Kent State University, Kent, Ohio 44242, USA;. Paul A. Farrell. Department of ...... Thus it might be expected that a classical shooting based method such as that ...
Numerical Solution of Nonlinear Singular Perturbation Problems Modeling Chemical Reactions Relja Vulanovic Institute of Mathematics, University of Novi Sad, 21000 Novi Sad, Yugoslavia, visiting: Department of Mathematics and Computer Scinece, Kent State University, Kent, Ohio 44242, USA; Paul A. Farrell Department of Mathematics and Computer Science, Kent State University, Kent, Ohio 44242, USA; Ping Lin Department of Mathematics, Nanjing University, Nanjing 210008, PR China, present address: Department of Mathematics, University of British Columbia, Vancouver, BC, Canada V6T 1Y4
Abstract
A linearization method is proposed for quadratic singularly perturbed boundary value problems which model some catalytic reactions. Each of the resulting linear problems is solved numerically by using non{equidistant nite{di erence schemes. Numerical results con rm that the method is accurate uniformly with respect to the perturbation parameter.
1 Introduction Singularly perturbed di erential equations arise as mathematical models for various phenomena in physics, chemistry, biology and other sciences. In this paper we shall consider their application to some chemical problems. We are motivated by [2], where the following problem has been considered: (1)
?"u00 + ur = 0; x 2 (0; 1); ?u0(0) = 0; u(1) + u0(1) = 1:
(2) The problem arises in catalytic reaction theory: it models an isothermal reaction which is catalyzed in a pellet. u is the normalized concentration of the reactant, x ispthe dimensionless distance from the center of the pellet (x = 0) to the mouth (x = 1), 1= " is the so called Thiele modulus de ned as K=D, where D is the di usion coecient and K is the reaction rate constant. Thus, when D 0 there are no
boundary layers, and thus, these problems are easier to solve numerically. Because of that, we shall consider only the more dicult case = 0. When = 0, u" has a layer at x = 1. The layer is exponential when r = 1, while for r 2 it is a power layer. There is a related problem for isothermal reactions in spherical catalyst particles, considered in [12]:
(4)
?"u00 + "(1 + u)? u0 ? 2"x? u0 + (1 + u)ur = 0; x 2 (0; 1); 1
2
1
subject to the conditions (1.2) with = 0. It has been concluded in [12] that the solutions of (1.3) and (1.4) are close when " and ti+1 q + 3h; 3. < ti+1 < q + 3h: In the rst two cases the following estimate of Ri will be used: (36) jRij M?1 [1 + m 1 ?xi+1 ]? ?1hiz(xi+1)r?1;
where z is as in (2.11). Firstly, let us prove (3.17). Using (2.9) we rst get: jRij Mhi" x max ju000(x)j + ru(xi)juk(xi+1)r?1 ? uk (xi)r?1j: xx ?1
+1
i
i
Next, on di erentiating (2.9), we obtain "ju000(x)j Muk (x)r?1 [ju0(x)j + ju0k (x)j]: Then from
u(x) uk (x) u0 (x) (1 + )z(x) Mz(xi+1 ); x xi+1 ; we conclude that
jRij Mhiz(xi )r? +1
1
max
xi?1 xxi+1
[ju0(x)j + ju0k (x)j]:
Finally, to show (3.17) holds, we require
ju0k (x)j M? [1 + m 1 ? x ]? ? ; k = 0; 1; : : ::
(37)
1
1
This follows immediately for k = 0:
0 u00 (x) = z 0(x) ? z 0 (0) z 0(x): For k = 1; 2; : : :, (3.18) follows after integration of (2.9):
ju0
k+1 (x)j
Z x Z x r ? 1 ? 1 u0 (s)rds uk (s) ds M" M"
0
M?1 [1 + m 1 ? x ]? ?1 :
0
Now recall that
uk (x) u0(x) z(x):
From this and (3.17) we see that to prove (3.16) it suces to show h ?1 [1 + m 1 ? xi+1 ] Mh: i
Turning to the special mesh, we conclude that the above inequality holds if we prove (38) S := 0(t )?1 [1 + m 1 ? (ti+1) ]? ?1 M: i?1
12
In case 1 this follows from and
0(ti?1 ) M;
1 + m 1 ? (ti+1 ) 1 + m 1 ?( ) M?p=(p+1) ;
on noting that by the choice of p in (3.11):
p ( + 1) 1: p+1
In case 2,
ti?1 ? q 13 (ti+1 ? q) h;
and
0(ti?1 ) '0(ti?1 ) M(ti+1 ? q)?p?1 : Using this and (ti+1 ) = '(ti+1 ) with ms 1, we get S M (ti+1 ? q)?p?1+p( +1) M; and (3.19) is proved in this case too. We shall use a di erent expression for Ri in case 3:
ju00(x)j + rz(xi )r : jRij 2" x ?1max xx +1 i
Then using (2.9) we get
+1
i
jRij Mz(xi )r ; +1
thus, in order to prove (3.16), we have to show:
z(xi+1 ) = z((ti+1 )) Mh:
(39) But, in case 3, we have
(ti+1 ) (q + 3h) = '(q + 3h);
and it follows that:
z(xi+1 ) Mhp Mh;
and thus, (3.20) is proved. Thus, in all three cases, (3.16) holds and, as indicated earlier (3.14) follows. 2 This result justi es our method. Moreover, from (2.14) we see that u0 is a good approximation to u" when " is small. Therefore, there is no need to seek a numerical approximation, unless its error is smaller than O( ). We consider this case in the next theorem. Theorem 3.2 Let fwig be the unique solution to the discrete problem
?"D00wi + ruk (xi)r? wi = (r ? 1)uk (xi)r ; i = 1; 2; : : :; n ? 1; ?D w = 0; wn = 1; 1
+
on the mesh (3.7-9) with
0
s m1 ; p 3 ; 13
and with (40) h c 3 ; where c3 is a constant independent of h and ". Then:
jwi ? uk (xi)j Mh; i = 0; 1; : : :; n: +1
The proof will be omitted since its technique is the same as that of Theorem 3.1. However, some comments are due. When (3.21) is not satis ed, then u0 is a O(h) approximation to u" . But on the other hand, (3.21) implies that only cases 1 and 2 from the proof of Theorem 3.1 are possible. We can handle these cases without the arti cial shift from (3.13), provided p 3= . Since there is no shift, we can even expect accuracy of second order. The rst order discretization of the left boundary condition, however, appears, in practice, to decrease that accuracy to rst order, particularly for large ".
4 Numerical Results In this section we shall present numerical results for problems (1.3) and (1.4). In all cases, the linearization is performed as indicated in section 2. The linearized equations are then solved using upwinded di erences, as given in the equations (3.3-5). Our numerical experiments show that in practice it is not necessary to take u0 very close to u" , as we did in the theoretical analysis when we chose u0 satisfying (2.11) for the problem (2.7-8). Thus, the initial guess in all cases is u0 (x) 1. The non-equidistant mesh described in section 3 was used except as indicated below. The values chosen for the parameters of the mesh were p = 1, q = 0:5, and, for the case r = 1, s = 1, and for r > 1, s = 1=m, where m is as in (2.11). Finally the parameter is determined from (3.10). In order for to exist the condition s q ? 1 1; given in section 3.1, must be satis ed. For large ", that is for values of " > 1=2 or " > 1=64, depending on r, this condition is violated. In these cases a standard equidistant mesh was employed. The uniform rate of convergence was determined using a variation of the double mesh method described in [4]. This involves calculating the double mesh error
end" = 0max jun ? u2nj in i which is the di erence between the values of the solution on a mesh of n points and the interpolated value for the solution at the same point on a mesh of 2n points. The uniform rate of convergence is then given by
pd = averagenpnd where and
pnd = ln(end ? e2dn )= ln(2) en : end = max " d" 14
"
1/ 2 1/ 4 1/ 8 1/ 16 1/ 32 1/ 64 1/ 128 1/ 256 1/ 512 1/ 1024 1/ 2048 1/ 4096 1/ 8192 1/ 16384 1/ 32768 1/ 65536 1/ 131072 1/ 262144 1/ 524288 1/ 1048576 1/ 2097152 1/ 4194304 1/ 8388608 1/ 16777216
pnd
8 1.16 1.22 1.31 1.44 1.82 1.64 1.92 2.05 1.98 1.86 1.75 1.70 1.71 1.74 1.76 1.78 1.79 1.81 1.82 1.82 1.83 1.84 1.84 1.84 1.66
Number of Mesh Points n 16 32 64 128 1.08 1.04 1.02 1.01 1.11 1.06 1.03 1.01 1.17 1.08 1.04 1.02 1.24 1.12 1.06 1.03 1.70 1.19 1.10 1.05 1.99 2.00 2.00 2.00 1.94 1.98 2.00 2.00 1.97 1.97 2.00 2.00 1.94 1.97 2.00 2.00 1.90 2.00 2.00 2.00 1.98 1.99 2.00 2.00 2.04 2.00 2.00 2.00 2.05 2.01 2.00 2.00 2.05 2.00 2.00 2.00 2.05 2.01 2.00 2.00 2.05 2.01 2.00 2.00 2.05 2.01 2.00 2.00 2.05 2.01 2.00 2.00 2.05 2.01 2.00 2.00 2.05 2.01 2.00 2.00 2.05 2.01 2.00 2.00 2.05 2.01 2.00 2.00 2.05 2.01 2.00 2.00 2.05 2.01 2.00 2.00 1.08 1.04 1.02 1.01
256 1.01 1.01 1.01 1.02 1.03 1.80 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 1.01
Average 1.05 1.07 1.11 1.15 1.31 1.91 1.97 2.00 1.98 1.96 1.95 1.96 1.96 1.97 1.97 1.97 1.98 1.98 1.98 1.98 1.98 1.98 1.98 1.98
Uniform Rate p+d : 1.14 Table 1: Double Mesh Rates of Convergence for problem (1.3) with r = 1; = 1 Some cautions on the interpretation of uniform rates of convergence derived in this way are also given in [4]. Table 4.1 gives in tabular form the values of the local rates of convergence, for a range of n and ", for the problem (1.3) with r = 1 and = 1. These are given by
pnd" = (ln end" ? ln e2d"n)= ln(2): The rightmost column contains the average rate of convergence for the row, that is for a xed value of ", given by averagen pnd" . The bottom row contains the values of pnd . This indicates a uniform rate of convergence pd = 1:14, although it should be noted that examination of the pnd , indicates that this is unduly in uenced by the rate for n = 8 and a better estimate would be 1:01. Although the derivation of the di erence schemes (3.3-5) in section 3, employed an upwinded di erence scheme, we remark that uniform convergence can be achieved with a 15
scheme which uses centered di erence approximations to the rst derivatives throughout. This is what is presented in Table 4.1. We emphasize that we use the centered di erence approximation instead of D0 in (3.4), as well as for u0k (xi ) appearing in P and F . We still use the directed di erence approximation
?D w = 0; however, for the boundary condition at x = 0. This appears to degrade the performance of the centered di erence scheme to rst order, when " is of order 1. For small ", the convergence is classically O(h ) however. It should be remarked that for problem (1.4) this +
0
2
e ect does not occur. This may perhaps be explained by considering that for problem (1.3), with small ", and for problem (1.4) the solution is almost constant near x = 0 and hence the upwind approximation D+ w0 attains second order accuracy. Tables 4.2 and 4.3 give the uniform rates of convergence, calculated using the double mesh method for problems (1.3) and (1.4) respectively. It is clear that both schemes
Scheme 0.0 Upwinding Centered 0.5 Upwinding Centered 1.0 Upwinding Centered 2.0 Upwinding Centered 3.0 Upwinding Centered
1 1.20 1.20 1.11 1.16 .99 1.14 .85 1.10 .77 1.07
r
2 1.25 1.25 1.21 1.24 1.08 1.23 .92 1.22 .87 1.21
3 1.59 1.59 1.55 1.57 1.52 1.56 1.37 1.54 1.29 1.52
Table 2: Uniform Rates of Convergence for Problem (1.3)
Scheme 0.0 Upwinding Centered 0.5 Upwinding Centered 1.0 Upwinding Centered 2.0 Upwinding Centered 3.0 Upwinding Centered
1 1.14 1.94 .84 1.96 .64 1.98 .67 1.99 .64 2.00
r
2 1.18 2.10 1.00 2.09 .74 2.08 .67 2.04 .61 1.96
3 1.52 1.69 1.38 1.67 1.20 1.66 1.00 1.65 .89 1.64
Table 3: Uniform Rates of Convergence for Problem (1.4) are uniformly convergent for both problems. Further, it should be noted that the centered 16
di erence approximation shows second order convergence for problem (1.4), in contrast to the case for problem (1.3). To compare our results with those given in [12], we calculate the E ectiveness Factor. This represents the actual reaction rate divided by the rate which would occur if the intraparticle reactant concentration were everywhere identical to that at the pellet surface. The e ectiveness factor, for an r-th order reaction with volume change in a spherical catalyst pellet, such as that treated by equation (1.4), is given by )jx=1 : = 3"(du=dx 1+ It is clear that this value can be calculated from the numerical solution of problem (1.4). The most direct way would be to estimate the derivative (du=dx)jx=1 by the directed di erence D?wn at the boundary point xn = 1. This is presumably a similar procedure to that used in [12], although no details are given there. The e ectiveness factors calculated using this method, on a mesh of 256 points, are given in Table 4.4. With the exception of the
4
1
"
1/9 1/36 1/100
r = 0
1.0 .9979 .9974 .9946 .9870 2.0 .9977 .9967 .9892 .9594 3.0 .9976 .9961 .9837 .9083 r = 1 1.0 .9663 .8885 .5627 .3342 2.0 .9518 .8488 .5000 .2903 3.0 .9382 .8153 .4565 .2613 r = 2 0.0 .9664 .8891 .5667 .3397 1.0 .9385 .8165 .4599 .2643 2.0 .9137 .7626 .4007 .2251 3.0 .8913 .7201 .3610 .1998
9308 .7666 .6608 .2135 .1841 .1649 .2189 .1679 .1420 .1254
Table 4: E ectiveness Factors for (1.4) using Di erences case r = 0 our results are in agreement with those in [12] to within 0:3%. To examine this discrepancy further one should note that the case r = 0 has an exponential layer as opposed to a power layer for r > 1. Thus it might be expected that a classical shooting based method such as that used in [12] might fare badly. As remarked in section 1, equation (1.4) can be rewritten in the form (1.5), and in the case of r = 0, becomes !0 2 0 x u ?" 1 + u + x2 = 0:
This may be solved exactly for u0 (1) and gives u0(1) = (1 + )=3". Thus the e ectiveness factor 1, independant of " and . Our results are signi cantly closer to this than the results in [12]. It is clear that the inaccuracies in our case are a result of using a di erence approximation to the derivative at a boundary with an exponential layer. To avoid this we 17
may make use of equation (1.5) to rewrite the derivative as an integral from 0 to 1 and then employ quadrature. x2 u0 ? x2u0 = 1 Z 1 x2ur (x)dx: 1 + u x=1 1 + u x=0 " 0 Using the fact that u0(0) = 0 and u(1) = 1 we get the folowing expression for the e ectiveness factor: Z 1 = 3 x2 ur (x)dx: 0 Table 4.5 gives the e ectiveness factor calculated using this formula with the integrals being evaluated using the trapezoidal rule. In this case the result is exact for the case with r = 0. Thus to summarize, the method described in this paper is in practice uniformly convergent
4
1
1.0 1.0000 1.0000 2.0 1.0000 1.0000 3.0 1.0000 1.0000 1.0 2.0 3.0
.9685 .9543 .9409
.8914 .8523 .8193
0.0 1.0 2.0 3.0
.9685 .9409 .9162 .8940
.8915 .8196 .7662 .7241
"
1/9 r = 0 1.0000 1.0000 1.0000 r = 1 .5664 .5046 .4617 r = 2 .5703 .4652 .4069 .3678
1/36 1/100 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .3366 .2933 .2645
.2151 .1861 .1671
.3434 .2696 .2312 .2063
.2213 .1714 .1460 .1296
Table 5: E ectiveness Factors for (1.4) using Quadrature for both problem (1.3) and problem (1.4). The paper also introduces a more accurate method for calculating the e ectivness factor numerically.
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[5] V. Hutson, J. S. Pym, Applications of Functional Analysis and Operator Theory, Academic Press, London, 1980. [6] L. V. Kantorovich, G. P. Akilov, Functional Analysis in Normed Spaces, Pergamon Press, New York, 1964. [7] V. D. Liseikin, V. E. Petrenko, On Numerical Solution of Nonlinear Singularly Perturbed Problems, (Russian), Preprint 687, SO AN SSSR, Novosibirsk, 1987. [8] L. B. Rall, Computational Solution of Nonlinear Operator Equations, Wiley, New York, 1969. [9] R. Vulanovic, On a numerical solution of a type of singularly perturbed boundary value problem by using a special discretization mesh, Univ. u Novom Sadu Zb. Rad. Prirod.{Mat. Fak. Ser. Mat. 13, 187{201 , 1983. [10] R. Vulanovic, On numerical solution of singularly perturbed boundary value problem with two small parameters. In: Numerical Methods and Approximation Theory II (D. Herceg, ed.), Institute of Mathematics, Novi Sad, 1985, 65{71. [11] R. Vulanovic, On a numerical solution of a power layer problem. In: Numerical Methods and Approximation Theory III (G. V. Milovanovic, ed.), University of Nis, Nis, 1988, 423{431. [12] V. W. Weekman, Jr., R. L. Gorring, In uence of volume change on gas{phase reactions in porous catalysts, J. Catalysis 4 (1965), 260{270.
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