Counting the number of solutions in combustion and reactive flow problems

Journal of Applied Mathematics and Physics (ZAMP) 0044-2275/90/040558-21 $ 1.50 + 0.20 Vol. 41, July 1990 �9 1990 Birkh~iuser Verlag, Basel

Counting the number of solutions in combustion and reactive flow problems

By Ed Ash, Brian Eaton and Karl Gustafson, Dept. of Mathematics, University of Colorado, Boulder, CO 80309-0426, USA

1. Introduction

Reactive flow problems, for example in chemical kinetics, are character- ized by rather complicated bifurcation curves. One reason for this is the Arrhenius rate law, which introduces an exponential nonlinearity into the reaction equations. Thus even the single homogeneous exothermic reaction equation

{ u A U = )t e ~/~ + ~) inf2 = 0 on Oft (1)

may possess anywhere from zero to an infinite number of solutions, depending on the values of the parameters 2 and ~ under consideration. The determination of the exact number of solutions is of importance to the associated reactive flow problems.

In an earlier paper [1] we obtained full bifurcation curves and solution profiles for (1) for the important case of n = 3 physical dimensions and a spherical domain. This answered a number of questions about the qualitative properties of (1) as compared to those of certain approximations. Because on spherical domains all solutions are radial, (1) reduces to an ordinary differential equation which can be integrated numerically.

Interest in [1] centered on the computation of the critical values (e.g., ~c = 0.238797 for the sphere) above which there is only one solution for all 2. However we noticed some difficulty in treating smaller e especially as u becomes larger. The case of very small e (i.e., very high activation energy

-J) is of interest in its own right and is the subject of the present paper. Shortly after [1] there appeared the paper [2] by B6rsch-Supan. BSrsch-

Supan was interested in the stability of the solution branches and showed that only the first and last branches are stable. He also computed the bifurcation diagrams for (1) for e = 0.05, 0.04, 0.03, 0.02, and 0.01, as long as solution size HuH- A - m a x ]ul remained small (e.g., see [2, Figure 2], A is between 0 and 25). For larger A the bifurcation curve suddenly dives to

Vol. 41, 1990 Counting reactive flow problems 559

values of )~ very close to zero (see [2, Figure 3], for e = 0.04). Computat ions then become difficult until A becomes very large when asymptotics take over (see the discussions in [2]). B6rsch-Supan estimates the last turning point, where the bifurcation curve finally turns upward, to be

A ~ 625 for e = 0.04

A ~ I 0 , 0 0 0 f o r e = 0 . 0 1

The purpose of the present paper is, following [1] and [2], to report the results of our recent investigations improving [ 1] and [2]. We were interested primarily in the equation (1) for the physically important case of the three-dimensional sphere (as were [1] and [2]), and for small positive e (which the Frank-Kamenetskii assumption of infinite activation energy, i.e.,

= 0, tries to approximate). For example, we can determine that the last turning points are

A = 1054.2805, 2 = 2.4383822 • 10 -7 for e = 0.04

A = 18178.9202, 2 = 1.0863 • 10 -38 for e = 0.01.

These were obtained numerically. We will also present here an analytical lower bound which is almost as good in both cases.

Beyond this focal problem, already mentioned above, of locating the last turning point (where a stable second high temperature solution branch occurs for very small exothermicity 2) on the bifurcation curve, a further goal of our investigations was to develop the capability to count precisely the exact number of solutions as one proceeds along the bifurcation curve. This amounts to schemes for locating very precisely all turning points on that curve, because it is at each turning point that the number of solutions increases or decreases by one. It turns out that for small e these bifurcation curves contain a rather flat intermediate portion, where a wiggle for example may occur in the seventh decimal place, thereby determining a change in the number of solutions there. We will give a striking example of this phenomenon, namely, the ~ = 0.01 test case, where we needed double precision on the CRAY 1 to correctly count solutions.

Full details of the numerical scheme may be found in [ 1, 3]. Here we will indicate the improvements we have made to that scheme to enable the accurate computat ion of the turning points for smaller epsilon. These improvements reveal in particular an interesting connection between the numerical method and analytic theory, providing for example some analytic bounds for the turning points.

These investigations have brought out a number of further interesting questions on which research is continuing [4]. For a survey of the history and further background on both numerical and analytical work on equation (1), see Gustafson [5].

560 E. Ash et al. Z A M P

The outline of the paper is as follows. In section 2 we discuss the qualitative theory as it pertains to this problem. In section 3 we give quantitative results for the case e = 0.01. Section 4 contains lower bounds for the values of 2 at the turning points on the bifurcation curve. Section 5 contains some additional remarks.

2. Qualitative theory

For a general discussion of the known qualitative theory for (1), we refer to the survey article [5]. Here we mention only a few key facts.

For simplicity we will restrict attention to (1) on the unit sphere in n = 3 dimensions. This is a so-called type A geometry much studied in the chemistry literature, and the most important case physically. For such spherical domains it is known Gidas, Ni, Nirenberg [6] that all positive solutions are radially symmetric. This reduces all considerations to the ordinary differential equation (n the space dimension):

f dZu (n - 1 ) du

u'(O) = u ( 1 ) = o

+ 2 e x p ( u / ( l + e u ) ) = O , O = < r < l

(2)

The most basic qualitative fact is that the Arrhenius equation (1) with > 0 and the Frank-Kamenetskii approximation (e --- 0) are fundamentally

different, the former possessing solutions for all positive 2, the latter only up to a finite 2c. This observation follows from a criterion of Keller and Cohen [7].

Related to this, we will see a second qualitative difference: for small e equation (1) will possess two stable branches for all 2 beyond a (very) small critical lower bound 2t,p. This means that for (1) at very small exothermicity values 2 there always exists a stable high temperature solution branch. This property is not shared by the Frank-Kamenetskii approximation.

Existence proofs are quite available for (1). These follow as in Keller and Cohen [7] for example. One may also use Nemitzkii integral operator formulations, upper and lower solution methods (as in Amann [6] of [5]), variational proofs (when the nonlinearity is H61der continuous), nonlocal (Nemitzkii) versions when the nonlinearity is not monotone, Schauder's fixed point theorem, etc. There is of course the usual regularity work to be done in these different proofs. For an excellent account of existence proofs for general domains see P. Bader [8].

We will therefore take the point of view here that the existence question is now relatively accessible and that the questions of (non)uniqueness, (in)stability, and the ultimate problem of counting exactly and determining


as precisely as possible all of the solutions for all parameter values (especially at critical points) are the most challenging and most important questions before us, at least in the present paper. This latter central question could be phrased generally as the study of the midport ion (between the two end branches) of S-shaped bifurcation curves. By S-shaped we mean basically S-shaped, with possibly, as in our case, a number of secondary turning points within the midport ion itself. It mus be admitted that analytical attempts to study the details of the intermediate portion of such S-shaped bifurcation curves have had only limited success, see for example Bader [8] and Shivaji [9].

We have found numerically (see section 3) that one reason for this analytical difficulty is that in the midport ion of the S-curve, solutions look rather like delta functions. These are very hard to estimate, even numerically.

To get some feeling for these qualitative features, let us borrow at this point our numerical results for e = 0.04. For e = 0.04 the exact values of the turning points are given in Table 1. The bifurcation diagram is shown in Figure 1, for A < 125. These results agree with those of [2, Figure 3], except that we have now located the last turning at A = 1054.2805. This is far to the right off the graph of Figure 1. Notice how fiat and small the bifurcation curve must be for 125 < A < 1055, after which the curve begins its inevitable final monotonic rise on the stable high temperature last solution branch.

Relative to results for smaller e which we will give later, the bifurcation curve possesses relatively few 'wiggles'. From it we may count the number of solutions for each exothermicity 2 exactly. This consists of "drawing" 2-sections horizontally on the bifurcation diagram. Thus, for e = 0.04, see Figure 1 and Table 2, we have for 2 < 21 ~2 .4 • 10 -7 exactly one solution, for 21 < 2 < 2 2 = 1.0772377 exactly three solutions, for 22 < 2 < 23 = 2.2038626 exactly five solutions, for 2 3 < 2 < 2 4 = 2.3736164 exactly seven solutions, for 24 < 2 < 25 = 3.3059654 exactly five solutions, for 25 < 2 < 2 6 = 3.4828675 exactly three solutions, and for 2 6 < 2 exactly one solution.

Table I Values of the turning points for e = 0.04.

Previous number of Turning Solution Exothermicity unstable modes point A = max u )~

0 1 1.77223940 3.4828675 1 2 12.776963 1.0772377 2 3 35.563049 2.3736164 3 4 59.911631 2.2038626 2 5 97.240239 3.3059654 1 6 1054.2805 2.4383822 x 10 -7

562 E. Ash et al. ZAMP

.3.~o4"'~176 i 5

2 . 0 0

1~~ I

"~176 I 2

~176 I 6 O OO , , , ~ f , , , , I , , , , I , , , , t L, ,

0 . 0 0 2~.0 ~0.0 75.0 100. 12"% NORM U

Figure 1 Bifurcation diagram and turning points for the full Arrhenius equation in 3 dimensions, for ~ = 004

1.0

0 9

0 .7

0 .6

II

o.5

0.3 6

0.~ 3

i 0.0 0.I 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

R

Figure 2 Scaled solution profiles. Arrhenius Model with e = 0.04 and n = 3. Numbers indicate the solution profile at 1st, 2nd, etc. turning points.

Vol. 41, 1990 Counting reactive flow problems

Table 2 Critical vatues for Arrhenius equation, n = 3, ~ = 0.01

Turning Number of Solution Exothermicity Extrema point unstable modes max u = A 2 type

563

1 0 1.6452 3.360192 2 1 7.5096 !,571710 3 2 13.3308 2.094349 4 3 20.1166 1.910666 5 4 27.6774 1.971815 6 5 36.3220 1,950904 7 6 46,3972 1.958063 8 7 57.7671 1.955584 9 8 71.3296 1.956461

10 9 87.5385 1.956144 11 10 107.2759 1.956254 12 11 131.8807 1.956214 13 12 163,4918 1.956229 14 13 205.7803 1.956223 15 14 265,7036 1.9562254 t6 15 358.6164 1.9562243 17 16 529.3647 1.9562248 18 17 1686.1076 1.9562244 19 16 2145,5515 1.9562254 20 15 2355.6125 1.9562230 21 14 2485.6552 1,9562289 22 13 2576.0195 1.9562137 23 12 2643.0873 1.9562541 24 11 2695.1134 1.9561437 25 10 2736.7768 1.9564507 26 9 2770.9603 1.955584 27 8 2799.5600 1.958063 28 7 2823.8493 1.950904 29 6 2844.7919 1.971815 30 5 2862.9407 1.910666 31 4 2879.1000 2.094349 32 3 2892.8672 1.571710 33 2 2906.6503 3.360192 34 1 18178.9202 1.0863 • 10 -38

max mm max rain max rain max mm max mm max mln max rain max mm max rain max mln max inlx max mm max mln max rain max mm max mm max mm

We note that generally one may assert that the maximum number of solutions is always less than or equal to the number of turning points plus one. The counting procedure itself, once the turning points have been located, is easily automated.

Figure 2 shows the solution profiles at each turning point. These have been normalized by dividing by their maximum value A. These may be compared for small A to those of [2, Figures 5 and 6] for e = 0 and E = 0.1, respectively. For larger A notice that for e -- 0.04 the most peaked temperature distribution occurs at the next-to-last turning point 5. At the last turning point 6 its qualitative profile resembles that at the first turning point 1.


3. Quantitative theory

We first outline the approach for obtaining solution curves and branch points that we used in [ 1, 3, 5]. Then rescaling of the initial value problem, following B6rsch-Supan [2], is employed to obtain a more efficient computation of solution curves and branching points. We also present an improvement to our previous method for locating all the turning points in a bifurcation curve at fixed e.

The Arrhenius equation (1) for spherically symmetric geometry becomes

~-r2-F--r d r + 2 e x p =0 , O _ < r > l

u(0) = A, u(1) = 0. (3.1)

The above equation is equivalent to the first order system

t U 1 ~I/2

' (~-) ( /"/1 Ul ) u 2 = - u 2 - 2 e x p 1-~

Ul(0 ) = A = Ilu, Iloo

u2~--0

(3.2)

where m = n - 1. Assuming the solution is continuously differentiable with respect to 2,

then expanding in terms of 2 at the boundary condition u(1) = 0, we have

~Ul(1) u, ( 1, 2) = u, ( 1, 20) + (2 + 20) T Xo

where u~(r, 2) denotes the solution to equation (3.1). The endpoint condition u1 (1 ,2 )=0 is obtained by adjusting 2 by the Newton-Raphson method

ul(1, 20) 2 = 2 ~ (tgUl(1)'~

The value of

can be found by integrating auxiliary equations obtained by differentiating the equations for u~ and u2 in (3.2) above. Assuming enough smoothness


with respect to 2 of the solution to (3.1) the auxiliary equations are

/

U 3 ~ U 4

,(m) [1 U 4 ~ - - U 4 - -

u3(0) = 0

u4(0) = 0

+ I (1 +eu l ) 2 exp

(3.3)

The system determined by (3.2) and (3.3) was solved using a multistep, variable stepsize, variable order, Adams method.

We now give some further details of the Higher-Order-Calculus Method employed in [1.3, 5]. Consider the one parameter differential equation

(y"(x) = f ( x , y, y' , )o), 0 < x < 1 < "~y(0)=0 , y ( 1 ) = 0

(m:, In the case of interest for us, f ( x , y, y ' , 2) = - + 2 exp . In

the bifurcation curve of 2 versus A = [lyll~ =y(0), branching points or "turning points" occur when the condition d2/dA = 0 is satisfied. If y(x, 2, A) denotes the solution to the equation

y"(x) = f ( x , y, y' , 2) y'(0) = 0, y(0) = A (3.4)

then the boundary condition F1 (2, A) -= y( 1, 2, A) = 0 implicitly defines 2 as a function of A.

By implicit differentiation

d2 (~FI/~A) dA (OF,/~2)"

Hence d2/dA = 0 gives the additional equation

F2(2, A) - OF,/~A = O.

Define the new variables

~y

0 @ d2

Then differential equations for ~b and ~k are obtained by differentiating


equation (3.4) with respect to A and 2:

• "= O + q"+~-2

with initial conditions

{ 4 ( 0 ) = 1, ~ ' ( 0 ) = 0

O(0) O'(0) = 0.

The solution of

{ F1 (2, A) = 0

F2(2, A) 0

is then obtained by the Newton-Raphson technique.

- -~ . AA = _ F F I ( 2 ' , A ' ) ]

~?F2 0F2 A2 [_F2(2 i, AOA -g2

The superscript i denotes the value obtained on the i-th iteration, and the updated values of the parameters are given by A i+~= A i+ AA.

An improvement in the efficiency of the numerical methods employed thus far was obtained using a scaling invariance. If y(x, 2, A) is a solution of (3.2), then for any fl > 0

y(x, ~22, A) = y(~x, 2, A).

This scaling invariance effectively transforms the boundary value problem into the simpler initial value problem

y" =f(x, y, y', 20) y'(0) = 0, y(0) = A

where 2o is an initial guess for 2. The above equation is integrated until a root, ro, for y is found, that is,

y(ro, 20, A ) = 0. The value for 2 is then corrected by 2 =(ro)22o. The problem of finding 2 is now a problem of finding the (only) root of y(x, 20, A). In addition, the Newton-Raphson method for finding the turning points is reduced to an iteration on a single parameter, i.e.,

A i + i = a i - - F2(ro, 20, A') OF2 - - (ro, 20, A 9 OA


Our original technique for determining all the turning points in a bifurcation curve for a given value of ~ was to plot the curve (i.e., solve /:1 (2, A) = 0 for a sequence of values of A, with AA chosen small enough to resolve the extremal points in the curve), to determine the turning points graphically, then to use these values as initial guesses for the Newton- Raphson Reration which converged to the turning points to within some prescribed tolerance. However, as e decreases and the number of turning points increases this technique becomes cumbersome and a more efficient procedure was sought. We were motivated by the following observations of B6rsch-Supan [2].

The number of unstable modes in the solution of

6--} - Au = 2 exp

is related to the number of turning points or branch points of the stationary problem

- A u = 2 exp .

At each turning point there must be a change (increase or decrease) in the number of the unstable modes. The number of unstable modes equals the number of zeros within the interval [0, 1) of the solution to the equation

~ r 2 + ~ dr + ( 1 +eu) 2exp ~ = 0

t7 (0) = 0, tT(0) = 1 (3.5)

where u is a solution of

~ +--r dr + 2 exp = 0 <

u' (0 ) = 0

u(1) = o

The new procedure devised for locating turning points is the following. As before, we solve F1 (2, A) = 0 at a sequence of values of A. But now, we solve (3.5) at the same time and thus obtain the number of unstable modes for solutions at each computed point in the bifurcation curve. Initial guesses of the turning points are made by interpolating between points where the number of unstable modes changes.

As was mentioned in the Appendix to [1], the coordinate singularity at x = 0 is removed by noting that due to the symmetry of the solutions, y'(x = O)= O, and hence, limx~o (y'/x)=y". Then any equation of the


1,0

0.9

0.8

0.7

Figure 3(a)

~d

>-

0.6

0.5

0.4

0.3

0.2

0.i

0.0

0.0

|

\

i , i

0.i 0.2 0.3 0.4 0.5

R

0.6 0.7 0.8 0.9 1.0

Figure 3(b)

1 ,0

0.9

O.B

0 . 7

0.6

S" II td o.5

o.~

0.3

0.2

o11

0.0

0.0

I I i ~ 1

|

@@

--L I l l l 0.i 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


Figure 3(c)

0 . I ' - I r F " I r --r f T T T I r T

0.09

0.08

0,07

0.06

II r ~ 0 . 0 5

002

0.00 0.0 0.I 0.2 0.3 0,4 0.5 0.6 0.7 0.8 0.9 1,0

R

Figure 3(d)

0.01

0.009

0.008

0 . 0 0 7

0.006

0.005

9-

>~ O .004

0. 003

3.002

0,001

0.00 u_ i-- L__A__~__ I ~ _ I l__] i ] B ~t I _._ 0.0 0.I 0.2 0.3 0.4 0.8 0,6 0.7 0,8 0,9 1.0

R


Figure 3(e)

0.01 r ~ i r , I I p T i ~ I

0.009

0,008 I

0.007 |

0.006

II DJ o.005 - >-

~ o . o o 4

0.003

0.002

0.0 0 ,1 0 .2 0.3 0 .4 0 .5 0.6 0 .7 0,8 0 .9 1 . 0

Figure 3(f)

1.0

0.9

0.8

0.7

0,6

S" H r'r! 0.5

>-,

~\ 0.4

0 , 3

0.I

0.0 _ _

0. 000E + 00

R

0.100E - 16

Vol. 41, 1990 Counting reactive flow problems 57t

N t~

>~ \

1 . 0 ,

0 . 9

0 . 8

0 . 7

0 . 6

0 . 5

0 . 4

0 . 3

0 . 2

0 . 1

@-| o.o r ,

0,0 0.I

T- r 7 --'~-- "I -~-- ]- -~ 1 --t-

r I i 0 I. 3 i OI ~ I i P 0,2 .4 0.5 0.6

Figure 3(a)-(g) Solution profiles at turning points 1 through 34, e = 0.01.

\

" x

J 1-- i I I I 0.7 0,8 0.9 1.0

t2

form y" + (m/x)y ' -2f(y)= 0 is replaced by the equation (1 +m)y"- 2f(y) = 0 at x =0 .

The difficult computational work required to implement the techniques discussed above is carried out almost entirely by the ODE solver. The ODE solver uses variable order, variable step size Adams methods. Root finding software is built in so that solving F1 (r0, 20, A) = 0 presents no additional difficulties. Typical computational results have already been presented for the e = 0.04 case in Section 2. The test case e = 0.01 was more interesting, as seen in Figures 3(a)-3(g).

The e = 0.01 test case brought a number of interesting precision questions to the fore. The exothermicity parameter 2 dives toward zero after the next to last turning point and the bifurcation curve remains very flat before finally rising to begin its climb to all values. With double precision on the CRAY 1 we were able to locate the last turning point (u34,234) (18.5 x 103 , 1.086 x 10-38).

To identify all of the turning points on this bifurcation curve we needed continuously the changes in the number of unstable modes, i.e., the changes in the number of zeros of the solution of the associated linear stability equation (3.5). Because the bifurcation curve itself wiggles out to the 7th decimal place, turning points cannot be identified visually, even in the middle of the S-curve, and an automated procedure was used.


The bifurcation curve itself is too long (2 will achieve the value 2 again near u ~ 3 • 1043) to reproduce here. In Figure 3(a)-(g) we show the solution profiles (height normalized as indicated) at the 34 turning points.

4. Bounds

A comparison approach appeared interesting to us and yields some insight into the local convexity regions of the bifurcation curve. We will also give an estimate 2o for where the "smallest" turning point occurs. That is, for 0 < 2 < 2o, then d2/dA 4= 0 for all A such that 2(A) = 2. In particular this implies uniqueness of solutions for all 2 less than 20.

First, we recall some aspects of the HOC (numerical) method discussed in the previous section. Consider the following initial value problem.

I n - 1 U"(r) + - - u'(r) = - 2f(u)

p

lu'(O) = o, u(O) = A

where f(u) = exp{ l - - ~ u } (4.1)

together with the boundary condition u(1)= 0. Denote a solution to the above equation by u(r) - u(r, A, 2). The condition u(1) = 0 implicitly defines 2 as a function of A since u(1, A, 2) = 0. So

d d2 dA u(1, A, 2) = uA ( 1, A, 2) + u~( 1, A, 2)

o r

d2 uA(I, A, 2) u (1, A, 2 )

From equation (4.1), uA(r, A, 2) can be obtained from the following initial value problem.

f u'j(r) + n ; l u '4(r)= -2f'(u)u'4(r) (4.2)

uA(0) = 1, uA(0) = 0.

Let co(r)= UA(r). We are led to the study of the pair of simultaneous

Vol. 41, 1990 Counting reactive flow problems

differential equations, for the case n = 3.

~r2 + r~r r + 2 exp =0 , 0 < r < l

u(0 )=A, u (1 )=0 , u ( 0 ) = 0

& ~ 1 7 6 u -~r2+r-~-r + ( l + e u ) 2exp ~ o~=0,

co(O) = 1, r = 0

By the above implicit differentiation, we have

d2 = 0 i f fog( l )=0 . dAA

We make a change of variables

y(r) = r~o(r), O <<- r < 1

in equation (4.4) to obtain

+ eu)2 exp ~ y(r) = 0

[ y ( 0 ) = 0 , y ' ( 0 ) = l , y(1)=co(1)

Again,

d A A = O iff y(1) = O.

573

(4.3)

O ~ r ~ l (4.4)

(4.4')

We now investigate the oscillatory (or lack of) nature of the function y(r) in (4.4").

We first recall a result of the Sturm theory, see for example Ince [10] for further details.

Consider

(A) L(y) = -~x K - Gy = 0

where K, G are continuous on [a, b]. Then the following Sturm's theorem, due to Picone, is useful.

Sturm's Fundamental Theorem. Suppose u is a solution of

( ~x K1 - G I u = O


and v is a solution of

--~x k, dxJ -G2v=O

where K1 >/s > 0, G1 >- G2. Then if/~1 and #2 are two consecutive zeros o f u(x), then v(x) has a least one zero v~ between #1 and #2- The theorem also holds if vl =/~l or v2 = #2. In this case we have another zero v2 of v(x) with

~/1 < /)2 < ].'/2"

Theorem 4.1. Let

/ 1 - 2e'~ /~2(A) = )t(A)4e 4 e x p ~ - - - - ~ ) .

Then d2/dA ~ 0 whenever/~2 < 7~2"

Proof. Sturm's theorem can be used to determine the oscillatory or non-oscil latory nature of solutions to equat ion (A). Let

k = min K(x), g = min G(x). X X

We now use Sturm's theorem with compar ison equat ion

(B) -gy=O

to obtain that the solutions of (A) do not oscillate more rapidly on [a, b] than the solutions of B.

In the problem at hand, f rom (4.4'), K(x) = 1 and

- 2 ( A ) { u(x, A, 2) } O < x < l" G(x) = (1 + eu(x, A, 2)) 2 exp 1 + eu(x, A, 2) '

We give two estimates for g = min0 ~ ~, ~ ~ G(x).

Estimate 1. Note that

u(x) exP{1 + ~u(x)}

m a x o~x~ , ( l + eu(x))2

Hence

g > --2(A) exp{ 1 AeA t" +

Vol, 41, 1990

Set

Counting reactive flow problems 575

The comparison equation

d2y (C,) d ~ + P l ( A ) y = 0 , 0 < x < l

s i n ( ~ x ) . The interval between consecutive has oscillatory solution zeros of s i n ( ~ x ) has length z ~ / ~ . Hence, whenever

- - > 1 ,

then no solution of (4.4') can have more than one zero in [0, l]. Thus

d2 7~ e 0

whenever

~: > ~ (A).

E s t i m a t e 2

Observe

max exp = max exp o <- x ~, (1 + ~U(X)) 2 1 + 8U(X) j O<=y <= ,4 ( 1 + Ey) 2

since y = u(x) is monotone decreasing on [0, 1]. Let

f ( Y ) = ( l + g y ) 2 e x p , 0 - < y < o o .

Then f(0) = 1, limy~ +oof(Y) = 0. Also,

f (Y) = (1 y -" exp

2~2 ,

Hence if

l - 2~ Jc

then

f ' ( y c ) = 0 and f ( y c ) = 4 e 2 e (~-2~)/~


Therefore f(Yc) is the maximum o f f ( y ) if

Let

1 0 < e < ~ and A - > - - 1 - 2g

2~2 �9

#2(a) = 2(A)4e 2 e ~ - 2~/~

Using the comparison equation

a2y (C2) ~ x 2 + # 2 ( A ) y = 0 , 0 < - x < l ,

we have

d2

whenever

~r 2 >/a2(A). []

Lastly, we compare the estimate dA/dA ~ 0 whenever ~z2> #2(A) to numerical results obtained when ~ = 0.04. The numerical computat ion (see Table 1) gave 2 = 2.438 • 10 -7 as the value of 2 where the last turning point occurs. The rough analytical estimate above gives for e = 0.04 that dA/dA ~: 0 whenever 0 < 2 < (g2/4e2) e (2'- 1)/, ~ 1.57 x 10 -7. This is a very favorable comparison. Moreover the analytical etimate for the more difficult test case e = 0.01 provides that d2/dA r 0 whenever 0 =< 2 < 6.78 x 10 -39. The numerical result (see Table 2) was )~ = 1.0863 • 10 -38. Again we see a very favorable comparison between the qualitative and quantitative theories.

5. R e m a r k s

1. The approximations alluded to in the Introduction were the well- known Frank-Kamenetski i approximation 1 + ~u ~ 1 and the Bazley-Wake [11] a p p r o x i m a t i o n 1 + ~u ,~ 1 + llull. A main point o f [ l ] was that by the proper use of modern ODE Solvers one may, numerically at least, "exactly" solve the " t rue" problem (1), thereby lessening the need to rely on informa- tion gleaned analytically from approximations. Indeed, the approximations can be misleading. The Frank-Kamenetski i approximation fails for 2 greater than a critical 2c, in the sense that beyond 2c solutions for it no longer exist. On the other hand the bifurcation curve of the Frank-Kamenetski i approximation (e = 0) wiggles along roughly in the same 2 interval (near the asymptote 2 = 2) as that for the bifurcation curve of the true equation with


small e, although the locations of the turning points, and hence the numbers of solutions, do not jibe. See Figure 4 of [1] and Figure 2 of [2]. The Bazley-Wake approximation, although providing solutions for all it as does the true equation, departs significantly from the correct bifurcation curve after the second branch. See Figure 4 of [2] for e = 0.1 and Figure 7 of [5] for e -- 0.01.

It should be noted that as u becomes large, all of these single local equations become less reliable in modelling the chemical reaction from which they were derived.

2. An early paper examining bounds and the relations between the stationary solutions and the associated time dependent reactive flow problems was Chandra and Davis [2]. A recent paper by Gray and Kordylewski [i3] examines possible relations between the multiple stationary solutions and traveling waves in the associated time dependent flow equations. The full connections between the unsteady and steady equations remain of interest.

3. The fact that for small positive e there is a stable high temperature solution branch for very small positive exothermicity values 2 is perhaps of independent interest. It allows the possibility of a new interpretation of "blowup": the jump not to 0% but to the high temperature branch. Of course one could object that the Arrhenius equation (1) holds less well as u departs from zero values (see [5] for a discussion of the physical derivation of (1); which equation to use depends on where one wishes to cut off a power series). But the Frank-Kamenetskii approximation is even truncated (see Section 2 of [5]).

4. B6rsch-Supan mentions (end of Section 2 in [2]) that it is not clear what happens (computationally) at the singular point r = 0. We addressed this question in the Appendix of [1] by using a modified equation for the first time step. This equation, as we mentioned in section 3 herein, amounts to the application of well known techniques for the removal of such singularities at symmetry points in the solution.

As was discussed in [5, Section 9], the resolution of the solution at the first time step is actually a much less important question than the resolution of the delta-function like profile in the ensuing time steps. Up to 10,000 steps were needed in some cases to resolve those solution profiles. In the most extreme cases the steepness of profile drop, up to 80% (from the value 1 to the value 0.2) occurred within 0 < r < 10 -15, a formidable computational problem.

5. Table 2 reveals a very interesting phenomenon of solution pairing at the turning points for e = 0.01. This qualitative behavior is quite different from that in the Frank-Kamenetskii approximation e = 0. A referee has pointed out to us that an investigation of such phenomena by use of formal asymptotic expansions was carried out in [14].


References

1. K. Gustafson and B. Eaton, Exact solutions and ignition parameters in the Arrhenius conduction theory of gaseous thermal explosion. Z. angew. Math. Phys. 33, 392-405 (1982).

2. W. B6rsch-Supan, On the stability of bifurcaton branches in thermal ignition. Z. angew. Math. Phys. 35, 332-334 (1984).

3. K. Gustafson and B. Eaton, Calculation of critical branching points in two parameter bifurcation problems. J. Comp. Phys. 50, 171-177 (1983).

4. E. Ash and K. Gustafson (to appear). 5. K. Gustafson, Combustion and Explosion Equations and Their Calculation, chapt. 7. In Computa-

tional Techniques in Heat Transfer (Eds. R. Lewis, K. Morgan, J. Johnson and W. Smith), pp. 161-195 Pineridge Press (1985).

6. B. Gidas, Y. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68, 209 (1979).

7. H. Keller and D. Cohen, Some positone problems suggested by nonlinear heat generation. J. Math. Mech. 16, 1361 1376 (1967).

8. P. Bader, On a quasilinear elliptic boundary value problem of nonlocal type with an application in combustion theory. Z. angew. Math. Phys. 35, 771-779 (1984).

9. R. Shivaji, Uniqueness results for a class ofpositone problems. Nonlin. Analysis 7, 223-230 (1983). 10. Ince, Ordinary Differential Equations, Dover Publications 1956. 11. N. Bazley and G. Wake, Criticality in a model for thermal ignition in three or more dimensions. Z.

angew. Math. Phys. 32, 594-602 (1981). 12. J. Chandra and P. Davis, Rigorous bounds and relations among spatial and temporal approximations

in the theory of combustion. Combustion Sci. Techn. 23, 153-162 (1980). 13. P. Gray and W. Kordylewski, Travelling waves in exothermi c systems. Proc. Roy. Soc. London

A416, 103-113 (1988). 14. A. Kapila, B. Matkowsky and J. Vega, Reaction diffusion system with Arrhenius kinetics: peculiarities

of the spherical geometry. SIAM J. Appl. Math. 41, 382-401 (1981).

Summary

Computational methods and comparison theory enable, when combined, an enhanced capability for counting the number of solutions in combustion equations. Very good lower bounds for the last turning point reveal a stable high temperature "explosion branch" for very small positive exothermicity.

(Received: May 30 1989; revised: November 7, 1989)

Counting the number of solutions in combustion and reactive flow problems

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