The ULTIMATE conservative difference scheme applied to ...klinck/Reprints/PDF/leonardComMeth1991.pdfresolution' shock-capturing or TVD schemes. The ULTIMATE third-order upwind scheme

Computer Methods in Applied Mechanics and Engineering 88 (1991) 17-74 North-Holland

The ULTIMATE conservative difference scheme applied to unsteady one-dimensional advection

B.P. Leonard Department of Mechanical Engineering, Center for Computational Mechanics,

The University of Akron, Akron, OH, USA

Received 13 October 1988 Revised manuscript received 4 May 1990

Modeling of highly advective transport is embarrassingly difficult, even in the superficially simple case of one-dimensional constant-velocity flow. In this paper, a fresh approach is taken, based on an explicit conservative control-volume formulation that makes use of a universal limiter for transient interpolation modeling of the advective transport equations. This 'ULTIMATE' conservative difference scheme is applied to unsteady, one-dimensional scalar pure advection at constant velocity, using three representative test profiles: a discontinuous step, an isolated sine-squared wave, and a semi- ellipse. The goal is to devise a single robust scheme which achieves sharp monotonic resolution of the step without corrupting the other profiles. The semi-ellipse is particularly challenging because of its combination of sudden and gradual changes in gradient. The ULTIMATE strategy can be applied to explicit conservative advection schemes of any order of accuracy. Second-order methods are shown to be unsatisfactory because of steepening and clipping typical of currently popular so-called 'high resolution' shock-capturing or TVD schemes. The ULTIMATE third-order upwind scheme is highly satisfactory for most flows of practical importance. Higher order methods give predictably better step resolution, although even-order schemes generate a (monotonic) waviness in the difficult semi-ellipse simulation. By using adaptive stencil expansion to introduce (in principle arbitrarily) higher order resolution locally in isolated regions of high curvature or high gradient, extremely accurate coarse-grid results can be obtained with very little additional cost above that of the base (third-order) scheme.

Introduction

In a landmark series of papers in the 1970's, Bram van Leer worked 'towards the ultimate conservative difference scheme' [1-5] for computational fluid dynamics (CFD). This work spawned a body of literature in the past decade involving advective modeling methods that are sometimes classified as 'shock-capturing' schemes or, m~re recently, as 'TVD' schemes, referring to the oscillation-suppression strategy of total-variation diminution [6]. The ultimate conservative difference scheme for CFD has proven surprisingly elusive. There have been notable 'successes' such as impressive demonstrations of nonoscillatory high resolution of steps and shock waves, for example; but progress has been unexpectedly slow. It seems that correction of one defect always introduces another debilitating side-effect. Unphysical oscillations inherent in classical second-order methods were eliminated by switching to first-order upwinding; but this merely replaced unacceptable oscillations with (what was ultimately realized to be) unacceptable global artificial diffusion. By devising methods with locally

0045-7825/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)

18 B.P. Leonard, The UL TIMA TE conservative difference scheme

varying (positive) artificial diffusion (small in smooth regions, larger in sharply varying regions), it is possible to achieve somewhat better step resolution than that obtained by global first-order upwinding without introducing spurious numerical oscillations. But step resolution is not very tight and other profiles, especially narrow extrema, may be seriously corrupted.

Some forms of shock-capturing (or TVD) schemes achieve their impressive results for step resolution by the use of locally varying positive artificial diffusion or viscosity (inherent in first-order upwinding) to suppress oscillations, combined with local negative viscosity (such as occurs in first-order downwinding) to artificially compress or steepen the front. Unfortunately, this inherent negative diffusion is responsible for artificial steepening of (what should be) gentle gradients, as well, as will be demonstrated. Because of the concomitant flattening of local extrema (due to the local positive artificial diffusion), this defect has become known as 'clipping', although the problem is initiated by the artificial steepening introduced to give high resolution to simulated fronts. In some cases (for example, a step-function followed by a ramp), inherent oscillations, rather than being suppressed, are converted into a series of small monotonic steps, a phenomenon known as 'stair-casing'.

As will be clearly demonstrated here, currently popular forms of TVD schemes achieve monotonic (although not always sharp) resolution of steps at the expense of gross (albeit nonosciUatory) distortion of simple smooth profiles. Thus, Mitchell's characterization of advective modeling as computational dynamics' ultimate embarrassment [7] is still appropriate, given the current state of the art. The ultimate goal of the current research program is to inject some self-confidence (as opposed to self-satisfaction) into computational fluid dynamics by developing a truly robust (i.e., universally applicable) framework on which further refinements can be constructed.

The present paper is the first in a proposed series presenting a fresh approach to simulating the advective term, which, being an odd-order (first-) derivative, is arguably the most significant aspect of CFD compared with other branches of computational mechanics [8]. An extremely simple form of 'monotonizing' universal limiter is described which can be applied to explicit conservative difference schemes, with no constraints on the order of accuracy and resolution. The universal limiter banishes unphysical overshoots and nonmonotonic oscillations without corrupting the expected accuracy of the underlying method. In particular, when used with (artificial-viscosity-free) third- or higher order base schemes, it does not induce artificial compression (steepening) or clipping, typical of so-called 'high resolution' (but actually, at best, still only second-order) shock-capturing methods.

Conservative explicit advection schemes of arbitrarily high order can be composed from 'transient interpolation modeling' (TIM); i.e., for pure advection of a scalar ~ at constant vector velocity v, the exact multidimensional transient solution over time At is

6 ( x , a t ) = - v a t , 0 ) , (I)

where the accuracy (in both space and time) of the approximate numerical method is determined solely by (multidimensional) spatial interpolation at the earlier time-level. The ULTIMATE conservative difference scheme then consists of using the universal limiter (UL) for transient interpolation modeling (TIM) of the advective transport equations (ATE).

Since accurate modeling of the advective term is one of the more challenging aspects of CFD (in addition to nonlinearities, multidimensionality, incompressible pressure-solvers, etc.), attention will be focused, in this initial paper, on the superficially simple but embarras-

B.P. Leonard, The ULTIMATE conservative difference scheme 19

singly difficult problem of unsteady one-dimensional pure advection of scalar profiles at constant velocity. Clearly it is important to make sure that simple lineer advection problems of this type are being adequately simulated before tackling more complex multidimensional real-flow situations. Three critical test profiles are considered: a step function, representing a basic discontinuity in value; an isolated sine-squared wave, representing smooth functions with a continuously turning gradient; and a semi-ellipse, combining discontinuous and continuous changes in gradient. At first, several (unlimited) explicit schemes are surveyed, together with some popular shock-capturing methods that are explained in terms of the Normalized Variable Diagram (NVD). The NVD is also used as the basis for development of the universal limiter, which is then explained in terms of unnormalized variables and applied to explicit polynomial TIM methods of second through ninth order. For most (relatively smooth) flows, the cost-effective third-order upwind ULTIMATE scheme gives excellent practical (coarse-grid) results. Local higher order resolution can be automatically built in with almost negligible additional cost by a technique known as adaptive stencil expansion, explained in Section 3.

1. Scalar advection using higher order polynomial schemes

Unsteady, one-dimensional, pure advection of a scalar 4, at constant velocity u is described by

o4, o,/, = - u (2) at t~x '

This equation can be integrated in time over a time-step At and in space over a control volume (CV) at station i from -Ax /2 to Ax/2, assuming a uniform grid. This gives the exact conservative difference scheme

~'+' = ~' - c(~*- ~), (3)

where the bars represent spatial averages and the asterisks time averages, the superscripts designate time-levels, c is the Courant number u At/Ax, and right and left time-averaged face values are indicated. The CV spatial averages can be written in terms of central node values plus a deviation term

~i = ~bi + DEV. (4")

Now assume that node values are related by an exact equation of the form

6;'+' - ,/,;' = -c(,/,, - 4,,), (5)

where the face values now include the effects of the deviation terms. The deviation terms themselves will then satisfy

DEVn+I _ DEV n = _c(DEVr _ DEVt), (6)

20 B.P. Leonard, The ULTIMATE conservative difference scheme

given suitable definitions of the terms on the right. Thus, from (3), (4) and (6),

n ?! ~b~ +' - ~b,. = -c[(~b*r - DEV~) - (~b 7 - DEV,)] , (7)

which can be put in the form of (5) provided the effective average face value at any face f is defined as

~b r = 4'~ - DEV; .

In this interpretation, (5) rewritten as

(8)

(9)

can still be considered exact for constant c, given appropriate definitions of the effective face values. Advective conservation is guaranteed, since ~r(i) = dpt(i + 1). In practical simulations, the face values are approximated and the form of the equation is assumed to be valid for variable advecting velocity (including sign-reversals), i.e.,

1#;!+1 ~--- 1#;!__ ( C r ~ r _ C l ~ l ) , (10)

where c,(i)= ct(i + 1).

1.1. Transient interpolation modeling

One method of generating advective algorithms in the form of (9) is based on (1), which in one dimension can be written, for constant u,

~bl '+n-- ~b(x, At) = dp(x- uAt, O)--dp"(x- uAt), (11)

where ~b"(x) can be considered to be a function of the normalized local coordinate, = (x - x~)/Ax, and given by

,t,"(x) = 6 ;' + f( ) . (12)

The homogeneous spatial interpolation function f can in turn be written as

f(~:) = ~gCs¢), (13)

where g might be a polynomial with coefficients depending on differences (of various orders) of local node variables. For a conservative scheme, these coefficients can always be written as follows:

o r g(~:) = (A,+,, 2 - A,_,,2) + (B,+,, 2 - B ,_ , , 2 )~ : + - . . ( 1 4 )

g(~:) = h ,+~,2(~)- h , -1 ,2(~) , (15)


where hi_l/2 c a n be obtained from hi+~ 2 by the indicated reduction in index by 1 on all involved node variables. Then, by defining

and ~b,(-~) = h/+l,2(~)

~pt(-~) = hi_l,2( ~) ,

(16)

(17)

(11)-(17) can be combined into a form identical to (9):

c n + l n , = ~b, - c[~b,(c)- ~bt(c)]. (18)

Note that the effective face values are in general functions of Courant number and that conservation is guaranteed by (16) and (17). Extension to variable velocity parallels (10), with face values being functions of their respective local Courant numbers.

For example, for first-order upwinding [9], transient interpolation modeling is based on a linear function

f ( ~ ) - - " ~ ( I ~ ; I - - 1#;1__1)

~- ~(~;1+1 -- 1#; 1)

for c > 0 (19)

for c < 0. (20)

Thus, for positive or negative c, the face-value ~br can be written in terms of quantities centered at the right CV face

1 ,, ,, n ( I ) (j~r(C) = ~ [(1#'i1+1 + t~ i ) - - SGN(c)(~bi+I - tbi )] = tbr , say (21)

and, of course, ~t is obtained by reducing all indexes by 1. Note that the first term in brackets is the sum of node values straddling the CV face, whereas the second term involves the difference of these values.

Similarly, the well-known Lax-Wendroff scheme [10] (second-order central TIM) corresponds to centered quadratic spatial interpolation

[ " ] f ( ~ ) = ~: (¢~1+1 --2 l~; '_l) + (lt)i'+l - 25,2 + ~b'~'_,) ¢ , (22)

and the corresponding right face value is

1 ~btr2~ 4~,(c) = ~ [(~b;'+l + ~b;')- c(4~'i'+1- ~b;')] = , (23)

valid for both positive and negative c values. Comparing (21) and (23), it is seen that these two schemes differ by the Courant-number-dependence of the coefficient of the first-difference term. Both give exact point-to-point transfer when c =- 1" 4,, = 4,1', 4't = ~bi'_ 1, 4"i '÷ ~ = 4,'i'-1; and similarly when c - - - 1.

The QUICKEST scheme [11] (third-order upwinding TIM), is based on upwind biassed cubic interpolation. As shown in Appendix A, the QUICKEST face-value can be written in

22 B.P. Leonard, The ULTIMA TE conservative difference scheme

terms of the canonical second-order (Lax-Wendroff) value together with an additional term

~b~3 , ,6(2 ' ( 1 - c 2 ) ( 2 SGN(c) 3) (24)

where 2 9 ~ is the (right-face-centered) sum of second-differences centered at i and i + 1 (or the difference of first-differences between i + 1/2 and i - 1 /2)

2 " - 2-h" " 2 ~ r = (q~i+2 "e'i+, "~- ~ i ) "~- (~tit+l t! t! t! tl

= ( 6 , + ~ . - 6 , + , ) - ( 6 , - 6 , _ , )

n tl t l t |

-- ~ i+2 -- ~ i + i - (~i "~" ¢~i-1

- 2~b;' + ~b" i - l )

(25)

and ~,3 is the difference of the above second-differences, i.e., the third-difference (centered at the right face)

3 ,, ,, " " ( 2 6 ) t$ r = t~i+2--3 ~ +3t~i - - ~ i - 1 i+1

Note that ( 2 @ ~ - SGN(c)8~)/2 is the upwind biassed second-difference, proportional to the upwind curvature. Conversely, for fourth-order central TIM (centered quartic interpolation), less weight is placed on 8 a. r

~4,=d,~a ) ( 1 - c 2 ) ( 2 c (27)

Second-order upwinding [12], based on upwind shifted quadratic interpolation, has a similar form to QUICKEST, but with a different coefficient of the upwinded curvature term:

' ~ ' 2 , ~ 8 , (28)

and Fromm's so-called 'zero-average-phase-error' method [12] is simply the average of (23) and (28), thereby cutting the curvature coefficient in half.

Fifth-order upwinding the sixth-order central follow a similar pattern (see Appendix A):

and

<b,s, = d~(4, + ( 1 - c2 ) (4 - c2) ( 4 S G N ( c ) ~ ) " 5! ~ ' 2 ~ (29)

dp(6) = &(4) + ( 1 - c 2 ) ( 4 - c 2 ) ( 4 c :) ' ~ ' 5~ ~ , - ~ ~ ' (30)

respectively, where 25~ 4 is the sum of fourth-differences

4 = 6'i'+3 vi+2 + 2~,+, + 24~;' 3~i'_ ~ + - 3 , h . . ,, 2 ~ , . - ~bi_ 2

and 85, is the face-centered fifth-difference

(31)

5=~b" -5~b" " -10qb i '+5 " " ~, i+3 i+2 + 10i+i ~b i - t - ,hi_2. (32)

B.P. Leonard. The UL TIMA TE conservative difference scheme 23

Seventh-order upwinding and eighth-order central are summarized by

and

dp(7,=q~(6,__(1-C2)(4--C2)(9-C2) ( 6 S G N ( c ) r 7 ) r r 7! 5~r 2 6 (33)

qb(8,=tb(6,_ ( 1 - c 2 ) ( 4 - c 2 ) ( 9 - c 2 ) ( 6 c 7) r ~-r 7! ~ r - - ~ 6 , (34)

respectively, where

and 2 ~ , 6 = ~b~,+4 _ 5tb",+3 + 94,,+2" - 5~b;'+, - 5~b;' + 9~b;'_, - 5 ~b,_ 2 ' ' + ~b,_ 3 " ( 3 5 )

1~i+2 35~b + 35~bi 21~bi_ l 2 ,+4- 7~b,+3 + 21 - i÷~ - + 74,1'_ - ~bi_3 • (36)

Note that for Nth-order central methods, the coefficient of 8~ -~ is proportional to c/N, as compared with SGN(c) /2 for the related ( N - 1)th-order upwind scheme.

Central or upwinded polynomial TIM methods of the above forms can, of course, be continued up to arbitrarily high order via straight-forward recursion formulas, based on binomial coefficients or Pascal's triangle (see Appendix A). Figure 1 shows the one- dimensional control-volume stencils involved as the order is increased. The stencils are shown for variable, possibility reversing, velocities, corresponding to (10), even though the test problems used in this study involve only constant velocity, positive to the right. For this reason, first-order upwinding, for example, involves the same three-point stencil as second- order central, corresponding to (21) and (23), respectively. QUICKEST, second-order upwinding, Fromm's method, and fourth-order central each involve a five-point stencil, when velocity reversals are allowed for. Four grid points are involved for each face, as seen from (24) and (28). This is the same stencil used by all (second-order) shock-capturing TVD schemes. Clearly, as the order of accuracy is increased, a wider stencil is necessitated: an Nth-order central scheme and its related ( N - 1)th-order upwind scheme require an (N + 1)- point stencil in one dimension.

,ram. ,ram, ram, ,m.

• [ . : . j • (a)

am,......

• . [ . i . • I - l u B n d

(b)

mmmm i • • O' • • • • b m ~ m ~ o

(e)

p ~ u m ' n ' I ! e • • • • a • i • • Q • ~ m mmma4

Id) Fig. 1. Control-volume stencils for varying, possibly reversing velocities. (a) First-order upwinding and second- order central. (b) Second- and third-order upwindin~.. Fromm's method, fourth-order central and TVD schemes. (c) Fifth-order upwinding and sixth-order central. (d) Seventh-order upwinding and eighth-order central.


1.2. Bench-mark test problems

The following three bench-mark test problems are selected on the basis of simplicity and ease of reproducibility, and are intented to represent basic characteristics of behavior that might be encountered in practice. A given numerical scheme is considered successful if it is able to simulate all three test problems to within some desired level of performance; if a scheme fails one or more of the tests, it is deemed unsatisfactory no matter how accurately it simulates any one of the other tests. Performance is judged on the basis of two basic criteria: (i) total absolute error (ABSERROR)

N

= E I ,1, (37) i=1

where e is the local error at each node

g = ~co rnpu ted - ~exact , ( 3 8 )

and (ii) the WAVINESS or total variation of error

N

E (39) i=1

In addition, it is usually desirable to monitor monotonicity; i.e., does an initially monotonic profile remain so? Strict monotonicity maintenance is a basic (in fact the only) constraint determining the nature of the universal limiter to be developed here. ,

The first test profile is a unit step change in ~ over one mesh widthS.. Initially, ~ = 0 everywhere to the right of a specified jump point; all other points, inc~;ding the upstream (left) boundary, are set at 1.0. The unit step profile is more fundamental ~han the isolated 'rectangular-wave', or box, used in some previous studies; e.g., Sweby [13] used an isolated rectangular box of width 20 Ax in addition to an isolated sine-squared function of the same width. The box profile, at best, merely gives twice as much information as the unit step; but for highly oscillatory methods, oscillations excited by the step-up interfere with those due to thc step-down, and the resulting complex wave-pattern is not as enlightening as that of the simple step simulation. The unit step is also a basic test of monotonicity, a fundamental aspect of advective modeling. A 'good' step simulation is one which monotonically resolves the step in a 'small' number of mesh widths- the smaller the (monotonic) 'numerical width', the better the method (for this test!).

The second test profile follows that used by Sweby [13], an isolated sine-squared wave of width 20 Ax

$(t=0)=sin2( wx ) 20 Ax

= 0 otherwise.

for 0 <- x <- 20 Ax,

(40)

This function represents a relatively smooth profile with a continuously turning gradient and a single local maximum; there is, of course, a step change in curvature on either side. In order

B.P. Leonard, The UL TIMA TE conservative difference scheme 25

to simulate practical situations, it is important to run the test problems over the same prescribed distance in all cases, irrespective of time-step (Courant number) or initial profile shape. In the tests described here, for example, the exact solutions advance by 45 mesh- widths. Two Courant numbers are used: c = 0.05 (900 time steps), representing 'small' At; and a 'moderate' value of c = 0.5 (90 time steps). The 25-time-step simulation used by Sweby was not long enough to see significant differences between the methods studied, or to allow their gross deficiencies to develop (which are typically worse at small Courant numbers, for a fixed distance).

The third test profile follows one used by Zalesak [14] that he attributes to B.E. McDonald. It consists of a semi-ellipse of width 2i w Ax, initially centered at ic,

tb i ( t = O) = ~ / 1 - ( i - ic )2 / i 2

= 0 otherwise.

if l i - icl < i,,,,

(41)

This is a rigorous test in that an initial (leading) step change in gradient is followed by a region of continuously changing gradient and finally by a trailing step. Methods which are oscillation- free in the simple step simulation may generate significant waviness ('stair-casing') just be'find the leading step or just ahead of the trailing step. The test used here differs slightly from that used by Zalesak in being 20 Ax wide to conform to Sweby's sine-squared profile, rather than 30 Ax; this does not have any significant qualitative effect on results.

Figure 2(a) shows the three initial profiles and the grid used for all tests. Numerical

mzm~ o

o •

o

(a)

o o o

o

. . . . . . . . . . . . .

0

X (b) I

0 1 2

Fig. 2. (a) Initial conditions, showing the unit step, an isolated sine-squared profile 20 Ax wide, and a semi-ellipse 20 Ax wide. (b) Exact solution after translating 45 Ax to the right.


boundary conditions are very simple: ~b~ = 1 at the first computed grid point. Higher order methods (above fourth) require the designation of pseudonode values (=1). In all cases, computation begins with grid-point 1. For convenience, the convecting velocity is normalized to u = 1 and Ax is taken as 1/100; but this is irrelevant to the computation, since the only parameter is the Courant number c. As mentioned, all tests are run so that the exact solution would translate a fixed distance, taken to be 45 Ax. Thus, the total number of update steps, N t, is related to the Courant number by

c = 4 5 / N t . (42)

The exact solutions are shown in Fig. 2(b).

1.3. Results for polynomial TIM methods

Figures 3-13 show the results of simulating pure advection of the three test profiles for first-order upwinding, the Lax-Wendroff scheme, second-order upwinding, Fromm's method, QUICKEST, fourth-order central, fifth-order upwinding, sixth-order central, seventh-order upwinding, eight-order central, and ninth-order upwinding, respectively. Each case is run at both a representative small Courant number ( c - 0.05, N t = 900) and a moderate Courant number (c = 0.5, N t - 9 0 ) . First-order upwinding is the only polynomial method which gives

(a) I x

_ L , % ._7 I (b) I

0 1

Fig. 3. First-order upwind results (a) c =0.05; step: ~ =5.21, ~//'= 1.88; sine-squared: I¢ =6.57, ~V= 1.88; semi-ellipse: ~? = 7.76, ~/~" = 2.00. (b) c = 0.5; step: ~ = 3.77, ~V = 1.83; sine-squared: i~ = 4.52, ~V = 1.46; semi- ellipse: i~ = 5.23, ~V'= 1.74.


**,e

(b) I x 0 1 2

Fig. 4. Second-order central (Lax -Wendro f f ) results. (a) c = 0 . 0 5 ; step: ~ = 5 . 2 0 , ~ ' = 4 . 5 4 ; sine-squared: ~g = 2.07, 'lip = 0.95; semi-ellipse: ~ = 4.17, '7#" = 3.23. (b) c = 0.5, step: ~ - 2.82, ~/' = 2.42; sine-squared: ~ = 1.44, ~ --- 0.65; semi-ellipse: ~ = 2.62, ~ = 2.08.

monotonic step resolution; but the width of the resolution is poor and the other profiles show the well-known effects of this method's inherently large global artificial numerical diffusion, which is worse at smaller Courant numbers, as seen in Fig. 3(a). The Lax-Wendroff method (Fig. 4) generates trailing oscillations (phase-lag dispersion) typical of central-difference methods; in these cases, phase lag is worse at smaller Courant numbers (compare Figs. 4, 8, 10 and 12). Clearly, these results are very disappointing.

Second-order upwinding gives rise to leading oscillations, as seen in Fig. 5. It was this observation that led to the idea of averaging the Lax-Wendroff method with second-order upwinding in an effort to cancel phase-lead and phase-lag, at least in some 'average' sense. Fromm's zero-average-phase-error method [12] indeed shows marked improvement (Fig. 6), with much less sensitivity to Courant number, as seen. Fromm's method actually cancels the leading dispersion term in the truncation error only at c - 0.5; in this case it is identical to third-order upwinding (QUICKEST), as seen by comparing Figs. 6(b) and 7(b). QUICKEST, which eliminates the third-derivative dispersion term entirely, gives markedly better performance at other Courant numbers. Note that the smooth (sine-squared) function is particularly well modeled. Fourth-order central is again highly oscillatory, with large ABSER- ROR and WAVINESS for each profile, as seen in Fig. 8. The fact that fourth-order central methods are substantially inferior to third-order upwinding is apparently not well-known, and


~- ................ ~ ~ ~ ;° o

(a) [ x

_ e ~ 6 A ~ - - _ , , , . - _ _ _

(b) [ x . . . .

0 1 2 Fig. 5. Second-order upwind results. (a) c = 0.05; step: ~ = 3.56, ~//= 2.32; sine-squared: ~g = 3.05, ~ ' = 1.22; semi-ellipse: ~ = 3.68, ~P= 2.22. ( b ) c =0 .5 ; step: ~ = 2.82, 74/.= 2.42; sine-squared: ~' = 1.44, ~/¢'= 0.65; semi- ellipse: ~ = 2.62, ~,~ = 2.08.

. _ _

O ~ (a)

O ~

(b)

x

. . . . . X

0 1 2

Fig. 6. Results using Frorfim's method. (a) c = 0.05; step: ~ = 1.51, 74/'= 1.89; sine-squared: ~ = 0.85, 0/4/" = 0.39; semi-ellipse: ~ = 1.70, 74"= 1.51. (b) c =0 .5 ; step: ~ = 1.41, 74P= 1.74; sine-squared: ~ =0 .26 , ~ ' = 0 . 1 5 ; semi- ellipse: ~ = 1.16, ~/V = 1.29,


(a)

(b) [ x 0 1 2

Fig. 7. Thi rd-order upwind ( Q U I C K E S T ) results. (a) c = 0.05; step: ~ = 1.66, ~" = 1.83; sine-squared: ~ = 0.41, W ' = 0 . 2 2 ; semi-ellipse: ~ = 1.41, W '= 1.40. ( b ) c = 0.5; step: $' = 1.41, ~ '= 1.74; sine-squared: ~ =0 .26 , ~//'= 0.15; semi-ellipse: ~ = 1.16, W '= 1.29.

o

(a) I

(b) ] x 0 1 2

Fig. 8. Four th-order central results. (a) c =0 .05 ; step: ~ = 3.22, W ' = 4 . 0 3 ; sine-squared: ~ =0 .21 , ~ = 0 . 2 1 ; semi-ellipse: ~ = 2.31, W" = 2.76. (b) c = 0.5; step: ~ = 1.57, W' = 2.09; sine-squared: ~ = 0.12, W" = 0.11; semi- ellipse: ~ = 1.21, W" = 1.55.


m (a)

t _ _. I x

(b) [ x

0 1 2

Fig. 9. Fifth-order upwind results. (a) c =0.05; step ~ = 1.26, ~1/'= 1.77; sine-squared: ~ =0.07, °/¥'=0.08; semi-ellipse: ~ = 0.95, off,= 1.31. ( b ) c =0 .5 ; step: ~ = 1.07, °/4/'= 1.68; sine-squared: ~ =0.05, ~E=O.06; semi- ellipse: ~ = 0.81, ~ = 1.22.

(a)

m (b)

-,¢p

..... J X

o 1 2

Fig. 10. Sixth-order central results. ( a ) c = 0 . 0 5 ; step: ~ = 2 . 4 9 , ~ ' = 3 . 8 3 ; sine-squared: ~ = 0 . 1 0 , ~F'=O.12; semi-ellipse: i~ = 1.70, ~/" = 2.50. (b) c = 0.5; step: ~ = 1.17, ~ - 1.93; sine-squared: f~ = 0.05, ~ = 0.06; semi- ellipse: f~ = 0.91, ~¢" = 1.40.


. . . . . . r . ~O ° .

., (a) [

o ,m ~n

(b) i x 0 1 2

Fig. 11. Seventh-order upwind results. ( a ) c =0.05; step: ~ = 1.05, ? ~ = 1.79; sine-squared: i~ = 0.04, ~ = 0.06; semi-ellipse: i ~ - 0 . 8 0 , ~W'= 1.28. ( b ) c =0 .5 ; step: ~ =0.93 , ~/#'= 1.64; sine-squared: ~ = 0.03, ~ ' = 0.04; semi- ellipse: Ig = 0.70, ~ = 1.16.

2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q

____,,_~.._8 ~,,,,,_,,..., o.._oo .o. 0 ~P -°o

, ......... ,_,.,,] i £. (a) [ x

2

- - - - , . ~ . . . I Z l . . ° o

(b ) I x

0 1 2

Fig. 12. Eighth-order central results. (a) c = 0.05; step: ~ = 2.23, ~¢" = 3.65; sine-squared: ~ = 0.06, ~ = 0.08; semi-ellipse: ~ = 1.46, ~/" =2.11. (b) c = 0.5; step: ~ = 1.02, ~f = 1.83; sine-squared: ~ = 0.03, ~¢/'= 0.04; semi- ellipse: i~ = 0.76, ~/" = 1.27.


_ ~ . n ° v

(a) x

. . . . . . . . . . . . ~ ~ - -~ _ __~_ee

ee~

(I)) I 0 1 2

Fig. 13. Ninth-order upwind results. (a) c = 0.05; step: ~---0.94, ~ - - 1 . 8 5 ; sine-squared: ~ ffi 0.03, ~V'ffi 0.05; semi-ellipse: ~ -- 0.78, ~/¢ - 1.28. (b) c -- 0.5; step: ~ -- 0.84, ~/~ -- 1.62; sine-squared: ~ -- 0.02, ~ = 0.03; semi- ellipse: ~'-=0.69, ~ - ~ 1.11.

one continues to hear of researchers who switch to central fourth-order schemes after experiencing 'difficulties' with second-order•

With fifth-order upwinding (Fig. 9), one begins to see a trend which continues to higher order: simulation of the smooth profile is excellent; as order increases, the step rise is steepest but the odd-order overshoots are larger and the even-order methods continue to be highly oscillatory, albeit with shorter wavelength; with the even-order central methods, significant waviness develops in the semi-ellipse simulation just behind the initial jump in gradient. These trends are seen by scanning across Figs. 7-13. Finally, note that central (necessarily even- order) methods are much more sensitive to Courant number. This is because the highest order term in the face-value expression is proportional to c rather than SGN(c); compare (23), (27), (30) and (34) with (21), (24), (29) and (33), respectively (also see Appendix A). All the methods considered here give exact point-to-point transfer at c-- 1, as seen from the formulas

• " " "+~ " For the higher order methods, for 6r, i.e., at c = l , ~ , = ~ , ~b t = ~ i _ l and ~ = ~ i - l . point-to-point transfer also occurs (over more than one mesh width) for larger integer values of c [15]; however, in the absence of modeled physical diffusion, these methods are not all stable over a continuous range, except for 0 < c ~ 1.


2. Normalized variable diagram and second-order 'monotonic' schemes

2.1. Normalized variables

Figure 14 shows a one-dimensional control-volume with attention focussed on one face (in this case, the left). In determining the effective face value, 4)s, the most influential nodes are the two straddling the face and the next upstream node, the latter depending of course on the flow direction at the face in question, i.e., the sign of u;. These three node values can be labeled (~D (downstream), ~u (upstream) and (~c (centrally located between the other two), as shown. Note the difference in definition of these nodes, depending on SGN(ur). In terms of original variables, there are clearly a very large number of cases to consider: combinations of positive or negative u r, positive or negative 4), and positive or negative values of gradient and curvature. Variations in sign, flow direction and scale can be normalized out by defining the normalized variable (at any point) as

$(x, t) = (6(x, t) - - (43)

Now, in a case when $I is a function of ~ , $c, ~b ~ and Courant number, the normalized face value is a function only of its adjacent normalized upstream node value and c

$/ " c) (44) C ) '

since the other normalized node values are constant:

$~j=O and , ~ " v'z) = 1. (45)

Equation (44) includes first-order methods, second-order central and upwind schemes and third-order upwinding, in addition to second-order (and third-order) shock-capturing algorithms. Higher order methods, involve more distant nodes but the normalized ~b I will still depend most strongly on 4)[.

2.2. Second.order time averaging

In general, for transient-interpolation-modeling methods based on (11), the time accuracy of the resulting control-volume algorithm, (18), is the same order as the degree of the spatial

eU

v ~ ~ ~ ~ ~ v

I I

uf - ,.. CV | uf . . , , - cv :

(a) uf =. o, (b) uf < O.

Fig. 14. Definition of upstream (U), downstream (D) and central (C) node-values, depending on the sign of u r


interpolation used in (12). Many advection schemes in common use (including TVD schemes) are based on second-order time averaging, which can be made explicit by straight-forward integration. Specifically, for spatially second-order-accurate methods, the deviation term in (4) is formally neglected; thus, from (8), the effective face values are just the estimated time averages. If these time averages are based on locally advected face-value linear behavior spatially, the method is second-order accurate in time, as well. For example, if ~; is the instantaneous face value, the time-averaged face value over At is

1 f a t . uAt ( oo) '~r=~'7 = ~ J 0 *A*)d*=~ ' : - - -~ ~ : ' (46)

where ~b} is the estimated face value at time-level n. In most cases, the spatial gradient is estimated as

(47)

so that

t l /'1

~b r = ~b: - c(~b~ - 6 c ) , (48)

or, in terms of normalized variables,

~r -- (I - c)#~ + c~:, (49)

where, now, the linear Courant number weighting is explicit and ~f depends only on ~c,

: =f~(~). (50)

Figure 15 shows the normalized variable diagram (NVD), plotting (in this case, linear) functional relationships in the form of (50), for

(I) First-order upwinding (IU):

~/ c; (51)

(2) The Lax-Wendroff method (2C):

/ = ½(1+ qbc), (52)

(3) Second-order upwinding (2U):

= ~ ; (53)

(4) Fromm's method (2F):

q~ = ~ + q~c. (54)


1D-x\

Fig. 15. Normalized variable diagram showing ~ as a function of ~ for several 'linear' schemes: first-order upwinding (1U); Lax-Wendroff (2C); second-order upwinding (2U); Fromm's method (2F); and first-order downwinding (ID).

Note that the latter three spatially second-order methods all pass through the point (0.5, 0.75); in fact, any method whose ~ ( ~ c ) function passes through this point with a finite slope is (at least) second-order accurate in space, since 4~ can then be written

~ n ~ n 'q"

6 : = ½(1- tbc)- CF(1- 26c) , (55)

where the curvature factor CF is a constant or, in the case of nonlinear schemes, a function of ~. ~c. This is more apparent in terms of unnormalized variables

or

n n n ~i = ½(~; + ~ c ) - CF(~; - 2~c + ~u)

@S = ½(t#D + @c) - CFt'~x2/: Ax2 + ""'

(56)

(57)

thus deviating from the second-order-accurate linear interpolation by second or higher order terms, provided CF is finite. Note that first-order upwinding, (51), and


(5) First-order downwinding (1D):

~j. = 1, (58)

cannot be written in the form of (55) for finite CF. The linear functions shown in Fig. 15 are in the form of (50), i.e., the estimated face value

at time-level n is a function of 4)c. It is also important to portray the time-averaged face value in the same way, ~b¢ (no superscript), according to the linear weighting, (49). Figures 16-18 show the NVDs corresponding to ~br(~bc, c) for the Lax-Wendroff, second-order upwinding and Fromm methods, respectively. The functions are shown for five different values of

! ~ 3_ and 1. The zero-value is distinguished by a dashed line, Courant number: c--->0, c = 4, ~, 4

since this can only be approached in an explicit calculation. The dashed line also represents ~} =fn(~c) , as seen from (49) and Fig. 15. Note that all these methods revert to 4,¢ = ~bc

P! (i.e., 4'¢ = tbc) when c = 1, giving exact point-to-point transfer, as discussed previously.

2.3. Nonlinear 'shock-capturing' schemes

For lack of better categorizing terminology, a number of currently popular algorithms have become known as 'shock-capturing' schemes. When applied to the pure advection problems

Sf

c - . 0 ~

~ C = I . 0

~ C = 0 , 5

1

Fig. 16. Normalized variable diagram (NVD) for the Lax-Wendroff (second-order central) method. ~r as a function of ~" = ~c for c ~ 0 , c=0.25, c 0.5, c=0.75 and c = 1.

Sf /

/ /

i /

c - 0 ~ \ ~ C = I . 0

\

~-C = 0.5

1

/ I

Fig. 17. NVD for second-order upwinding. ~: as a function of ~c for c - , 0 , c=0.25, c=0 .5 , c=0 .75 and c = 1.


~f

c - 0 - - ~

X _ c = 1, 0

' "X- -c = 0.5

Fig. 18. NVD for Fromm's method. ~i as a function '~ ' t ! of 4~c for c---> 0, c = 0.25, c = 0.5, c = 0.75 and c = 1.

Cf

1

//

i

ZY, !: ~ / ~ c = 1.0

k..- c = 0 .5

I

Fig. 19. NVD for the Minmod scheme. ~r as a function of ~ . and c.

studied here, these second-order methods (in space and time) can be portrayed either in the ~ NVD or in the ~y NVD, corresponding to (50) or (49), respectively. However, in contrast

to the linear functions of Fig. 15 (or of Figs: 16-18), these methods are distinguished by nonlinear functional relationships between ~b l and ~', although they remain linear in c according, to (49). All schemes revert to first-order upwinding outside the monotonic range,

" ¢ PI

i.e., for ~ < 0 or ~c > 1. They all pass through the origin (0, 0) and the point (1, 1) in either NVD. For the 4~(~bc) NVD, they all pass through (0.5, 0.75), as required for second-order methods.

Figure 19 shows the so-called minimum-modulus (Minmod) method [6], which is seen to follow second-order upwinding for 0~ < ~b c ~<0.5 and Lax-Wendroff for 0.5 ~< ~c < 1. A related method used by Chakravarthy and Osher [16] consists of second-order upwinding combined with first-order downwinding at time-level n. This is shown in Fig. 20. The MUSCL scheme of Van Leer [5] is shown in Fig. 21; it consists of Fromm's method in the 'smooth' (small curvature) region near ,7,,, -,c "> 0.5, with piecewise linear deviations to pass through (0, 0)

t ! and (1, 1). In terms of ~bl, the prescription for this method is (1) A sufficient (although not necessary) 'monotonic' limiter:

~ --2~c for 0 ~< ~" 1 , c ~ , (59)

(2) Fromm's method in 'smooth' regions:

n

4 , (60)


Sf

7

c - 0-~ / ~

~ - c = 1.0

=0.5

w

Fig. 20. NVD for the Chakravarthy-Osher scheme. ~r as a function of ~ , and c.

Sf

c - 0 7

~--C = 1.0

= 0 .5

1

Fig. 21. NVD for MUSCL. ¢~ras a function of ~c and C.

(3) First-order downwinding:

~" = 1 for ~ <~ ,7,,, r . 'c ~<1; (61)

with first-order upwinding elsewhere, as usual. Another scheme due to Van Leer [2] can be described by replacing the piecewise-linear function of MUSCL with a curved line (a parabola) for ~! in the monotonic range; this has the same slope as MUSCL for ,7," = 0 and 1 as seen w f + - , in Fig. 22. For convenience, this scheme will be referred to as Van Leer's 'curved-line advection method' (CLAM); in the Sweby diagram (see Appendix B) the function forms portion of a hyperbola, thus the scheme is sometimes referred to as Van Leer's 'harmonic' advection method. A scheme developed by Roe~[0], nicknamed 'Superbee', is shown in Fig. 23. Again piecewise linear, the ~ function (as ~ increases from 0) consists of a portion of slope 2, a portion of slope ½ (following Lax-Wendroff), a portion of slope ] (second-order upwinding), and a portion of zero slope (first-order downwinding). This is considered to be one of the most 'compressive' of the second-order schemes with respect to its narrow step resolution, as will be seen. This idea can be taken to extremes, however; another formally second-order scheme, which might aptly be called 'super-C' (for 'compressive') is shown in Fig. 24; and an extremely 'compressive' limited first-order downwinding scheme, 'Hyper-C', is shown in Fig. 25. The Super-C scheme requires ¢br to be the smaller of the upper universal limiter boundary (developed in the next section) and Lax-Wendroff, i.e., the smaller of

~r=~clc and ~ r = ½ ( l - c ) + ½ ( l + c ) ¢ ~ c (62)


Sf

7

n

.y//.c:,o

f ~ - c =0 .5

1

Fig. 22. NVD for CLAM. ~r as a function of ~c and c.

Cf

//

c - 0---xx

I! "7../ ~ c = 1.o

zt 7 " C=0.5

23. NVD for Superbee. ~r as a function of ~c Fig. and c.

m ,

Of

c - 0 - - I

~ C = 1 . 0

~--C = 0 .5

Sf

C -" 0 -J]

I I I

~ c = 0,25 ~ c =0 .5

\ \ r c = o.z5

~-C = 1.0

Fig. 24. NVD for Super-C. ~I as a function of g ~: and c.

Fig. 25. NVD for Hyper-C. ~r as a function of ~ ~. and c.


" ~ ' t l for 0 ~< ~b c ~< ½; with limited second-order upwinding, i.e., the smaller of

q~I=l and t~ I = ½ ( 3 - c ) ~ c (63)

"~' t l for ½ < 4~c ~< 1, with first-order upwinding elsewhere. Hyper-C (limited first-order downwinding) simply requires ~b I to be the smaller of

~c/C and 1 (64)

in the monotonic regime, with ~; = ~c elsewhere, as usual. Finally, note that the Courant- number-dependence of Super-C and Hyper-C is no longer linear.

2.4. Results for the nonlinear schemes

Figures 26-32 show the results of simulating the three test problems with each of the nonlinear schemes just described; again, the two representative Courant numbers (0.05 and 0.5) are used. As seen in Fig. 26, although Minmod resolves the step monotonically, the structure is not at all sharp and the other profiles are rather diffusive, especially at smaller Courant numbers. In the Chakravarthy-Osher simulations, shown in Fig. 27, the leading-edge steepening effects of the first-order downwinding are seen, with concomitant clipping due to first-order upwinding; again, profiles are more diffusive at small c values. The two Van Leer schemes, MUSCL and CLAM (Figs. 28 and 29), give similar results to each other, as perhaps

.,%--- (a)

I

~ . . . . . . . .

(b) x

o I 1 2

Fig, 26. M i n m o d results, (a) c = 0,05; step: ~ = 2,34, ~/V = 1,70; s ine-squared: ~ = 1.87, ~ = 0.75; semi-e l l ipse: ~e = 2,51, ~lf'= 1.37. (b) c = 0.5; step: ~ = 1.85, ~/t p = 1,60; s ine-squared: ~' = 1.05, ~" = 0.54; semi-el l ipse: ,~ = 1.86, 71 r = 0,27,

B. P. Leonard, The ULTIMATE conservative difference scheme 41

1 i

0 (a)

K I

_ I k . . . . . \ _

( b ) [ x

0 1 2

Fig. 27. Charkravar thy-Osher results. (a) c = 0.05; step: ~ = 2.09, ~ = 1.73; sine-squared: ~ = 1.55, ~W" = 0.74; semi-ellipse: ~ = 2.16, og/,= 1.59. (b) c = 0 . 5 ; step: ~ = 1.60, ~ = 1.64; sine-squared: ~ = 0.87, ~ / ' = 0.64; semi- ellipse: ~ = 1.66, ~ = 1.42.

1 ( a ) I - x

(b) I x

0 1 2 Fig. 28. MUSCL results. (a) c = 0.05; step: ~ = 1.45, ~ = 1.60; sine-squared: ~ = 0.58, ~P= 0.40; semi-ellipse:

= 1.44, °W'= 1.36. (b) c =0 .5 ; step: ~ = 1.18, wW= 1.51; sine-squared: ~ =0 .26 , ~ ' = 0 . 2 0 ; semi-ellipse: ~ = 1.09, o/#,= 1.19.


(a)

(b) [ x

0 1 2

Fig. 29. C L A M results. (a) c = 0.05; step: ~ = 1,67, ~/' = ] .63; sine-squared: ~ = 0.83, ~/" = 0.50; semi-ell ipse: = 1.59, ~ ' = 1.35. (b) c =0 .5 ; step: ~ = 1.36, ~1~'= 1.53; sine-squared: ~ =0.49, ~ ' = 0.39; semi-ellipse: ~ =

1,23, ~" = 1.23.

(a) ] x

n

(b) _ I x

0 1 2

Fig. 30. Superbee results. (a) c = 0.05; step: ~ = 0.87, ~P = 1.34; sine-squared: ~' = 0.40, ~/" = 0.40; semi-ellipse: = 1.44, '/V= 1.39. ( b ) c = 0 . 5 ; step: ~ =0 .85 , ~/V= 1.35; sine-squared: ~ =0.31 , ~# '=0.31; semi-ellipse: ~ =

1.36, 7t/'= 1.34.

B.P. Leonard, The ULTIMA TE conservative difference scheme 43

(a) [ x

(b) I x

1 2

Fig. 31. Super-C results. (a) c = 0 . 0 5 ; step: ~ = 0.60, °ttr= 1.18; s ine-squared: ~ = 0.28, ~ f ' = 0.32; semi-el l ipse: ~g = 1,23, o///, = 1.37. (b) c = 0,5; step: ~ = 0.60, ~ = 1.20; s ine-squared: ~ = 0.27, ~f" = 0.32; semi-ell ipse: ~ =

1.38, ~ = 1.39.

m

(a)

(b)

x

I

n flip

. . . . . . . . J = =

I 1

X

Fig. 32. Hype r -C results . (a) c = 0.05; step: ~ = 0.00, ~ = 0.00; s ine-squared: ~ = 2.69, o/~ = 2.38; semi-el l ipse: = 1.34, ~ = 2.12. (b) c = 0.5; step: ~ = 0.00, ~#" = 0.00; s ine-squared: ~ = 2.69, ~/" = 2.76; semi-el l ipse: ~ =

3.10, ~ = 2.89.


expected from the qualitative similarity of their NVDs (Figs. 21 and 22). The MUSCL scheme, in particular, is one of the more successful of the well-known second-order non-linear schemes, considering overall performance for all three test profiles. Once again, both MUSCL and CLAM deteriorate at small Courant-number values, due primarily to the unnecessarily restrictive TVD limiter, (59).

Superbee (Fig. 30) gives significantly sharper results for the step simulation at all c values. The smooth-function (sine-squared) simulation is slightly better than that of MUSCL; however, there is a degree of artificial steepening inherent in this method, as seen in the semi-ellipse computation. In the presence of rapid changes in gradient (large curvature)- near the leading and trailing edges of the profile- the scheme has a tendency to convert all gentle slopes into sharp steps followed by plateaus. This defect is purposely taken to extremes with Super-C (Fig. 31) and Hyper-C (Fig. 32). Clearly, in terms of step performance, Super-C

A

N . . J .==I

N

1.00

i !

w I l l ¢/)

.10

.01 i

\

I I I l

3-~

[ I

I

2

I ' l i l ,

I , I , i i I i l i l t 1,0 10.0

ABSERROR (STEP)

Fig. 33. Absolute error for the semi-ellipse profile (upper curves) and sine-squared profile (lower curves) plotted on a log-log scale against absolute error for the step, with Courant number as a parameter ranging from 0.01 to 0.978, with values shown at c =0.1, c = 0.5 and c =0,9. Curves show results for (1) Minmod, (2) MUSCL, (3) Superbee and (4) Super-C, Arrows show direction of increasing Courant number.


supersedes Superbee and Hyper-C supersedes Super-C. Super-C also does an excellent job of simulating the sine-squared profile; however, the semi-ellipse results are rather bizarre, showing stair-casing for small c values and gross artificial steepening at c = 0.5. Although Hyper-C gives virtually exact step simulation at all c values, the other profiles are totally corrupted. These schemes were included to show the futility of designing a method on the basis of a single criterion (in this case, sharp monotonic resolution of steps); consequently the almost incredible step simulation of these artificial-compression methods should not be used as a standard for judging the overall performance of truly robust schemes.

In order to show the effect of Courant number over a wider range, Fig. 33 gives a log-log plot of ABSERROR for the sine-squared profile (lower curves) and the semi-ellipse (upper curves) versus ABSERROR for the step simulation for Minmod, MUSCL, Superbee and Super-C, with Courant number as a parameter ranging from 0.01 to 0.978, with points shown at values of 0.1, 0.5 and 0.9 on each curve. At very small c values, Minmod produces large errors for all test profiles. The other schemes' semi-ellipse errors are comparable, with step errors decreasing in the order: MUSCL, Superbee, Super-C. The sine-squared error follows the same order. Superbee and, in particular, Super-C, show much less sensitivity to Courant number than the other schemes. Although all methods' trajectories in this plane approach the origin (exact point-to-point transfer) as c---> 1, the tendency is rather slow, and even at c = 0.9, the sine-squared error in particular is unacceptably large, compared with what can be achieved with higher order methods.

3. The universal limiter for arbitrarily higher order schemes

,¢•F! The normalized-variable plane, with ,,c as abscissa and ~r (no superscript) as ordinate, can be used to construct a very simple diagram representing constraints on the effective time- averaged normalized face value so as to guarantee maintenance of monotonic profiles, thereby suppressing extraneous overshoots or nonmonotonic oscillations, but allowing arbitrarily high resolution depending on the formal order of accuracy (in both space and time) of the base method. As seen in the previous section, nonlinear second-order schemes (linear in c) can be represented by a single curve in the ($c, St) plane for any fixed value of the Courant number. This is true for the nonlinear third-order scheme as well (to be described), but the Courant-number dependence is then also nonlinear, as will be seen. All the nonlinear second-order schemes (except Super-C) satisfy Sweby's sufficient TVD criteria [13] (translated into present notation)

and ~r =~" ;=~"c for~"c ~<0 and ~">~c 1

" "

~< '* " 1] forO~<~bc~<l y ~ min[24~c,

(65)

(66)

In particular, all nonlinear functions pass through (0, 0) and (1, 1). These criteria are in terms of &~; they are thus limited to temporally second-order schemes linear in c, according to (49).

To allow higher order accuracy, it is extremely important to work directly with the effective time-averaged normalized face value &;, rather than 4'~. By imposing simple monotonicity- maintenance criteria, much less restrictive constraints- the universal l imiter- are placed on


the allowable face values. The computational strategy is then, in principle, as follows: " v i i

(1) formulate Sy by some desired high-order method; (2) compute the actual normalized $c value and the intended normalized £/i I value, (3) if Sy falls within the allowable universal limiter range, simply proceed; (4) if ~b I lies outside this range, reset (limit) its value to that of

~ " n the nearest constraint boundary at the given ~bc value; (5) reconstruct the new ~b t. from ~I; (6) repeat the procedure for all other faces; (7) update according to (10). In actual practice, it is less expensive computationally to carry out this strategy directly in terms of unnormalized variables, as will be described in detail.

3.1. Monotonicity-maintenance criteria a~'l, I

Figure 34 shows normalized node values with ~b c in the monotonic range, 0 ~< ~ ~< 1. As suggested by the cross-hatching, monotonic behavior requires necessary conditions on ~t'"

~c ~ ~I. ~ 1 (67) N

and on the corresponding face value of the adjacent upstream CV face, 4~u:

o (68)

Consider (9), written for ~ in normalized variables

c = 4 , c - C (69)

In order to maintain monotonicity, the new Oc value must be constrained by

¢ • , . + t .7.,.+I .7,.+I (70) U ~ w C ~ W D ,

For pure advection at constant velocity, the right-hand inequality is less restrictive than (~r ~ d)~:, but the left-hand inequality results in

.- ~ 1 .-, .7.,+t ~r ~ ~. + c ( ~ c - , - u ) . (71)

£ - - - ' -

. . , / : ]°' I

I, I I f . . = w , ~ . . = ~ I I , m , . = , . I

I CONIROL I I ~ uf I VOLU~E I I, . . . . . . . . . J

Fig. 34. Normalized node-values in the case of locally monotonic behavior. Hatching shows necessary conditions on the face value of interest, ~t, and on the corresponding upstream CV face value, ~u.


~' ,r~n+ 1 And since ~b u is nonnegative and ,p. u a n d ~n+l U = 0 , i . e . ,

nonpositive, the worst-case condition is given by ~u = 0

~ r < ~ c l c for 0 < ~ c ~ < l . (72)

This, combined with (67)

4~c ~< 4~r <~ 1 for 0~< 4)c~< 1, (73)

N/I n constitutes the universal limiter in the monotonic range of ~bc. For ~c < 0 or >1, numerical experimentation has shown that the simple condition

=4)c for ~bc<0 or ~bc>l (74)

gives satisfactory overall performance; this, of course, is equivalent to first-order upwinding as used by other nonlinear (second-order) TVD schemes. It does not erode the accuracy of the overall scheme, which is determined solely by behavior in the smooth region, near ~bc ~ 0.5.

The universal limiter, (72), (73) and (74):is shown in diagrammatic form in Fig. 35; the Courant-number-dependent boundary, ~; = ~bc/C, is shown dashed to stress the fact that its

/,]1

,L

Fig. 35. Normalized variable diagram showing the universal limiter boundaries. The dashed boundary has a Courant-number-dependent slope of 1/c. The case shown is for c = 0.2.


slope changes with different values of c. Note that fer c--->0, this boundary approaches the vertical axis; while for c = 1, it degenerates into ~b¢ = q~ c (everywhere), corresponding to exact point-to-point transfer as usual. For reference to previous work, the corresponding criteria in terms of Sweby's variables, r and ~p, are given in Appendix B.

3.2. U L T I M A T E strategy (normalized variables)

For clarity, the precise steps in applying the universal iimiter to transient interpolation modeling of the advective transport equations are given, as follows, using normalized variables. (1) Designate upstream (U), downstream (D), and central (C) nodes on the basis of SGN(u/)

for each face in turn. f l • , n (2) Compute DEL = ~ - ~btj, if IDELI < 10 -5 (say) set 4~: = ~bc and proceed to the next

face. (3) Otherwise, compute ~c = (~bc - ~b~j)/DEL; if this is less than 0 or greater than 1, again

set ~b¢ = ~b c and proceed. (4) If not, compute ~b r =(d~/- O~)/DEL, where ~¢ is based on a desired high-order TIM

method. ~ ,~n ~ ~/. ~n (5) If~ ~¢< vc, reset (the lower limit) Or = ~c; if .> 6c/C, reset (the upper limit)

6: = if >1 , reset (the absolute upper limit) d~: = 1. (6) Reconstruct ~b: = Of DEL + ~b~j (this is the face value used in the update algorithm). (7) After finding all face values in this way, explicitly update according to (10).

33. U L T I M A T E strategy (unnormalized values)

In order to avoid divisions and multiplications involved in constructing normalized variables and reconstructing the unnormalized variables, it is better to work directly with the unnormalized variables. (1) Designate upstream (U), downstream (D) and central (C) nodes on the basis of SGN(u:)

for each face in turn, and compute DEL = ~b~- ~ and ADEL = [DEL I for each face. (2) Compute ACURV=Iff~-24~=+4~j I for each face; if ACURV~>ADEL (non-

monotonic), set ~b: = ~b c and proceed to the next face. (3) Compute the reference face value 4~R~F = tb~ + (4~C -- tb~)/C for each face. (4) Set up some desired high ordel face value ~:. (5) If DEL>0, limit 4,: by 4'c below and the smaller of ~Rm: and ~ above. (6) If DEL < 0, limit if: by ~bc above and the larger of ~t~EF and ~ below. (7) Update according to (10).

In order to get some feeling for the ULTIMATE strategy, Figs. 36-39 show normalized -.]h(4,c, c) for the universal limiter applied to Lax-Wendroff, second- variable diagrams, ~ : - n

order upwinding, Fromm's method and the third-order QUICKEST scheme, respectively. Note similarities (and differences) between Figs. 37 (ULTIMATE second-order upwinding) and 20 (Chakravarthy-Osher), and between Figs. 38 (limited Fromm) and 21 (MUSCL). Also note qualitative siniilarities between the limited versions of Fromm's method and QUICKEST (Figs. 38 and 39), respectively; they are identical at c = 0.5 (as well as at c = 1, of course).


Sf

C - - 0 - 7 I

~-C = 0,5

L-C = 1.0

sf

C - - 0 - ! i ~ " L.-c = 1.0

L c = 0 .5

Fig. 36. NVD for ULTIMATE-Lax-Wendroff. Fig. 37. NVD for ULTIMATE-second-order- upwinding.

if

C - 0 - /

L c = 0 , 5

~-C = 1.0

Sf

c 0 , / / / / , ~ - c = 1.0

/

" . . , / / / / ~ - c - - 0, s

Fig. 38. NVD for the ULTIMATE-Fromm method. Fig. 39. NVD for ULTIMATE-QUICKEST.


3.4. Results for the ULTIMATE schemes

Figures 40-49 show the results for the three standard test problems obtained by applying the universal limiter to the Lax-Wendroff scheme, second-order upwinding, Fromm's method, QUICKEST, fourth-order central, fifth-order upwinding, sixth-order central, seventh-order upwinding, eighth-order central, and ninth-order upwinding, respectively, each for the two Courant numbers c =0.05 and 0.5. Note the inadequacies of the limited Lax-Wendroff scheme (Fig. 40) and limited second-order upwinding (Fig. 41) similar in some respects to the nonlinear shock-capturing schemes. Especially note the poor performance at the smaller c value; the reversed asymmetry of the profiles corresponds to the respective asymmetry of these two methods' NVDs (Figs. 36 and 37).

The limited Fromm method (Fig. 42) is clearly an improvement over the simple second- order schemes, and has qualitatively similar performance to limited QUICKEST, as expected from the similarity of their NVDs (Figs. 38 and 39). The limited QUICKEST scheme (Fig. 43) gives results which are probably entirely adequate for most practical situations. The sine- function error, in particular, is now within tolerable bounds. Although the limited fourth- order method (Fig. 44) gives lower ABSERROR for each profile, there is a clearly discernible increase in WAVINESS in the semi-ellipse simulation, especially at small Courant number values; this is a typical shortcoming of even-order methods. Overall, the limited-QUICKEST scheme seems to be the best of the schemes using the five-point stencil of Fig. 1; it is certainly far superior to any of the second-order shock-capturing schemes (which involve the same

(a) I

(b)

I X

1 2

Fig. 40. ULTIMATE-Lax-Wendroff results. (a) c=0.05; step: ~ = 1.79, ~1/'= 1.69; sine-squared: ~ = 1.09, ~¢/'= 0.70; semi-ellipse: ~ = 1.79, ~//'= 1.53. (b) c = 0.5; step: ~ = 1.57, ~W = 1.64; sine-squared: ~ = 0.88, W'= 0.60; semi-ellipse: ~' = 1.63, ~///" = 1.47.


_

(a) I x

(b)

0 1 2

Fig. 41. U L T I M A T E second-order upwind results. (a) c = 0.05; step: ~ = 2.09, o/4/, = 1.73; sine-squared: ~ = 1.57, °/4/'=0.73; semi-ellipse: ~ =2 .17 , o/¢,= 1.59. (b) c = 0.5; step: ~ = 1.57, °/4/' = 1.64; sine-squared: ~ = 0.92, o/~= 0.50; semi-ellipse: • = 1.60, 'If/' = 1.49.

2

1

m

(b)

(a) I x

x

0 1 2

Fig. 42. U L T I M A T E Fromm method. (a) c = 0.05; step: ~ = 1.41, ~ = 1.60; sine-squared: ~ = 0.53, ~ = 0.36; semi-ellipse: ~ = 1.42, o/¢,= 1.34. (b) c = 0.5; step: ~ = 1.12, ' IF= 1.50; sine-squared: * = 0.13, ~ =0 .09 ; semi-

ellipse: ~ = 1.01, ~ = 1.16.


(a) [ x

(b) j

0 1 2

Fig. 43. U L T I M A T E QUICKEST results. (a) c : 0.05; step: ~' = 1.30, ~ : 1.60; sine-squared: ~ : 0.23, ~W: 0.14; semi-ellipse: ~ = ].23, ~ = 1.24. ( b ) c : 0 . 5 ; step: ~ : 1.12, ~ : 1.50; sine-squared: ~ =0.13, ~ = 0 . 0 9 ; semi-ellipse: ~ = ].01, ~ : 1.16.

_ (a)

A I

G ,

O - -

(b) I X

0 1 2

Fig. 44. U L T I M A T E four th-order central results. (a) c -- 0.05; step: ~ - 1.01, ~/" - 1,45; s ine-squared: ~ = 0.20, = 0.20; semi-el l ipse: f~ = 1.04, ~ = 1.29. (b) c = 0.5; step: i~ = 0.92, ~ = 1.39; s ine -squared : ~ = 0.12, ~ =

0.13; semi-ell ipse: f~ = 0.90, ~/" = 1.18.


t A ...... (a)

0 m

(b) j x

0 1 2

Fig. 45. ULTIMATE fifth-order upwind results. (a) c =0.05; step: ~ = 0.91, ~//'= 1.38; sine-squared: ~ = 0.14, ~¥'=0.14; semi-ellipse: ~=0 .81 , ~/V= 1.12. ( b ) c = 0 . 5 ; step: ~=0 .83 , ~#'= 1.32; sine-squared: ~' =0.09, ~W'= 0.09; semi-ellipse: ~ = 0.72, °/4/" = 1.07.

(a) [ x

(b) x

I 1 2

Fig. 46. ULTIMATE sixth-order central results. (a) c =0.05; step: i~ =0.82, ~ = 1.32; sine-squared: i~ = 0.16, ~f '=0.17; semi-ellipse: ~ =0.83, ~W= 1.18. (b) c=0 .5 ; step: f~ =0.77, ~/f'= 1.27; sine-squared: i~ =0.08, °/4/'= 0.09; semi-ellipse: ~ = 0.72, ~ = 1.10.

54 B.P. Leonard, The ULTIMATE conservative difference scheme 2[ 1

(a) x

(b) I x

1 2

Fig. 47. U L T I M A T E seventh-order upwind results. (a) c = 0.05; step: ~ = 0.78, °/V = 1.28; sine-squared: fg = 0.12, ° / ¢ - 0 . 1 3 ; semi-ellipse: ~ =0 .72 , ~t/'= 1.09. ( b ) c = 0 . 5 ; step: ~ = 0 . 7 3 , ~¢'= 1.23; sine-squared: ~ = 0 . 0 9 , ~W= 0.12; semi-ellipse: ~ = 0.66, ~¢" = 1.05.

R

(a)

I

'1" 1

(b) I x

o 1 2

Fig. 48, U L T I M A T E eighth-order central results, (a) c = 0.05; step: ~ = 0.73, ~/' = 1.24; sine-squared: i~ = 0.15, ~/" = 0.19; semi-ellipse: ~' = 0.75, ~#" = 1,16. (b) c = 0.5; step i~ = 0.70, ~" = 1.20; sine-squared: g' = 0.09, °/4/" = 0,12; semi-ellipse: ~ = 0.67, ~V' = 1.10,


(a) I x

2 m J' 0

(b) x I 1 2

Fig. 49, ULTIMATE ninth-order upwind results. (a) c = 0.05; step: ~ = 0.71, ~//~= 1.20; sine-squared: ~ = 0.I1, °///'=0.12; semi-ellipse: g' =0.69, ~W'= 1.08. (b) c =0.5; step: ~ =0.69, 7t.'= 1.16; sine-squared: ~ = 0.09, ~ ' = 0.13; semi-ellipse: ~ = 0.63, ~" = 1.06.

stencil). The artificial waviness of the limited fourth-order method (which also uses this stencil) detracts from an otherwise excellent performance.

If better step resolution is required, the limited fifth-order upwinding scheme (Fig. 45) gives highly satisfactory results, although of course this requires a seven-point stencil in general (allowing for velocity reversals). The limited sixth-order central method (which uses the same seven-point stencil), Fig. 46, gives slightly better step resolution, but worse performance for the semi-ellipse (in terms of both ABSERROR and WAVINESS) and a slightly higher ABSERROR in the sine-squared simulation at smaller Courant numbers (due to slight asymmetric clipping, typical of even-order methods).

The higher order schemes follow a predictable pattern, with better step resolution, and almost exact smooth-function simulation, but with annoying waviness in the challenging semi-ellipse problem- now even noticeable in the upwind schemes, but still much worse with the even-order central methods, especially at small c values. Clearly, the ULTIMATE strategy could be continued to arbitrarily high order, either with polynomial TIM schemes of the type considered here or with alternate forms of interpolation such as splines, for example. The only stipulation is that the base method must be explicit in determining the intended 'h i - step (4) of the ULTIMATE strategy.

Once again, to see the effect of Courant number over the complete stable range, Fig. 50 shows ABSERROR of the sine-squared and semi-eUipse profiles plotted on a log-log scale against ABSERROR of the step for the ULTIMATE Fromm, QUICKEST, fifth- and seventh-order upwind schemes, with Courant number as a parameter ranging from 0.01 to


Q . . . .=. . . I

1 .00 i . u

P ~

' " .10

LOJ Z N ¢J')

~ e

.01 |

I ' I ' I i I ' I ' I ' I I I '

3-~dl 4-,, t

~ - 2

I i l t I l l n l t I i I , I ~ 1 ,0 10 ,0

ABSERROR (STEP)

Fig. 50. Absolute error for the semi-ellipse profile (upper curves) and sine-squared profile (lower curves) plotted on a log-log scale against absolute error for the step, with Courant number as a parameter ranging from 0.01 to 0.978, with values shown at ~.---0.1, c = 0.5 and c =0.9. Curves shown results for (1) ULTIMATE Fromm, (2) ULTIMATE QUICKEST, (3) ULTIMATE fifth-order upwinding, and (4) ULTIMATE seventh-order upwinding. Arrows indicate direction of increasing Courant number.

0.978, with points shown at c = 0.1, 0.5 and 0.9 on each curve. For clarity, the ULTIMATE even-order schemes have been omitted. The limited version of Fromm's method is perhaps of academic interest (being slightly better than MUSCL in the region near c = 0.5); but since the limited QUICKEST third-order upwind scheme requires the same stencil and essentially the same number and type of computations, it is clearly a more attractive method. Figure 50 should be compared with Fig. 33 giving the corresponding results for second-order shock- capturing methods. The obvious global characteristic for the higher-order ULTIMATE schemes is their much lower error for the smooth-function simulation, due to their lack of artificial steepening and concomitant clipping. As expected, step-simulation error decreases monotonically with the order of the underlying base method. The semi-ellipse and step errors of the higher order ULTIMATE schemes are strongly correlated, whereas for the second-


order artificial compression methods of Fig. 33, the semi-ellipse error is roughly the same for each scheme, again reflecting the artificial steepening and clipping of these methods.

3.5. Simplified UL TIMA TE QUICKEST strategy

Refering to Fig. 39, it is seen that in the range

0.2 ~< ~b c ~< 0.8 (75)

the ULTIMATE QUICKEST scheme is in fact identical to the unconstrained QUICKEST scheme. Thus, considerable simplification can be made in the algorithm without any approxi- mation or effect on results. Inequality (75) can be rewritten as

4,c- 0.51 ~0.3 (76)

or, multiplying by 2,

l l -2gcl ~<0.6. (77)

Rewriting in terms of unnormalized variables results in

ICURVI ~ 0.6[DELI, (78)

where CURV is the upwind-biased second-difference

CURV= ~ - 2~b~ + ~ (79)

and DEL is the normalization difference

D E L = ~ - ~ " u . (80)

Thus, if (78) is satisfied, the unconstrained QUICKEST scheme can be used directly, with no need for testing of universal-limiter constraints. In any practical flow, this criterion will be satisfied in the overwhelming bulk of the flow domain, being violated (if at all) only at a small fraction of grid points near where sharp changes in gradient occur.

Of course, if

$c~<0 or ~b c~>l , (81)

all ULTIMATE schemes (of any order) will use (in terms of unnormalized variables)

~br = $ c . (82)

Inequalities (81) are equivalent to

1 -o.51 o,5 (83)


or, equivalently,

ICURVI >I IDELI • (84)

Thus, the simplified ULTIMATE QUICKEST strategy is as follows: (1) Designate 'D', 'C' and 'U' nodes based on SGN(us) in the usual way, for each face. (2) Compute IDELI and ICURVI, (3) If (78) is satisfied, use the unconstrained and unnormalized QUICKEST face value [15]

1 ,, " ~-~ " - th" 1 &j = ~ (&D + &C)- (4'0 C) - g (1 -- cZ)CURV. ( 8 5 )

(4) Otherwise, if (84) is satisfied, use (82). (5) Otherwise, compute the limited QUICKEST face value according to the (unnormalized)

ULTIMATE strategy. (6) Proceed to the next CV face. (7) Update in the usual way.

Note that Step (5) occurs only in the small ranges

0< ~,a c <0.2 or 0 .8<&c <1 (86)

3.6. Adaptive stencil expansion

The ULTIMATE QUICKEST scheme is a simple, robust algorithm using the same stencil as second-order shock-capturing schemes but with much better global accuracy on a coarse grid. If higher resolution of near-discontinuities is deemed necessary, it is clearly possible to use higher order (upwind) schemes globally; but, of course, this would be more costly. However, since the need for higher resolution occurs only in small localized regions, a cost-effective strategy is to use the efficient simplified ULTIMATE QUICKEST scheme as a base method, automatically switching to a higher resolution ULTIMATE scheme only where needed, by monitoring the solution as it evolves.

Numerical experimentation has shown that the need for adaptive stencil expansion can be determined by monitoring two (solution-dependent) parameters at each CV face: the absolute first-difference across the face, GRAD; and the absolute average second-difference, CUR- VAV. For the right face, these would be

and GRAD = I$;'+1-$;'] (87)

CURVAV= ~1($" - " " " . ,+., 6 , + , ) - ( , / , , - 4 , , _ , ) 1 • ( 8 8 )

If both of these quantities are smaller than preassigned thresholds, the simplified ULTIMATE QUICKEST scheme is used. In most cases of practical interest, this will account for the overwhelming bulk of the flow domain. In large gradient or large-curvature regions, thresholds will be exceeded locally and the algorithm will then automatically branch to a higher order ULTIMATE scheme at the particular face-values involved. Note that this


necessarily involves only very few grid points at any one time, since (because of the high resolution) large gradients or sudden change in gradients occur in localized narrow regions.

Since GRAD and CURVAV are dimensional quantities as defined in (87) and (88), a change in scale requires a change in the threshold constants. This problem can be resolved by making the threshold 'constants' proportional to some typical (constant) reference value of 14,1 occurring in the flow domain. Alternatively, GRAD and CURVAV could be nondimensional- ized with respect to such a value. Note that one cannot use local values of t o

nondimensionalize GRAD and CURVAV (because they would become anomalously large in regions where In any case, some experimentation is needed in choosing optimal threshold constants in order to achieve excellent resolution where needed (in localized regions) without expensive overkill (in terms of using higher order methods globally). Figure 51 shows results of using a third-, seventh- and ninth-order adaptive-stencil-expansion scheme. In this case, the reference value of is taken to be 1.0. If thresholds are not exceeded, the third-order upwind base scheme is used. If CURVAV exceeds 0.025, the algorithm automatically switches locally to ULTIMATE seventh-order upwinding in computing the respective face value. If CURVAV exceeds 0.15 or GRAD exceeds 0.2, ULTIMATE ninth-order upwinding is used locally. In Fig. 51 open circles indicate nodes for which the base third-order scheme (with no stencil expansion) is to be used in computing the adjacent right-face value in the subsequent update step; half-shaded circles show where seventh-order is to be used; and

(a) I x

0 m

(b) j x

1 2

Fig. 51. Results for the adaptive-stencil-expansion ( 3 / 7 / 9 ) method. (a) c=0 .05 ; step: ~ = 0 . 7 2 , ~ ' = ].22; sine-squared: ~ = 0.12, ~ = 0.]3; semi-ei]ipse" ~ = 0.71, W" = 1.09. (b) c = 0.5; step: ~ = 0.70, W" = ] . ]8 ; sine- squared: ~ = 0.09, ~" = 0.12; ~emi-ellipse: ~ = 0.64, ~" = ].05.

60 B. P. Leonard, The V L TIM A TE conservative difference scheme

the fully shaded circles indicate ninth order. It should be clear that the higher-order-stencil points move along with the solution as it evolves.

. Some aspects of generalization

Clearly in this initial paper outlining the principles of the universal limiter, attention has been narrowly focussed on the idealized academic (yet still very challenging) problem of one-dimensional pure advection at constant velocity on a uniform grid. This was done purposely, of course, to identify the basic problems associated with the advection term, the modeling of which is by far the most difficult numerical aspect and major pacing item in the development of computational fluid dynamics. One should get the superficially simple cases right before tackling the complex ones. It makes absolutely no sense, logically or practically, to simulate a complex multidimensional flow using highly sophisticated multi-equation turbulence theory, for example, with an advection method which essentially ignores the turbulence models (except as a diagnostic to switch off their own effects) and replaces modeled physical viscosity with inherent (or explicit) artificial viscosity throughout the bulk of the flow domain. But this is still the ‘state of the art’ in a disturbingly large (and growing) number of commercial and research codes, especially in heat transfer and related industries. But, obviously, in order for the universal limiter to be of practical value in a general-purpose code, several generalizations will need to be made. Space in a single (already lengthy) article does not permit detailed expositions of such generalizations or verification using classical test- problems. This will be taken up in future papers - both by the author and presumably by other researchers who may wish to extend and apply the theory in various ways. However, some generalizations are fairly straight-forward and will be sketched in this section, without showing specific results. Other generalizations of a more obvious nature (diffusion, nonuniform grids) or a more controversial one (systems of nonlinear equations) will be briefly addressed in the closing section.

4.1 e Variable adveeting velocity

The ULTIMATE strategy is easily extended to unsteady one-dimensional advection, where the advecting velocity is a function of space and time. For simplicity, assume that the right-face Courant number is positive

C,.>O, (89) and that in a local region + is increasing monotonically,

The update algorithm is based on (lo), repeated here for convenience

4 :,+I = +I’ + c,+, - c,& , (91)

where the face Courant numbers are considered to be known quantities at time-level yt. As


usual, ~b, is first estimated using some desired high order method. This is limited by the adjacent node values

n n 4~i ~< 6, <~ 4~i+l • (92)

Now require, conservatively, for monotonicity maintenance

n ~ + 1 ~ i - 1 " ( 9 3 )

This can be rewritten, using (89), as

f! n c,~b, ~< c,~b t + ~b, - 4',-1 (94)

and using a worst-case estimate of ~b~, this becomes

n n n c,6, ~< c,4~,_] + ~b, - ~b,_] (95)

for c t >0. (If c t < 0 , it may not be appropriate to require persistence of monotonicity.) Of course, for constant velocity, (95) is equivalent to (72). One further condition is necessary in the variable-velocity case, guaranteeing

~"+' " ( 9 6 ) i ~ ~ i + 1 '

which results in a condition on ~t, for control-volume (i), given by

I! cl~ l ~< c,~, + ~'+t - ~ , (97)

or, in terms of 4~,, viewed as the left face of CV(i + 1), using a worst-case estimate for the 'far-right' face value,

t l f ! t ! c,~, ~< c,,~+1 + - (98)

assuming c,, > 0. In the constant-velocity case, this is superseded by the right-hand inequality of (92). The equivalent restrictions for monotonic decreasing regions should be clear. Local extrema are treated in the usual way.

4.2. Nonlinear advection

Consider one-dimensional nonlinear advection

+ = 0 , (99) Ot Ox

where f(ck) is monotonic increasing. This can be rewritten in control-volume form as

dp i"+' = dp," + A ( f t - f , ) , (100)


where A = At/Ax, ft = f(~bl) and fr = f(~b,). Again, for definiteness, assume that

~ ) n tl n n ,-, < 6, < 6,+, < 6 (101) i + 2 •

First, estimate ~b r by some desired high order method. As usual, this will be limited by interpolative monotonicity

,, n (102)

which is equivalent to

Af(~#~') <~ Af(tbr) ~< Af(~#~'+~). (103)

Then, to assure (91),

hf, ~< hf(~b~'_,) + ~b"i - - O i - l n , (104)

which is the generalization of (95). The condition analogous to (98) is superseded by the right-hand member of (101). Clearly if

f ($ ) = uO (105)

for constant u, the limiting conditions revert back to (72).

4.3, Multidimensional algorithm

Because of the control-volume formulation of the one-dimensional algorithm, it is a straight-forward procedure to extend the ULTIMATE strategy to two and three dimensions. For two dimensions, the explicit update step analogous to (10) is

~ n + ! n i = ~ i -- ( C r ~ r - - CI¢~, ) - - ( C b ¢ b -- C t ~ t ) , (106)

where 'bottom' and 'top' CV faces have been introduced, in addition to 'left' and 'right'. Because of strict conservation,

and &,(i, j ) = ~bt(i + 1, j)

&t(i, j )= ¢~b(i, j + 1)

(107)

(108)

(and similarly with the Courant numbers). Cell-averaged source-terms can be added to the right-hand side of (106), if appropriate.

Focusing attention on one CV face (say the left), the first step is to compute some explicit high (third or higher) order multidimensional estimate for ~t. This is then limited in a manner similar to Fig. 35, where ~c is based on three node values in a direction normal to the face: the two straddling the face together with the next upstream-biased node in the normal


l I / / "

I \ \ \ i I \% [ \ \

- " / t--':-.-:-7 t . . . . . . ' - . ," t V~tl - \ ' o~1 ,' - "

I ' "'-_ t lv , [ I / T ~I ~ , - - I ~'"- / i___l__'_,,' ", i t___~ / " " ' \ \ [ / ~'~ "" , \

\

ca~ u~-o, v~ :-o. (b) u e~O. v /< O.

Fig. 52. Two-dimensional control-volume showing node stencils involved in estimating the left-face value using the ULTIMATE uniformly third-order polynomial interpolation algorithm. Solid dots indicate nodes involved in applying the universal limiter.

direction, determined by SGN(u,,), where u, is the normal component of velocity at the face in question. Thus, in general terms, the limiting step is 'locally one-dimensional' in the normal direction for each face, even though the high-order base method is mult;,~ "r:ensional.

For example, Fig. 52 shows the nodes involved for the left face (for ~, and v~ as shown) using the author's uniformly third-order polynomial interpolation algorithin" (UTOPIA) as the base method, with the limiter-nodes shown as black dots and the remaining nodes as open circles. The UTOPIA scheme is the multidimensional generalization of QUICKEST based on vector transient interpolation modeling according to (1). Allowing for all velocity directions on all four faces results in a 13-point stencil (the same as that used for the second-order discrete biharmonic operator). Higher order schemes can be designed in a similar way, using adaptive stencil expansion. Making use of additional nodes in the normal direction for each face appears to be more effective than involving other nodes in the transverse direction. Extension to three dimensions follo~v~ an enthely similar procedure. Details and results of two-dimensional bench-mark test problems have been l~csented elsewhere [17].

5. Discussion and forecast

The universal limiter, portrayed in Fig. 35, represents an extremely simple way to produce explicit 'monotonic' advection schemes of arbitrarily high order. Potential numerical oscillations are suppressed without corrupting the expected resolution of the underlying scheme. It has been seen that second-order methods (including well-known shock-capturing or TVD schemes) are significantly inferior to the third-order ULTIMATE QUICKEST scheme, which uses the same five-point stencil. Better resolution of sharp changes in gradient can be achieved by using a higher order base method, and concomitantly wider computational stencil. However, higher (even) order central methods are susceptible to a degree of waviness under some circumstances, as seen with the semi-ellipse simulation. In terms of overall performance, there seems little to be gained beyond the ULTIMATE fifth-order upwind scheme, or the highly cost-effective adaptive-stencil-expansion technique that uses ULTIMATE QUICKEST


in smooth regions and automatically switches to a higher order ULTIMATE upwind scheme in isolated regions of high curvature or gradient.

More precise resolution of narrow local extrema can be built into the algorithm by using an automatic discriminator that switches to an u~limited higher order scheme in such regions [18]. Figure 53 shows results using the automatic discriminator in conjunction with the adaptive variable-order scheme. The small arrows indicate where the universal limiter is automatically switched off. Note the better resolution of local maxima, especially in the case of the sine-squared profile. This technique is even more important where very narrow extrema are involved [18].

The great advantage of the ULTIMATE strategy is that it can be used with explicit schemes of (in principle) arbitrarily high order, giving concomitant resolution of sudden changes in gradient without artificial steepening and clipping. The limited polynomial schemes reviewed here already give excellent results (above second order), but clearly the ULTIMATE philosophy can be used with other more sophisticated and more appropriate ~orms of spatial interpolation. One such method currently under investigation involves exponential ('tension') splines applied locally; this technique offers possibilities of very high coarse-grid resolution on a compact stencil.

Extension to multidimensional flow is straightforward and, as mentioned, this is being explored elsewhere [17]. Although not addressed in this paper (since attention has been focused on the critical problem of pure advection) it should be clear that the introduction of

(a) j '* ' " x

m

(b)

! I

)

m

m I ' * * ' - v

0 1 2

Fig. 53. Results for the adaptive-stencil-expansion (3 /7 /9 ) method using an automat ic discriminator. (a) c = 0.05; step: ~ = 0.73, ~/f" = 1.23; sine-squared: ~ = 0.02, ~ = 0.03; semi-ellipse: ~ = 0.61, ~ = 1.02. (b) c = 0.5; step:

= 0.69, ~r = 1.16; sine-squared: ~ = 0.01, ~f' = 0.02; semi-ellipse: ~ = 0.58, ~f" = 1.01.


modeled physical diffusion to an order consistent with the advection scheme presents absolutely no problems. The inclusion of physical diffusion in fact puts less demands on the advective algorithm; in most cases of practical interest, the ULTIMATE 3rd/7th/9th-order advection scheme combined with a simple second-order diffusion model gives results which are graphically indistinguishable from the exact solution (when known). The second-order diffusion model is appropriate because in smooth regions this is consistent with the third-order base scheme [15], and in sharply varying regions diffusion is very small, so that the model used is not important. In a similar way, extension to spatially varying grids presents no problems. The higher order accuracy of the base method can be formally maintained [11, 15]; or, provided the expansion ratio does not exceed about 125 percent, the simpler uniform-grid formulas for third- and higher order methods can be used directly without reducing the practical order of accuracy-although the formal order is, of course, reduced [19].

One of the most important questions concerns how well the ULTIMATE strategy general- izes to handle systems of nonlinear equations such as the Euler (or Navier-Stokes) equations of gasdynamics, and the 'shallow-water' and related equations of oceanography and meteorol- ogy. This is by no means a trivial generalization and will undoubtedly form the subject of future papers; however, it is fair to predict that ULTIMATE simulations will be at least as successful as the better shock-capturing schemes, since the underlying philosophy is very similar. In particular, because of natural physical compression, shock-wave or hydraulic jump (or atmospheric front) resolution can be expected to be narrower than the scalar step resolution studied here. And because of the ability to use arbitrarily high order resolution, unsteady gasdynamic simulations can be expected to give extremely reliable results, even at very high Math numbers and nonideal conditions. Because of the conservative control volume formulation, multidimensional gasdynamic and geophysical algorithms are expected to be similarly robust. The general philosophy here is to model the control-volume advective modes precisely and let the physics (as reflected in the governing equations) take care of the 'acoustic' or wave modes. This idea seems not to work well with currently popular second-order flux-limited schemes because of these schemes' addictive reliance on locally varying (positive or negative) artificial viscosity, requiring precise knowledge of characteristic speeds and directions. But the principle has great potential when used with higher order ULTIMATE multidimensional algorithms, which, by design, require only advective velocities at the CV cell faces, as prescribed by the governing equations.

Finally, the ULTIMATE strategy can be applied to steady-flow multi-dimensional simulations (even though the acronym might be somewhat out of place). Quite simply, the steady-state limiter consists of ULTIMATE with the Courant number set of zero. This results in the extremely simple condition in the monotonic regime

¢c ~< 4'y ~ 1 for 0 < ¢c ~< 1, (109)

with ~b s = ~c, or similar function passing through (0, 0) and (1, 1), in the nonmonotonic range. In other words, for locally monotonic behavior in the direction normal to a given CV face, after a high-order multidimensional estimate of ~b s is made, the face value actually used is constrained simply to lie between the adjacent upstream and downstream node values: ~b u and ~o, respectively. The author's multidimensional SHARP algorithm [20], a simple high- accuracy resolution program, is a third-order nonlinear scheme conforming to these require-


ments; it is based on an exponential upwinding or linear extrapolation refinement of the multidimensional steady-state QUICK scheme [15]. But, of course, the principle can be extended to arbitrarily high order using the universal limiter for tight resolution and accuracy, thus giving ULTRA-SHARP simulation of steady multidimensional flows containing near- discontinuities [21]. In a practical algorithm, it is important to construct a single-valued upper boundary for the allowable NVD region near ~c"> 0+. This is most conveniently achieved by requiring

~bf ~< const q~c for 0 ~< ~b c , (110)

in addition to (109), with suitably large values of the slope constant (---10 or 100). The piecewise linear NVD function of the steady two-dimensional SMART code of Gaskell and Lau [22] is also of this form. Steady-state methods based on second- and third-order schemes (including TVD iimiters) can be solved with straight-forward ADI scalar pentadiagonal matrix algorithms using Gaskell-and-Lau's modified (nonlinear) curvature-compensation technique, based on an equivalent of the curvature factor, CF(~bc), appearing in (55). Alternatively, arbitrarily high order limited schemes can be solved by popular ADI tridiagonal methods using the author's downwind weighting factor technique [21], thus rendering the ULTRA- SHARP very-high-order multidimensional nonoscillatory steady-flow schemes immediately compatible with many of the general-purpose commercial and research steady-state elliptic solvers currently in use, that are currently typically based on low-accuracy blended combinations of first- and second-order advection methods.

6. Acknowledgment and perspective

Much of the programming and computer graphics manipulation for the large number of test cases shown (and countless others not included) was ably handled by H.S. Niknafs. As is often the case in scientific investigations, the research reported here was developed independently of other similar earlier work, in this case during the latter part of 1987 while the author was an ICOMP Visiting Research Engineer at the NASA Lewis Research Center (work funded under Space Act Agreement C99066G). The idea of the universal iimiter stems from the author's development in 1974 of exponential upwinding [23, 24], a third-order nonlinear nonoscillatory advection scheme for steady and unsteady flows which predated [25] QUICK and QUICKEST [11], and from extensive experience over the past decade or more with the normalized variable diagram and associated monotonicity criteria [23]. The author is indebted to Dr. S.H. Chang, another ICOMP researcher, for pointing out some of the more intelligible shock-capturing and TVD literature- particularly several articles by Yee [26] and the 1984 survey paper by Sweby [13]. As described here (in Appendix B), Sweby works with a much more restrictive 'TVD region', conforming to current practice in operational schemes, although the less restrictive conditions would have been known to him, since they had appeared in GAMM proceedings papers by Roe and Baines in 1982 [27] and 1983 [28]. The author's attention was directed toward the latter papers by S.T. Zalesak in November 1987 and by P.L. Roe in May 1988. Interestingly enough, Roe's 1986 review article [29] makes no mention of the less restrictive limiter boundaries, but rather describes instead only some of the more restrictive limiters


(such as Minmod, 'CLAM' and Superbee), even though the earlier proceedings papers are referenced.

Zalesak's IMACS paper [14] describes work by his colleague, J. Lyon, at the Naval Research Laboratory, who appears to be the only other researcher to have used arbitrarily high (up to eighth) order advection schemes constrained by an equivalent of the universal limiter. Finally, Gaskell and Lau, at the University of Leeds, have (again independently, following from their steady-flow work [22]) developed an equivalent limiting strategy for unsteady flow applied to a quasi-third-order advection scheme [30]. They report highly accurate simulations of standard shock-tube benchmark problems without resorting to flux vector splitting, Riemann solvers, or any of the other complexities associtated with currently popular forms of first/second-order shock-capturing schemes. This confirms the author's belief that characteristic decomposition is not a necessity, but rather a symptom of poorly designed advection schemes, based on variable artifical viscosity, which require detailed explicit knowledge of all characteristic velocities and amplitudes. If the advection (macroflux) terms appearing in the conservation equations for mass, momentum and energy are carefully modeled with suitably limited higher order methods, and microflux terms (pressure tensor, heat flux, etc.) treated appropriately, the governing equations themselves will automatically generate the correct additional wave modes. The far-reaching implications of this, especially for multidimensional flows, should be clear.

Appendix A. Arbitrarily higher order polynomial schemes

In this appendix, formulas are derived for recursively computing arbitrarily high order face values based on polynomial interpolation. Attention is focussed on the right face of a control-volume cell. First, define

½(6. +61,), t ~ r i + 1 (A.1)

i.e., linear interpolation across the right face. Let the first-difference across the right face be represented by

ti fl 8~ = ~b,+, - ~b, • (A.2)

The average second-difference centered on the right face is given by half the difference of first-differences across two cells

~ 2 1 n __ n n _ . _

, = 6 , + , ) - ( 6 7 - 6 , - , ) ] ' " " " " (A.3)

Higher order differences, centered on the right face, can be computed recursively, from Pascal's triangle for the odd-order differences and a related triangle for the average even- order differences. These are shown in Tables A.1 and A.2, respectively. Any entry in either table is obtained from the previous row by subtracting the adjacent left value from the adjacent right value.

For definiteness, derivation of the third-order upwinding scheme will be shown in detail.


Table A.I

N Coefficients of odd differences

0 0 0 0 0 (+1) 0 0 0 0 1 0 0 0 0 +1 - 1 0 0 0 0 2 0 0 0 (+1 - 2 +1) 0 0 0 3 0 0 0 +1 - 3 +3 - 1 0 0 0 4 0 0 (+1 - 4 +6 - 4 +1) 0 0 5 0 0 +1 - 5 +10 -10 +5 - 1 0 0 6 0 (+1 - 6 +15 -20 +15 - 6 +1) 0 7 0 +1 - 7 +21 -35 +35 -21 +7 - 1 0 8 (+1 - 8 +28 - 5 6 +70 -56 +28 - 8 +1)

Table A.2

N Coefficients of even differences (x2)

0 0 0 0 0 +1 +1 0 0 0 0 1 0 0 0 (+1 0 - 1 ) 0 0 0 2 0 0 0 +1 - 1 -1 +1 0 0 0 3 0 0 (+1 - 2 0 +2 - 1 ) 0 0 4 0 0 +1 - 3 +2 +2 - 3 +1 0 0 5 0 (+1 - 4 +5 0 - 5 +4 - 1 0 6 0 +1 - 5 +9 - 5 - 5 +9 .-5 +1 0 7 (+1 - 6 +14 -14 0 +14 -14 +6 - 1 ) 8 + 1 - 7 +20 -28 + 14 + 14 -28 +20 - 7 + 1 E n E n m m m E m n m ~ ~ m m m m ~ ~

For positive convecting velocity (or Courant number, c), the third-order algorithm is based on interpolation of a cubic polynomial through eke_z," tk~_t," ~" and ~+~." Using a local normalized coordinate

= ( x - x , ) / A x , (A.4)

Lagrangian interpolation on a uniform grid gives

0'~.(~) = A_2(~: + 1)~:(~ - 1)+ A_,(~ + 2 ) ~ ( ~ - 1)+ Ao(~: + 2)(~ + 1 ) ( ~ - 1)

+ At(~: + 2)(~ + 1)~. (A.5)

The coefficients are evaluated by equating node values at integer values of ~::

and

t ! p! 0b + ( - 2 ) = A -2( - 1 ) ( - 2 ) ( - 3 ) = tk,_2,

$ ~ ( - 1 ) = A _ t ( 1 ) ( - 1 ) ( - 2 ) = " ~ i - ! '

$ +(0)= Ao(2)(1)(-1) = ¢,

(A.6)

(A.7)

(A.8)

$'2,(1)= A~(3)(2)(1)= " ~ + l , (A.9)


so that (A.5) becomes

1 ,, 1 . ~b+(~:)= 0[3l ~ b ' - 2 ( ~ : + l ) ~ : ( ~ - l ) + 1.-~.w ~b'-1(~:+2)~:(~:-1)

1 1 2!1! 4~'(~: + 2)(~: + 1 ) ( 6 - 1) + ~ ~b~+l(~: + 2)(~ + 1)~:. (A.10)

Transient interpolation modeling, based on

n c~ '/ + l = 4p + ( ~ = - c ) , (A.11)

thus gives, in this case,

¢ ~ 7 + l " -" -- ~b~_2c(1 - C 2) "~" ½4'~'+~C(1 + C)(2-- C) + ½~b'i'(2 -- C)(1 -- C 2) 1 n - ~4~,_lc(1 - c ) ( 2 - c) ( A . 1 2 )

when c is positive. It is not difficult to manipulate this into

4,,+1 " ½c($" ,, ,_2,_,,, - 2 5 " + ~b" i = {~i - - i+1 -- l ~ i - l ) + ~t." [ { P i + l i i - l )

+ ~c(1 - c2)(6;'+~ - 36;' + 36;'_, - 6 ; ' - 2 ) • (A.13)

Note that the right-hand side includes the three symmetric terms of the Lax-Wendroff scheme together with an upwind-biased third-difference term.

The algorithm can be rewritten in conservative control-volume form

~, ,+t ,, ( A . 1 4 ) , = - -

where ~bt(i ) = ~ , ( i - 1), by identifying, in this case,

(3,_ 1 C ( 1 - C 2 ) ( ' 2 (~'a'+, + ~'~')- ~ (~ ; '+ t - 6'~')- 6 6" " "

- , + , - 2 e , + $ , - t ) . ( A . 1 5 )

This can be rewritten as

c , (1 c l[ :] , ~-r 2 6r - 6 9 2 1 - - , - ~ 6 . (A. 16)

A little reflection will reveal the fact that the corresponding formula for negative c would involve a change in sign of the 6 ~ term. Thus, for positive or negative c-values, the formulas could be combined to give

$,3 ,= t3 ,0)_ c ' (1- -C2)[ r - r r 2 ¢ ~ , - - 6

2 SGN(c) ~] ' 2 6 . (A.17)

Note that the first two terms on the right-hand side of (A.17) represent the Lax-Wendroff


(second-order) contribution, 4 't2~r . Thus, the third-order upwind scheme can be written

( c ~ - 1 2 ) [ ~ SGN(cr) 63] ' (A.18)

where it is assumed that the formula is valid for variable c. It is of interest to compare (A. 18) with the corresponding formulas for first-order upwinding

(21)

4~ '' = ~b(r °) - ~' SGN(c,)a'r (A.19)

and second-order central differencing (23)

1~(2) = r ~(0)__r lCrt~ " (A.20)

An entirely similar procedure leads to the fourth-order central-difference formula

~(4) ~. ~}(2).~_ (C~-- 12) [~2 Cr ~] r r 3l , - ~- a , (A.21)

which should be compared with the third-order formula, (A.18). Similarly.

i~(5) = d~(4)+ ( C ~ - l " ) ( c ~ - 2 2) [ ~ 4 - SGN(c,) ] ' ~-' 5! ' 2 6 s, , (A.22)

and

tb{.,,}=tb(4)+ (c~ -12)(c~ - 22) [~4 c, ~] "" 5! , . - ~ 6 , (A.23)

~b,7, = ~b~/,, + (c~- 12)(c~- 2'-)(c~- 32) [ (, SGN(c,) ] ' 7! ~" - 2 6, ~ (A.24)

d}~s' = ¢fcr6) + 7l ~ , - -ff 6 . (A.25)

Clearly, a pattern has emerged, so that for an Nth-order upwind scheme, for example (N odd)

(N- !)/2

I-I ( c ~ - k 2 ) [ ~ _ SGN(c,) N]

N! - ' 2 (A.26)

and for an Mth-order central scheme (M even)

M/2- 1 I-[

1-, ( M - 1)l ~r _ ay-']

M (A.27)

B.P. Leonard, The U L T I M A T E conservative difference scheme 71

Appendix B. Relationship to the Sweby diagram

In a well-known paper [13], Sweby introduced sufficient conditions for schemes to have total-variation-diminishing properties, and portrayed these conditions in what has become known as the 'Sweby diagram', a plot of the flux-limiter factor, q~, against the local gradient ratio, r. For constant Courant number, the flux-limiter factor is defined as

t I t l ~ = ( , ¢ > r - , c ) / ( , 7 w - , ~ ) , (B.1)

where ~b s is the limited face value, ~bc is the adjacent upstream node value and 6} w is the face value according to second-order central differencing (Lax-Wendroff)

1 67w ~ , ,~_ ~ ,, _ ~ (,/>" = D "~" C# ( ~ D @ C ) ' n (B.2)

where 4'D is the adjacent downstream value, as usual. All of the schemes considered by Sweby are linear in Courant number in the form of (49), rewritten for c <> 0, as

&s = (1 -Icl),¢>7 + Icl6" C " (B.3)

The Lax-Wendroff face value can also be written in this form

L W n ~b s, - ' ( 1 - 1<1)~7 ~ + Icl6c, (B.4)

where "hCEN is the linear interpolation ~(0D + ~bc) Substituting (B.3) and (B.4) into (B 1) "t'f • results in an equivalent expression for the flux-limiter factor

, ,

~ = (&7'- 4bc - O c ) , (B.5)

which can be rearranged as

~ = ~ , + ~(~c~.. ,, s - ~bc), (B.6)

interpreted as first-order upwinding modified by adding a term proportional to the difference between linear interpolation and first-order upwinding.

In terms of the normalized variables as defined in the present paper, it is not difficult to show that

. . (-~ - n n

( 6 s - 6c) - 6 c ) = (B.7) = ½(1 - ~c) ½(1 -Icl)(1- ~) '

Similarly, the gradient ratio used by Sweby is

@ t! ~ tt t!

r = ( ~ ~ , , _ ,, " - c) (1 4~c)

(B.8)


The so-called TVD region proposed by Sweby consists of first-order upwinding (tp = 0) for negative r, followed by

0<~ ~0 <~2r for 0<~r<~l, (B.9)

and then limited by first-order downwinding at time-level n, ~ = 1 in (B.7), giving

0~<tp<~2 forr~>l . (B.10)

Note that (B.9) can be interpreted as

~"=2@c for0<4~c<½ (B.11)

Figure B.1 compares Sweby's TVD region (shaded) interpreted in the (~c, ' ~ ) plane and in the (r, ~0) plane. For reference, Roe's Superbee scheme is shown, passing through the 'second-order' point, (0.5, 0.75) in Fig. B.l(a), and (1, 1) in Fig. B.l(b). By contrast, Fig. B.2 gives the 'extended Sweby diagram' corresponding to the ULTIMATE strategy of Fig. 35; recall that that figure involved ~bj rather than ~ , as used in Fig. B.l(a). Note that the sloping boundary OB in the extended Sweby diagram has a slope of 2/Icl as compared with the slope of 2 for OC in the original Sweby diagram. Also note that the upper bounday of the extended diagram increases in height as Icl--, 1; in fact

~o. = 2/(! -Icl), (B.12)

where, in the original diagram, Fig. B.l(b), the upper boundary is independent of c:

~Oc ~ 2 .

As the Courant number varies, the point B in Fig. B.2 'slides' along the line ~P8 = 2(1 + ra),

- n ~)f / 2

I

q , -

(a)

(-A)

C~ (A - )

- ~.-!i!~:iiiii~:.i~:~iiiii!~i!!i!~

ii!iiiiiiiii : :~:~:~:~:~:~:~:~:~:~: ~:!:!:!:~:~:~:~:!:~:~:~ :~:~:~:?:~:':~:~:i.- 1 2 r

(b)

Fig. B. 1. Sweby's so-called 'TVD region' (shaded). (a) NVD. (b) The (r, ~0) plane. For reference, the Superbee scheme is shown by the heavy piecewise-linear function.


(,

0

j 2

• i i i .. ii.:i..iiiiiiiiiiii iiii i i!iiiiiii !i ! i i ..iiiii..iiiiiiiii ! ..i

.. liiiiiiiiiii!iiiii!iiiiiii!iiiiiiiiii::....::!iiiiiiiiiiiiiiiiiiiiiiiiii!ii!iiiiiiiii!i!i!iiiiiiiiiiiiiii:.

- t:iii~iiiiiii:iiiiiigiiiiiii:iig:." ~:~.ii:iiiiii!:~iiii:i:i:~iiii:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i::. g.'."!i:.::i:."."!ii:i:iiiii:!:::" ..:~!!i:i:ii...:!:!:i......:i:i!iii!i.....:iii!iii~iiiiiiiiiiiiiiiiiiiii~iiiiiiiii!iiiiii. ~j:!:~:i:i:!:!:i:i:i:i:i::" ..::!:i:i:i:i:!:i:!:i:i:i:i:i:!:i:i:!:i:!:i:i:i:i:i:i:!:i:i:~:i:i:i:i:i:i:i:i:i:i:!: ~ t:iiiiiiiiiiiiiiiiiii::....::iiiiiiiiiii~iiiiiiiiiiiii!i!iiiiiii!iiiiii!iiiiiiiiiiii!ii!ii!iiiiiiiiiiii~iiiiiii:

!iiiiiiiiiiiiili ,. 1 2 r

Fig. B.2. 'Extended Sweby diagram' in (r, ~) coordinates, corresponding to the universal iimiter portrayed in Fig. 35. For reference, the Super-C scheme is shown by the heavy piecewise-linear function.

where r n = [ c [ / ( 1 - [c[). For reference, the Super-C scheme is shown in Fig. B.2. Hyper-C simply follows the upper boundary: A O B A . As is immediately obvious, Sweby's TVD region is grossly over:yestrictive, resulting in predicted (normalized) face values which are too small, especially for 4~c values slightly larger than 0 or slightly smaller than 1. This is one reason for the poor performance of currently popular second-order TVD schemes.

References

[71

[8l

[91 [10l

[1] B. Van Leer, in: H. Cabannes and R. Temam, eds., Proceedings of the Third International Conference on Numerical Methods in Fluid Mechanics, Vol. 1 (Lecture Notes in Physics, Vol. 18) (Springer, New York, 1973) 163-168.

[2] B. Van Leer, Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme, J. Comput. Phys. 14 (1974) 361-37{).

[3] B, Van Leer, Towards the ultimate conservative difference scheme. III. Upstream-centered finite-difference schemes for ideal compressible flow, J. Comput. Phys. 23 (1977) 263-275.

[4] B. Van Leer, Towards the ultimate conservative difference scheme. IV. A new approach to numerical convection, J. Comput. Phys. 23 (1977)276-299.

[5] B. Van Leer, Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method, J. Comput. Phys. 32 (1979) 101-136.

[6] P.L. Roe, in: E. Engquist, S. Osher and R.J.C. Sommerville, eds., Large Scale Computations In Fluid Mechanics, Pt. 2 (Lectures in Applied Mathematics, Vol. 22) (Amer. Mathematical Soc., Providence, RI, 1985) 163-193. A.R. Mitchell, in: J. Noye and C.A.J. Fletcher, eds., Computational Techniques and Applications: CTAC-83 (Elsevier, New York, 1984) 2-14. B.P. Leonard, in: J. Noye and C.A.J. Fletcher, eds., Computational Techniques and Applications: CTAC-83 (Elsevier, New York, 1984) 106-120. C.A.J. Fletcher, Computatio:lal Techniques for Fluid Dynamics, Vols. I and II (Springer, New York, 1988). P.D. Lax and B. Wendroff, Systems of conservation laws, Comm. Pure Appl. Math. 13 (196{)) 217-237.

74 B.P. Leonard, The UL TtMA TE conservative difference scheme

[111

I121

[13]

1141

[151

1161

[171

[181

1191

[2Ol

1211

1221

1231 1241

I251

1261

1281

1291

13Ol

B.P. Leonard, A stable and accurate convective modeling procedure based on quadratic upstream interpolation, Comput. Methods Appl. Mech. Engrg. 19 (1979) 59-98. J.E. Fromm, A method for reducing dispersion in convective difference schemes, J. Comput. Phys. 3 (1968) 176-189. E K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Numer. Anal. 21 (1984) 995-1011. S.T. Zalesak, in: R. Vichnevet.~ky and R.S. Stepelman, eds. Advances in Computer Methods for Partial Differential Equations VI (IMACS, Rutgers University, 1987) 385. B.P Leonard, in: W.J. Minkowicz, E.M. Sparrow, G.E. Schneider and R.H. Pletcher, eds., Handbook of Numerical Heat Transfer (Wiley, New York, 1988) 347-378. S.R. Chakravarthy and S. Osher. High resolution applications of the Osher upwind scheme for the Euler equations, AIAA Paper 83-1943 (1983). B.P. Leonard and H. Niknafs, Cost-effective accurate coarse-grid method for highly convective multidimensional unsteady flow, in: Proceedings of the CFD Symposium on Aeropropulsion, NASA Lewis Research Center, Cleveland, OH (April 1990). B.P. Leonard and H. Niknafs, Sharp monotonic resolution of discontinuities without clipping oil narrow extrema, Comput. & Fluids 19 (1990) 729-766. I.E Castro and J.M. Jones, Studies in numerical computations of recirculating flows, Internat. J. Numer. Methods Fluids 7 (1987) 793-823. B.P. Leonard, Simple high accuracy resolution program for convective modeling of discontinuities, Internat. J. Numer. Methods Fluids 8 (1988) 1291-1318. B.P. Leonard and S. Mokhtari, Beyond first-order upwinding: The ULTRA-SHARP alternative for nonoscil- iatory steady-state simulation of convection, Internat. J. Numer. Methods Engrg. 30 (1990) 141-154. EH. Gaskell and A.K.C. Lau, Curvature compensated convective transport: SMART, a new boundedness preserving transport algorithm, Internat. J. Numer. M..thods Fluids 8 (1988) 617-641. B.P. Leonard, in: AIAA Computational Fluid Dynamics Conference, 4th (AIAA, New York, 1979) 226-233. B.P. Leonard, in: C. Taylor and K. Morgan, eds., Computational Techniques in Transient and Turbulent Flow (Pineridge, Swansea, UK, 1981) 1-23. B.P. Leonard, G.J. Vachtsevanos and K.A. Abood, in: C.A. Brebbia, ed., Applied Numerical Modelling (Wiley, New York, 1978) !13-128. H.C. Yee, Upwind and symmetric shock-capturing schemes, NASA TM-89464 (1987). EL. Roe and M.J. Baines, in: H. Viviand, ed., Conference on Numerical Methods in Fluid Mechanics, 4th, Proceedings (Vieweg, Braunschweig, 1982) 281-290. EL, Roe and M.J. Baines in: M. Pandolfi and R. Piva, eds., Conference on Numerical Methods in Fluid Mechanics, 5th, Proceedings (Vieweg, Braunschweig, 1983) 283-290, EL. Roe, in: M. Van Dyke, J.V. Wehausen and J.L. Lumley, eds., Annual Reviews of Fluid Mechanics 18 (Annual Reviews Inc., 1986) 337-365. P.H. Gaskell and A,K.C. Lau, in: C. Taylor, P. Gresho, R.L. Sani and J. Haiiser, eds., Numerical Methods in Laminar and Turbulent Flow 6 (Pineridge, Swansea, UK, 1989) 51-62.

Documents

The ULTIMATE conservative difference scheme applied to ...klinck/Reprints/PDF/leonardComMeth1991.pdfresolution' shock-capturing or TVD schemes. The ULTIMATE third-order upwind scheme