6.2) by a sampling with respect to a variable that 

is uniformly distributed in the cell, Le. 

I. - 

where 07 is uniformiy distributed in [-v2,1/2]. Following the reasoning of [A 

it is clear that the resulting random-choioe method is non-datory aud that its 

limits are weak solutions of (1.1). Although the random-choice approach has 

many attractive computational features, it has been our experience that in many 

application it is possible to accomplish the same computational goals with a selfadjusting moving grid. In this case the use of the latter is preferable as it offers 

gain in resolution without a loss in pointwise accuracy that is associated with sampling. 
7. Numerical Illnstration. In this sdon we compare the new uniformly 

second-order non-oscillatory scheme of this paper (to be refered to .as uM02) to 

the typical second-order TVD scheme (to be refered to as lYD2). Both schemes 

can be written in the form (6.3, Le. 

(7. IC) si" = m(Sj',ST); 

here %+m is (6.4a) and m(x,y) is the min mod function (3.3). Szf are different for Tva2 and uNo2: I! 

. 

(7.2) TvD2: s;t =dj*mv" 

where dn+m and DIAn aredefined h(3.2). 

UNO2 and 'p7rD2 are both second-order accurate Godunm-typc schemes 

that differ dy inthe rcumstructicm step (4.2a): 

where the slopes of the lines are calclllated by (7.3) and (7.2), respectiv&ly. 

Therefore we start our cumft8lisoL1 on the approximation level. 

In Table 1 and Fig. 1 we present aggroximationS to 
u(x) = sinm, -1 5; x S 1. We divide [-l,l] into N equal intervals and 

defme 

* Osj5N. xj = -1 + p . - 2 

N (7.4b) 

The symbols in Fig. 1 denotes values of uj = sin.rrxj for M = 10 in (7.4b). In 

Fig. la we show the piecewise-parabolic interpolant Q(r;u) (see section 3). In 

Fig. lb we show the piecewise-hear apFroximation LI”Mo2f.9c;u> which is (7.4a) 

with (7.1~) and (7.3). In Fig. IC we show the piecewise-hear approximation 

L-(xp) which is (7.4a) with (7.1~) and (7.2). We make the following observations regarding Fig. 1: (i) Q is a better approximation than L~~; L- is a 

better approximation than L-. (ii) ~(~~02) > TV(U) >. TV(L~. ~n 

table 1 we quantify the first observation; we list the L,-error and the L1-error of 

these approximations to sinax for a rehnement sequence of N = 10,20,40,80 

in (7.4b). Clearly Q is an 0(h3) appdimation, while L 

O(h2>. The error in LUNa is about a 1/3 of the error in L-. 

and L- are 

In Table 2 and Fig. 2 we present solutions _- of UNO2 and TVD2 €or the constant coefficient case 

with periodic boundary conditions, at P = 2 with dh = 0.8. Figs 3a and 3b 

show UNO2 and W, resgectively, with N = 20 in (7.4b). In table 2 we list 

the L,-error and L1-enm for a refinement sequence with N = 20,40,80,160. 

Clearly UNO2 is second-order accurate in both L, and Ll, while WTX is 

second-order accurate in L1 but only first order accurate in E,. Fig. 3b demonstrates that the degeneracy to fmt order accuracy at local extrema in the TVD 

- 36- 
scheme adversely affects the accuracy everywhere (Because the scheme is TVD it 

cannot switch abruptly to seoond-order accuracy as this would introduce oscillatious; consequently it takes quite a while to reawer the second order accuracy). . 

Next we approximate the discOntinuous €unction 

-xsin(3nx2/2) , -I-< x < -m 

Isin(Znx)/ , kI < m 

- 1 - 1/6~i11(3nx) , < x < 1 

(7.7) 

which we extend to have period 2 outside [-1,1]. 

In Eg 3 we present apBroximations to +(x), using N = 20. Fig 3a shows 

Q(x;q, Fig 3b shows LuNo2(x;~, and Fig 3c shows L-(x;q. We again 

observe that Q is better approximation then Lmm, while Lmm is a better 

approximationthen L TVM . 

In Flg 4 we present solutions of UNd and TVD2 for the constant cueffiuent problem (7.6), initial data given by (7:2), and periodic boundary conditions. 

We take t = 2 and dh = 0.8. Figs 4a and 4b show UNO2 and TVDZ respectively with N = 40. Fig 4b shows the damping effect that the TVD scheme 

imposes due to its degeneracy to first order accuracy at local extrema. 

_.- 

In Fig 5 we solve she same problems, except we imposc boundary conditions. 

At x = -1 we impose the given function (7.7) evaluated at -1 - t. No boundary conditions are imposed at x = 1. We implement this numerically using 

UNO2 and TWX? except at the boundary points. There we are in general, unable 

to construct non-oscillatory piecewise parabolic interpolants Q(x,u'), so we construct the only possible parabolic interpolant thru xj,xi+l and the point to either 

the left or right which lies in the region. The analogous procedure is carried out 

- 37 - 
at the reconstruction stage. Figs Sa and 5b again show the results at t = 2 with 

T/h = 0.8. 

The possible introduction of oscillations through the boundary conditions 

does not seem to have degraded the per€ormance of either schene (in fact the 

opposite is observed). Again the TVD2 scheme shows a damping effect. 

In Table 3 and Fig. 6 we present results for Burgers' equation 

with periodic boundary conditions and dh (1 + lal) = 0.5. The solution to (7.7) 

is smooth for r < UT; at t = UT it develops shocks. In Table 3a we list the 

L,-mor and L1-errm of UNO2 and mat r = 0.15 for u = 8 = 0 in 

(7.7). This table shows the same asymptotic behavim as Table 2, except that 

because of the large gradients it shows for a smaller h. 

In Figs 6a and 6b we show results of 'hO2 and TVD2 for (7.8) with a = 2 

and B = 1 at r = 0.35 with N = 20. In this case the solution to (7.8) 

develops a shock moving with speed 2 beginning at time r = l/?r =: 0.318. _.- 

In Table 3b and Figs 6c and 6d we repeat the previous calculations for the 

schemes (2.13): 

(7.9C)f'j.cm = 

- 38 - 
As we have remarked in section 2, h (7.9a) is not a cell-average Q€ 

Vn(X,T), but Only an approxhnation to ite Therefore it is not necessary to take 

dj+m in (7.8~) to be (6.44. We choose djAm SO that (1.9~) is continuous at 

bj+n = 0: 

We denote the schemes (7.9) with Sj" defined by (7-IC) and either (7.29 or 

(7.3) by FWX and €NE, respectively. We note that (7.9) is identical to (7.1) in 

the constant coefficient case, and consequently IFVIYZ and FNO2 are noIIoScillatory in the constant coefficient case. Figs SC and 6d show that FMolE and FVD2 

are also non oscillatory in the case (7.8). Furthermore, Table 3b shm that 

FNo2 is much more accurate than UNO2 (FVD2 is about the same as "WX). 

In all pvius examples we have petynted pintwise calculations; namely, 

we have initialized the numerical solution by taking vy to be the value of the initial data at xi, and we have considered vy to be an approxima~on to u(xi,t,,j. 

(Surely this is an acceptable practice for kd order accurate schemes.) In TaMe 

3c we repeat the caiculation €or UNO2 in Table 3a, but now in a sense of cellaverages and denote it by ANOZ. Now we initialize UNO2 for (7-8) with 

a = p = 0 by cell-averages of the initial data, !.e. 

and regard vy to represent cell-averages of u(x,t,,). To obtain a pointwise 

approximation to u(x,t,> we first compute pint values Y;+z of its indefinite 

- 39 - 
(7. lob) 

~ext we obtain a global piecewise-hear approximation v(x,t,) to u(x,t,> by 

where Q is the piecewise-parabolic interpolant of section 3. Finally we get 

Thw the only difference between AN02 in Table %, and UNO2 in Table 3a 

is the initialization (7.10a), which itself differs only slightly €rom the mesh values 

of the initial data (since sin(n&Z)/(?ritn) = 1 - l/d(~hn)~ + O(h3). 

We remark that cell-averages do play a significant role when the initial data 

iS discontinuous (she they provide infarmation about the loation of she discontinuity) and in higher-order Godunm-ty& schemes; tbis will be described else- 

-? 

where. 

We turn now to prtsent calcdahcms with a formal extension of UNO2 *and 

TVD2 for systems of amstmation laws. We consider a Riemann problem for the 

Euler equations of gas dynamics 

(7.11b) u = (p,m,EIT, f(u) = (m,m2/p + P, m(E + P)/P>~, 

(7.11c) 

Here p, m, E and P are the density, momentum, total energy and pressure, 

-40- 
. 

respectively; we take y = 1.4. 

In the following we apply the extension technique of [3]1 to UNO2 and TVD2. 

The idea is to extend UNO2 and TVDZ to systems in such a way that will be 

identical to (7.1) in the Scapar case, and will decouple into (5.3) €or each of the 

characteristic variables in the constant coefficient system case. To accomplish 

that we :ISC Roe’s averaging €or (7.11) (sa [13]) 

(7.1%) vj+m = Wy’ vy+1> 

for which 

and define local charactetistic variables with respect to the right-eigenvector sys- 

(7.13a) = vy - A@+, - 4-d 

(7.13b) 

(7.13d) 

- 41 - 
Likewise 

teristic field, and is defined as follows: 

denotes the component of the vector of slopes in the k-th Charac- 

(7.14) TVD2: S$ = df*mw 

Df*mw = m(df+xw - df+mw, df+mw - df-mw) . 

In Figs 7.8 and 7.9 we show numerical solutions of UNO2 and IWX, respectively, for the Riemann problem (7.1b) with 

Ut = (1,0,2.5)', V' = (O.P25,QYO.25) . I: 

These figures demonstrate that the formal*extension to systems is nonoscillatory 

in this case. since the solution to the Riemann problem is just constant states 

seperated by waves we do not get to see here _- the extra resolution power of UNG2, 

except that its numerid solution is somewhat '*&peru* than that of WDZ. In 

this dculation we have not employed any artificial compression in the linearly 

degenerate field and therefore the contact discontinuity smears like nU3* as 

expected. The interested reader is referred to [4], [SI and [lo] €or a detailed 

description of such annpsion techniques, as well as for details of entropy 

enforcement mechanisms. 

- 42 - 
References 

[l] S.R. Chakravarthy and S. Osher, "A new class of Mi& Accuracy TM) 

Schemes for Hyperbolic Conservation Laws," Ah4 gaper 85-0363, Reno, 

NA. (1985). 

[Z] P. Colella and P.R. Woodward, The piecewise-parabolic method (PPM) for 

gas-dynamical simulations, J. Camp. Phys. v. 54, 174-201 (1984). 

[3] A. Harten, On a class of High Resoluiton Total-Variation-Stable FiniteDifference schemes, SINuM, v. 21, pp. 1-23 (1984). 

141 A. Harten, High resolution schemes for hyperbolic consevation laws, J. 

Camp. Phys., 49(1983), p~- 357-393. 

[5] A. Harten, "On second order accurate Godunov-type schemes". to appear. 

[6] A. Harten and J.M. Hyman, "A Self:Adjusting Grid for the Computation of 

Weak Solutions of Hyperbolic Conservation Laws", J. Camp Phys. Y. 50 

- 

(1983), pp. 235-269. 

[7] A. Harten and P.D. Lax, "A Random -Choice finite difference scheme for 

hyperbolic conservation laws", SINUM, v. 18, (1981), 289-315 

181 A. Jameson and P.D. Lax, "'canditions for the Construction of Multi-point 

Total Variation Diminishing Difference schemes.", Princeton U. MAE 

Report 1650 (1984). 

[9] S. Osher, Gmvergence of Generalized MUSCL Schemes, NASA hgley 

Contractor Report 172306, (1984), STMJM (to ap). 
[IO] S. Osher and S.R. Chakravarthy, "High-resolution schemes and the entropy 

condition", SINUM, v. 21, (1984), pp. 955-984, 

[11] S. Osher and S.R. Chakravarthy, "Very high order accurate TvlD schemes", 

ICASE Report #84-44, (1984). 

[12] S. Osher and E. Tadmor, "On the convergence of difference approximations 

to conservation laws", submitted to Math-Comp. 

[13] P.L. Roe, ApproBma te Riemann solvers, parameter vcctofs, and differmce 

schaes, J. Comg. Phy~., V. 43 (1981), ~p. 357-372. 

[14] P.K. Sweby, '*High resolution schemes using flux limiters for hyperbolic amservation laws, SINUM, v. 21, (1984), pp. 995-1011. 

[IS] B. Van k, Towards the ultimate conservative scheme, JI. Monotanicity 

, and conservation combined in a second der scheme, J. Comp. Phys. 

I: 14(1974), p~. 361-376. 

- 

[16] 8. Van Lecr, Towards the ultimate Conservative difference scheme, IV. A 

new approach to numerid amvdon,_J. Comp. Phys., 23(1977), pp. 276- 

298. 

-44- 
Tabk I: Approximations to u(x) = sin(m), -1 S x 5 1 , with periodic bounw caditiom. 

N 

20 

40 

80 

- 

UNO2 Tvm c2 mm TvD2 

7.097 X 8.119 X 8.944 X 6.778 X 

1.6~ x 10-3 3.477 x 10-2 2.044 x 10-3 2.033 x 10-2 

Lm-ERROR 

80 11 3.870 X 

L I-ERROR 

1.453 X 4.926 X 5.626 X 

e 

160 

I I 

1.971 x 1 1.231 x 3.558 x 1.802 x 1 5.576 x 1.525 x lo-' I i 

I 

I 9.201 x 10-5 5.975 x 10-3 1.172 x 10-4 1.528 x 10-3' 

I 1 I 

2.476 x 10-4 3.083 x 10-3 9.163 x 10-3 2.148 x 10'~ 1.355 x 10-3 3.902 x 10-2 

3.104 x 7.710 x low4 2.308 x 2.617 x 3.351 x 9.787 x I 

I 

T&k 2: Numeriml SO~U~~OIIS af Ut + U, = 0, u(x,O) = sin PX, -1 5 x s 1 at t = 2 

with periodic boundary amditions and ~/h = 8.8. 

I L1-ERROR II &ERROR II 

- 45 - 
Table 3a. Numerical solutions of u, + z+ = 0, u(x,0) = shm, at t = 0.15 and ~/h = 0.5 - 

UNO2 and TVD2 

6.938 x 10'~ 

L,-€RROR I II 

I 

2.091 x io-2 3.726 x 10-~ 1.322 x io-2 

40 5.712 x 1.054 x I 

1.959 x 

5.106 x 10-4 

I I1 I 

I 1.054 x 8.869 x 3.835 x 

4.424 x 10-3 2.163 x 10-4 1.072 x 10-3 

Tab& 3b. Same as Table 3a for FNO2 

1.m x 10-4 ' 1.837 x 10-3 I 

- 

N 

5.270 x 10-5 2.946 x 10-~ 

- 

40 

- 

80 

- 

160 

1.965 x 10-~ 3.621 x 10-4 

and FVlX. 

&ERROR L1-ERROR I 

-46- 
Table 3c. Same as table 3a for ANO2. 

40 6.623 x 3.243 x 

160 4.597 x 10'~ 2.079 x 10'~ 

- 47- 
t 

PIECEWISE PRRABOLIC 

N 

. - 

m 

h 

0 

m 

cv 

Q 

3 

u: 

(I: 

c 

I 

v: 

CI 

c 

I 

U 

0 

I 

c 

I I I 1 1 

-1 .o -0.6 -0.2 0.2 0.6 1 .o 
ZD 

'0 

N 

0 

X 

N 

? 

W 

0 

I 

0 

d 

I 

SL'O SZ '0 sz 0- 

n 

- 

A 

0 

CI 

m 

0 

N 

0 

X 

cv 

0 

Lo 

0 

I 

. 

0 

I 

d 

.rl 

2 
L //' 

d 

6 

? 

0 

"! 

0 

x 

4 

1 

k 

0 

Q 

r: 

0 

r( 

0 

a. N 
0 

m 

N 

0 0 

ro 0 

t0 
c L1 , 

I 

I 

LO 

I 

la 

I 

,o 

io 

I 

cx 

IN. 0 

0 

0 

N 

? 

0 

0 

? 

0 

9 - I 

X 

0 

Lo 

0 

m 

? N - Lo 0 Lo 0 

”. p1 

0 ? 

0 la 

m c‘f c. - 

P I 

r I 

Y 

1 

-. 0 0 

Y) 0 

1 

? 

0 

m 

? 

? 

0 

0 

0 

0 

X 

0 

0 

N 

0 

0 

0 

9 

0 

8 - I 

I’ 
0 

0 

0 

0 

0 

". 

X 

0 

0 

0 m 

0 

0 

9 

L 

I 

S 

c. 

0 

0 

m 

p! 

0 

0 * 

0 

c. .I 

I 

3- c 

.o 0 

h 

0 

0 

P 

c: 

0 
I 

s 

? 

0 

0 

W 

1 

0 

w 0 

0 

I 

1 

0 

m 

Y 

0 

3 - 0 

9 

c. 

8 

0 

c. m 0 

0 

1 

P 

- 

P I 

0 

m 0 

0 

0 

0 

0 

Y 

8. 

? 

0 

0 

9 - I 
$7 
r 1 I I II 

SZ'S Sf *z SZ'Z St'l SL'O 

n 
X 

6'0 L'O S'O 

OHY 

S'O 1'0 

In 
r 4 I I I 

0'1 8'0 S'O 3'D ZOO O'I 

n 

6 

Q, 
d 

DRXGINAI; PAGE IS 

OF POOR QUAT'TY 

. 

n 

P 

u