N EIJqEVIER ('~nnpm. Mctlmds Appl. Mech. Engrg. 129II 9961 255-269

Computer methods in applied

mechanics and enginemlng

Linearized O-methods I. Ordinary differential equations

J.I, R a m o s ' , C,M, G a r c i a - L d p e z Deparmmemo de l,ent, uaj,,s y ('i,,ncm~ d,, la ()m~lmmcifm. I-. 12i. Ing,'niero~ Imlus~ciah's. Univ,,r.~idad de M~ihlga. Plaza El Ejido,

.~tn. 2¢~Ol,~-,~hiht,~,t, Sp,¢in

Received 27 April 1995: revised 3 July 1995

Abstract

Fully-linearized O-methods for autonomous and non-autonomous, ordinary differential equations are derived by approximating the non-liuear terms by means of the lirst-degree polynomials which result from Taylor's series expansions. These methods are implicit but result in explicit solutions, A-s!abtc. consistent and convergent; however, they may be very demanding in terms of both computer time and storage because the matrix to be inverted is, in general, dense. The accuracy of fullyqinearizcd f-)-mcthods is comparable to that of the standard, implicit, iterative O-methods, and deteriorates as the value of O is decreased tram (-) .= 11.5, for which both O- and fuily-linearized O- methods are second-order accurate. P~mially-linearized O-methods based on the partial linearization of non-linear terms have also been developed. These methods result in diagonal or triangular matrices which may be easily solved by substitution. Their accuracy, ho,xever, is lower than that of fullv-linearized O-methods.

I. Introduction

The spatial discretizalion of evolutionary, partial differential equations by finite difference or finite element methods results in a compiex system of ordinary differential equations for the time derivatives of the dependent variables at the grid points or the nodal amplitudes, respectively. This system of ordinary differential equations may be discrelized by means of a variety of finite difference methods, e.g. explicit, semi-implicit arid implicit techniques. A fully implicit discretization, however, may require the evaluation of a Jacobian matrix at each iteration, and may be very demanding in terms of both computer time and storage. Therefore, it would bc desirable to develop implicit methods for the solution of ordinary differential equations which do not require iterative techniques. The linearized O-methods proposed in this paper fall into this category, and are based on the lincarization of implicit O-techniques. This linearization may be carried out with respect to all or a few dependent variables; in the first case, the Iinearization yields a fully-linearized method, whereas in the second one, a partially-linearized technique is obtained, Thus fully-linearized O-methods employ the exact Jacobian matrix, whereas partially-linearized techniques use an approximate one.

The iinearized, implicit methods proposed here can be applied to a variety of evolution problems governed by ordinary or partial differential equations. In particular, they may be used for the numeri- cal solution of non-linear, one-dimensional, reaction-diffusion equations with linear diffusion operators which are frequently solved by means of finite difference [1], linite element [2] and decomposition [3]

"Corresponding author.

01145-7825/96/$15.00 @ 1996 Elsevier Science S.A. All rights reserved 5;SD! t11/45-7825195 H)11915-9

256 J.L Ramos, C.M. Garcia-L6pez/Comput. Methods AppL Mech. Engrg. 129 (1990) 255-269

methods. These equations may also be solved by means of operator-splitting techniques [4. 51 which decompose the reaction-diffusion operator into a sequence of reaction and diffusion operators. The reaction operator represents a system of non-linear, first-order, ordinary differential equations whose implicit discretization yields a system of non-linear algebraic ones which are frequently solved by means of the Newton-Raphson method until a user's specified convergence criterion is satisfied.

Non-iterative, implicit methods for reaction--diffusion equations may be obtained by linearizing the non-linear reaction terms with respect to time [4, 5]. However, since the linearization should be performed with respect to all the dependent variables, the resulting discretized system of linear algebraic equations has a banded structure of N P blocks of N x N matrices, where N P and N denote the number of grid points and the number of reaction-difft,sion equations or dependent variables, respectively [4, 5]. However, a system of N linear, algebraic eq,J~Llions may be easily obtained if the reaction terms are only linearized with respect to the dependent vari~:ble whose equation is being solved at the expense of some loss in accuracy.

In a recent paper, Twizell et al. [6] introduced an implicit diseretization for ordinary differential equa- lions which results in explicit expressions for the values of the dependent variables at the current time level, and used it to study the numerical solution of two, non-linearly coupled, one-dimensional, reaction- diffusion equations.

In this paper, we present fully-!inearized, implicit, O-methods for ordinary differential equations which require ihe inversion of a matrix at each time step. We also develop partially-linearized methods which result in explicit expressions for the values of the dependent variables at each time step, and compare them with the method proposed by Twizell et al. [6]. In contrast with the schemes of Twizell et al. [6], the partialty-linearized methods presented in this paper are derived from the partial time linearization of the non-linear terms. Depending on the order in which the equations are partially linearized, it is shown that, for a system of N non-linear, ordinary differential equations, there are N! + 1 partially-linearized, implicit methods which result in explicit expressions for the values of the dependent variables and whose accuracy is O(h), where h denotes the time step.

Both the fully- and the partially-linearized ¢9-methods presented in this paper may be used with finite difference or finite element discretizations of reaction-diffusion equations. If finite difference methods are employed, the partially-linearized methods presented in this paper result in a sequence of linear algebraic equations which have a triangular matrix structure for each dependent variable, whereas fully- linearized, implicit methods yield a block tridiagonal structure for all the dependent variables.

2. Fully-linearized O-methods

Consider the following initial value problem

x' = f (x , t), x(to) = x0, (1)

where f is a function in C t (I~ s × ~., ~N) and x C ~.s (-)-methods for Eq. (l) can be expressed as

Xn+ I - - X n h (gf(x,,,t,.) + (1 - ~9)f(x,.~,t, . .~), ( 2 )

where h = t,,+~ - t , is the stepsize. Eq. (2) is. in general, non-linear due to the non-linearity off, therefore. its solution requires an iterative technique. Iterations may, however, be eliminated by approximating f (x ,÷l , t,j+l) by the Taylor polynomial of degree 1 of f(x,,+t, tn÷l) around (x,,, t,,) as

[ o, ] x,,.lh- .r,, _ Of{x,,, t,,) + (1 - O) f (xn , t,,) + ~-(x,,, t,,)(t,,+! - t,,) + ~ . , f,,)(x,,+l - - Xn) + O ( t l 2) (3)

Denoting f ( x , , tn) by f,,, Or~Or (x,,, t,,) by T,, and Of/Ox (x,,, t,,) by J,,, and substitution of Eq. (3) into Eq. (2) yields

x,,.l - x,, = hq~f(t,,, x , , h) = hAl, l((,, + T,,h) , (4)

where A,, = I - h(1 - O)J,,, J is a Jacobian matrix, and I is the N x N identity matrix.

J.l. Ramos. C.M. Garcia.L6pez /Comput. Method.s Appl. Mech. Engrg. 129 (1996)255-269 257

Eq. (4) provides an explicit expression for x,,÷l, corresponds to an implicit method which has been obtained by keeping the linear terms in the Taylor's series expansion of f ( x , t), and is here referred to as fully-linearized (0-methods. The characteristic ooivnomial of the~e methods is p(~¢) = ¢: - 1. Therefore, fully-linearized (0-methods are zero-stable and are consistent when f is Lipschitz-continous, i.e.

p( l ) = 0, q)r(t,,,x(t,,),h) = p'(l) f(x(t , , ) , t,,) , (5)

where the prime denotes differentiation. Since fully-linearized (')-methods are consistent and zero-stable, they are also convergent [7]. Therefore,

these methods are A-stable because they coincide wit.~t ~-.~-mcthods for linear problems. Fuily-linearized (-)-methods are only second-order accuratc when (-) = 0.5, and coincide with the explicit Euler forward method for O = 1 [7].

The solution of Eq. (4) requires the inversion of the matrix A,, at each time step which may be a very demanding task if N is sufficiently large since the matrix A,~ is, in general, dense. If this matrix is approximated by either a diagonal or a triangular matrix, one may substantially reduce the computational time required to determine x,,+l at the expense, of course, of some loss of accuracy. Linearized (9-methods based on approximating the matrix A,, by either a diagonal or a triangular matrix are here referred to as partially-linearized (-)-methods and are the subjcct of next section.

3. Partially-linearized O-methods

As indicated in the previous section, fully-linearized (-)-methods are based on the linearization of f ( x , t) with respect to t and all the components of x. If the linearization of f ( x , t ) is performed with respect to only some components of x. one obtains partially-linearized O-methods. In this section, we consider partially-linearized t'-)-methods which replace the matrix A, by either a diagonal or a triangular matrix so that its inversion is an inexpensive task since the unknowns may be easily determined by forward substituti.on.

3.1. Partially-linearized diagonal O-methods

If the matrix A,, in Eq. (4) is replaced by the diagonal matrix whose elements coincide with those in the main diagonal of A,,. then Eq. (4) becomes

x,,+t = x . + hE.(f , , + I~T,,), (6)

where E,, = [l - h(I - (0)diag(J,,)] -t and diag(L,) denotes the N × N diagonal matrix whose elements are those of the main diagonal of J,.

Eq. (6) provides an explicit expression for x,,+~ which corresponds to the linearizalion of ~(x, t) with respect to t and xi, i = 1,2 ..... N. This partial linearization results in an accuracy of O(h) even for (-) = 0.5 since the linearization has only be performed with respect to a dependent variable in each equation (cf. Eq. (3)).

3.2. Partially-linearized triangular O-methods

If the matrix A, in Eq. (4) is replaced by the lower tri~mgular matrix Ln whose elements are such that lij = aij if i ~< ] and lii = 0 if i > j, Eq. (4) may be solved by forward substitution from the first to the last equation.

Since the N Eqs. (1) may rearrange in any order, it is clear that there are N! partially-linearized triangular (-)-methods. The accuracy of these methods is O(h) even for O = 0.5 since the linearization has only be performcd with respect to some, but not all, dependent variables in each equation (cf. Eq. (3)).

Since the partially-linearized diagonal and triangular ('/-method'; presented in this section provide the solution of the elements of x,,÷l in a sequential manner, they may also be referred to as sequential or implicit methods which yield explicit solutions.

258 JJ. Ramos, CM. Garcia-Ldpez/Comput. Methods AppL Mech, Engrg. 129 (1996)255-269

The partially-linearized diagonal O-methods should be less accurate than the partially-linearized tri- angular ones because the linearization is only performed in each equation with respect to only one dependent variable in the former, whereas it is performed with respect to several dependent variab~.es in the latter. Furthermore. the accuracy of both partially-linearized diagonal and triangular O-methods should be lower than those of both the standard and the linearized O-techniques.

There is some analogy between the partially-linearized triangular and diagonal ~)-methods presented in this and the previous section, respectively.; and the iterative Gauss-Seidel and Jacobi. respectively. techniques. In partially-linearized triangular O-method!s. the linearization of the jth equation is per- fo:med with respect to x~ for i ~ j. i.e. the iinearization of f/(x, t) is performed with respect to x, with i --: 1,2 .... , j, whereas, in partia!ly-linearized diagonal O-methods. the linearization of the/ th equation is only performed with respect to x i, i.e. the linearization is only performed with respect to the dependent variable whose equation is being solved.

3.3. C o m p a r i s o n s be tween par t ia l l y - l i near i zed 6 ) -me thods a n d o ther impl ic i t -erp l ic i t t echn iques

In an interesting paper. Twizell et al. [6] considered the following system of equations

dx dy dt - x y , d---[ = x y - k y , x(0) = .r~, y(0) = y~,,

where k is a constant, and proposed the following implicit technique

x , , i - x , = - h x , , . l y , , Y , . t - Y,, = h(x , , , l y , , . l - ky,+t ),

which results in the following explicit expressions

x,, y , ( 1 + by,, )

x,,~ = 1 +It3',,' Y"+~ = (1 +hk)(1 + b y , ) - h x , , "

The partially-linearized diagonal (-)-method presented in this paper yields

X n ~- I ~ X . 1 - ( ' )hy .

1 + (1 - O ) h y . '

1 + h d ) ( x , - k ) Y"'~ = Y"I - h(l - (-))(x, - k ) "

while the partially-linearizcd triangular (-)-method yields

1 - O h y , x,+ i = x , 1 + (1 - O)hy , , '

(7)

(8)

(9)

(zl))

( | l )

1 + h O ( x , - - k ) + (1 - O)hy, , &r Y,,, I = Y, 1 - h( I -- 6))(x,, - k) ' (12)

where

AX -~ .~n ~ I - X , = - t l X,).', 1 + t1(1 - O)y,,'

if the partial linearization is performed in the order of the equations. Eq. (11) coincides with Eq. (ga) for 6) = 0. whereas Eq. (t2) becomes, for O = O.

(13)

1 + by,, - hZx, ,y ,

Y,,+l = Y,, (1 + hy,)(l + h k ) - h.r, - hZx,~y,, ' (14)

which agrees with Eq. (Oh) except for the terms proportional to h-" in the denominator and numerator.

4. P r e s e n t a t i o n o f r e s u l t s

The linearized O-methods presented in this paper have been used to obtain the numerical solution of ordinary differential equations with O = 0 and 0.5 as indicated in the next sections. Note that. for O = 1, the O-method and its linearization coincide. For the sake of convenience, the O-methods corresponding

J.1. Ramos, C.M. Garcia-l.61w.~lComlmt. M,'thod.~ AppL Mech. Engrg. 129 (1996)255-269 259

to O = 0 and 0.5 are referred to as TH0 and THS, respectively, while the corresponding linearized, partially-linearized diagonal, and partially-linearized triangular O-methods are denoted by TI-IL0 and THLS, THD0 and THDS, and THTK0 and THTK5, respectively. For systems of two ordinary differential equations, K = 1 and K = 2 correspond to the partial linearization of the equations in the order (1, 2) and (2, 1), respectively, while for systems of three ordinary differential equations, K = 1, K = 2, K = 3, K -- 4, K = 5, and K = 6 correspond to the partial linearization of the equations in the order (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 2, 1) and (3, 1, 2), respectively.

The results of the fully- and partially-linearized O-methods presented in this paper have been com- pared with those obtained by means of the Runge-Kutta methods of order 2 and 4, i.e.

• EM = Modified Euler Method = Runge-Kutta method of order 2 given by

xc~ --- x(tc3,

h .'c,,~1 = x, + ~ (/(x,,, t,) + f(x, , + h.f(x,,, I,i), 1,, + h)) + O(h 2 ),

• RK = Runge-Kutta method of order 4 given by

.r. = x(t.),

ti .'t,,+l = x,, + g (kl +2k_, +2k.~ + k4) + O(h 4)

where

kl = f(x, , , t,,)

k ~ = f x , ,+-~kq , t , ,+

(h k, = f .r,, + g 1c2, t,, +

k4 = .f(x, + hk:,, t, + tl)

4.1. Single equations

The following ordinary differential equations were used to assess the accuracy of the linearized O- methods developed in this paper.

• Pl:

• P2:

• P3:

y' = -y , y(0) = 1. Exact solution: y(t) = e ' V 3 1

Y' = - 2 ' y(0) = I. Exact solution: y(t) = ,¢q +------~t

y ' = y cos(t), y ( 0 ) - 1. Exact solution: y(t) = e ' ' '~ .

4 ( Y ) y ( 0 ) = 1. Exact solution: y ( t ) - 20

• P4: y ' = 1 - ~ , l + 1 9 e ,,4"

Problems P 1 - P 4 have been taken from DETEST [8] and have been analyzed in the interval [020] using h = 0.04. Since the analytical solutions to these problems are known, the accuracy of the numerical methods has been assessed by determining the error E(t,,) = ]Ix,, --x(t,,)ll. The non-linear algebraic equations for the TH methods were solved by means of the Newton-Raphson technique until t]x ~'*t - x~]] ~ 10 ~ t-~ where k denotes the kth iteration within the time step.

In Table 1, we show the mean, Emc.n, and the maximum error, E,~x, where Emea, is the arithmetic mean of the errors E(t,,), and Em~,x is their maximum value. Table I indicates that the accuracy of both iterative and linearized O-methods deteriorates as O is decreased from O = 0.5 for which these methods are second-order accurate. Table 1 also indicates that the accuracy of the RK method is higher than

260 J.I. Ramos. CM. Garcia.L6pez/Comput. Method~' Appl. Mech. Engrg. 120 (1996)255-269

Table i Errors for probtcrns PI-P4

PI P2 Emcan Ernax Em~;m Em:,~

EM 1.36869e-05 1.01105e-04 9.13376c-06 3.7820gc-05 RK I A)9829¢-09 8. I 1423¢-09 8.91903e- 11 5.28782c- I 0 THL0 9.89375e-04 7.23736e-03 1.35924e-03 3.71173e-03 TH{; 9.89375c-04 7.23736e-03 t.33270c-03 3.60789¢-03 TltL5 6.63992c.116 4.90590e.05 2.28068c-06 9.44210e-06 TH5 6.63992c-06 4.90590e-05 1.12610c-(15 4.64888c-05

P3 P4

EM 3.02539c-04 8.83862e-04 8.33584c-05 1.35436c-04 RK 1.19555c-08 3.18968e-08 2.90156¢-10 4.59301Je-I11 TH L0 9.13838e-02 4.54 ! 91 c -01 2 242450 -02 4.17542c -02 TH0 1.59072e-01 6.13724e-01 2.22843c-02 4.~ 5282e-02 THL5 4.75336e-02 1.39215c-01 7.687(18c-(15 1.35163e-04 TH$ 2.46970e-02 7.25259c-02 8.88872c-06 2.09088c-(15

those of the EM, T H and T H L techniques. Except for problem P3, the accuracy of both the T H 5 and the T H L 5 methods is higher than that of the EM scheme, and the accuracy of the T H technique is higher than that of the T H L method .for the same value of 19 as it should be expected, because the T H L schemes correspond to the linearization of the T H techniques.

4.2, Sys tems o f ordinary dif ferential equat ions

The following systems of ordinary differential equat ions were employed to assess the accuracy of both the fully- and the partially-linearized O-methods deve loped in this paper.

• S I

YPl = 2(yl - Y,Y2), Yl (0) = 1,

Y~ = - (Y2 - Y~Y2), y2(0) = 3, • $ 2

Y'1 = -YlY", Yi ,,,,Jm" = 1, )"2 = YlY2 - 0.9y2, y2(O) = 0.! .

• $3

Y'l = -Yl +Y2, yl(0) = 2, Y~ = Yt - 2y2 + Y3, y2(0) = 0, Y~ = Yz - Y3, y.~(0) = l ,

• S4

Y't = Y:Y3, Yt (0) = 0, Y~ = -YlY3 , y2(0) = 1, y~ = -0.51y~y2, y3(0) = 1,

• $5

Y'l = ( l . ] - 0.01yt)y~ - 0.056yly4, yl(0) = I0, y~ = -5.59y2 - 0.056y2y4 + 0.34y2y,, y2(0) = 10, y.~ = (0.036 - 0.0036y3)y~ - 0.017y3y~, y3(0) = 10, y~ = 0.056yty4 + 0.056y2y4 - 0,94y4 - 0.017y4y6, y4(0) = 10, y~ = (0.12 - 0.003ys)y5 - 0.005ysy7, Y5 (0) -- 10, y~ = 0.017y3y~ + O,O17y4y6 - 0.72y~, - 0.005y6y7, y6(0) = 10, y~ = 0.005ysy7 + 0.005y6y7 - 0.2y7, y7(0) = 10, y~ = 0,01y.s + 0.2y~ + 0.08y7 - 0.25ys, y8(0) = 10, y~j = 1.67yl - 0.34y2y~ + 0.11y3 + 0.01y5 + 0.24y6 + 0.03y7 + 0.05y8, y g ( 0 ) = 10.

J.I. Ramos, CM. Gurcia-L6pez t('mnput. Methods Appl. Mech. E, grg. 12q (1996)255-269 261

Problems Sl, S3 and S4 have been taken from DETEST [8], while $2 was taken from [6]; all these problems have been s~ived in the interval [0,20] using h = 0.04. Problem $5 was taken from [9], and has been solved in the interval I0,120] using h = 0.05. The Newton-Raphson method was used to solve the system of non-linear algebraic equations resulting from the Tit methods until IIx k+l - x k ]1 <~ 10 :- where k denotes the kth iteration within the lime step, except for $3 which is a linear system. Therefore, the Ttl and THL methcds should yield identical results for problem $3.

The accuracy of the numerical methods considered in this paper for problems SI-$5 has been assessed by comparing the results with those of the Runge-Kutta method of order 4, i.e. RK. For problems $1-$4, the arithmetic mean ,,ff these errors is denoted by Em~,,n(j) which is equal to the arithmetic mean of Ixi ~t - "¢i t~t~ , where .til¢~: is the solution determined by the RK scheme, .rff is the solution calculated by the method under consideration, and / denotes the jth component of the vector x. The maximum error has been denoted by E ...... (j) and corresponds to the maximum value of Ix: f - ~tCK at/. ,

Tables 2 and 3 show the errors for the systems of two and three, rcspectively, ordinary differential equations considered in this paper. The trends of the 1"11 and THL methods are similar to those already discussed for single ordinary differential equations (,:f. Table 1 ), i.e. the accuracy of the TH scheme is, in general, higher than that of the THL one, the a:curacy of both the TH and the 'I'HL algorithms decreases as 6) is decreased from 6) = 0.5. the accur~cy of the RK technique is higher than those of the EM and TH ones, and the accuracy of both the 31"tt5 and the TILL5 methods is, in general, higher than that of the EM one, even though the order of the truncation errors of these methods is the same, i.e. O(h:). The error of the RK scheme presented in Tables 2 and 3 has been determined by means of the technique described in il0].

Table 2 shows that the accuracy of partiaUy-linearizcd (-)-methods is, in general, lower than those of the TH and TIlL ones for O = 0.5, as it should be expected owing to the loss of accuracy introduced by the partial linearization of the differential equations. As indicated previously, the accuracy of partially-

"lhblc 2 Errors for problems S! and $2

Si E,~;,, I I ) t-',~ ....... (21 I-,,,,,( I ) E,,,,~ {2)

EM 7.71561e-()3 5,29974c-()3 7.33127c-02 4,59665e-02 RK 6.85775c-07 ,1.5543()c-07 7,6 t 992c-06 4.19262c-06 TH0 6.85055c-01 4.62735c-01 3.7838t)e+{l{) 1,87894e+00 THIL0 6.90532c-(H 4.67236c-01 3.714982c+00 1.8879%+00 THD0 14.731 file-01 8.7579qc-01 7A6335c-~ {X) 6.11674e+00 THTI0 ~,~.92518c-(E 8.22564c-q2 9.~7536e-01 7.21653e-01 THT20 1.55629c-¢11 9.37657c-02 1.62751e+{K) 8.831)55e-01 THS 7.10598c-()3 4.8'1836c-03 6.69506e4)2 4.08129e-02 THL5 1.44596c.()3 1.11~35q5c-(13 i .63775e-02 7.26886e.03 THD5 t).919fi8c-{)1 8.7569(~e-0~ 8.14662e.00 5.99710e+00 THTI5 5.7323qc-I11 4.52113 It-Ill 5.514894c+~R~ 2.80,502e+00 THT25 5,4549t~e-Ol 4.4(~486c-()1 5.42q16e+0{I 2.77404e+0fl

SZ I:-nl~.:kn ( l ) Hrl,c,,n {2) /'-',u ,x ( I ) Em,,x (2)

EM i .f~q006c-0('~ 4.20987e 4)7 3.13235c-(~,6 1.48879e-06 RK 6.43827c-13 2.(JL466c+13 2.39~95¢-I 2 3.18469e-13 Till) 1.33928c 4)3 1.33325c-()4 1.83384e-0"~ 3.28471 e-0,, i TILL0 1.34701e-{13 1.33188c4}4 1.8421ge-03 3.270230-04 THDU 1.79355¢-1}3 1.83788¢-{}4 2.51 {}{M}¢-(}3 5.2 ;75 i e-04 TILT10 3.37976¢4|4 9.24241 ¢-(}5 7.69233e-{)4 3.09124e-{)4 THT20 5_36546e 4}4 2.33027e-()4 ~,~. 17()75e-.(14 5.527{}8e-04 TH5 2,69.H9c-116 3.36699c-117 3.6110113c-{}6 1.29632e-06 TH L5 1.62614c-0fi I .~8474c-O7 1.86052c-06 4.31 i 99e-07 THI)5 1.56401 c-t}3 1.05556c-04 1.95472e4}3 4.11526e-04 THTI 5 6.26735c-04 2.75221 c-(15 8.08009c-04 6.45933e-05 THT25 9.43481 c 314 1.07435c-0J, 1.14981 c-03 4.060115e-04

262 J i Ramos, C,M. Garcia-L6pez/Comput. Methods Appl. Atech. Engrg, 129 (1996)255-269

Table 3 Errors for problem 53

83 g~=.. ( I ) E,-ae,,. (2) E~,,. (3) E,,,,,,,( 1 ) E .... (2) E m~ (3)

EM 2.87150c-05 4 . 3 7 1 5 8 e - 0 5 2 . 0 1 3 0 7 e - 0 5 5 . 1 6 4 6 6 e - 0 4 9 . 6 5 8 9 6 e - 0 4 4.51020e-04 RK 1.03133e-09 1.99368e-09 9 . 7 4 0 7 5 e - I ( ) 2 . 2 1 7 1 6 c - 0 8 4 . 4 0 2 2 6 e - 0 8 2.18509e-08 TH0 9 . 8 4 6 8 4 c - 0 4 9.781117e-04 5 . 2 3 4 5 3 e - 0 4 1 . 3 0 1 1 2 c - 0 2 2.10138¢41~. 8.25196e-03 TItL0 9.84686e-04 9.781) 18e-04 5.23450e 414 1 . 3 0 1 1 2 e - 0 2 2 . 1 0 1 3 8 c - 0 2 8.25196e-03 THI)0 1 . 1 8 9 5 3 e - 0 2 1 , 2 5 1 9 0 e - 0 2 1,28867e.f12 1 . 2 6 5 8 2 e - 0 2 1 ,28137e412 1.57109e-02 THTI0 1 .2 6 5 5 9 e -02 1 . 3 2 9 6 6 e - 0 2 1 . 2 1 7 8 2 e . 0 2 1.3(Yd87c-02 2 . 0 8 7 3 4 e - 0 2 1,30464e-02 Tlt'I~0 2 . 4 7 3 2 7 e - 0 2 2 . 6 0 4 7 3 e - 0 2 2 . 5 7 2 4 0 e - 0 2 2.59740e-02 3. I 1705c-02 2.83636c-02 THT30 1 .3 5 1 9 3 e -02 1 . 2 3 7 4 6 e - 0 2 1 . 2 5 2 7 9 e - 0 2 1.89230e-02 1.29870e 4;2 ~ .29870c-02 THT,40 1 .3 5 1 9 3 e -02 1 . 2 3 7 4 6 e - 0 2 1 . 2 5 2 7 9 e - 0 2 1 . 8 9 2 3 0 e - 0 2 1 . 2 9 8 7 0 c - 0 " 1.29870e-02 %HTS0 1.41672e.03 5.45981 c-04 1 . 5 7 0 6 3 e - 0 3 1.57022e-02 I. 18756e-0.! t.42693¢-02 THT60 2 . 4 7 3 2 7 e - 0 2 2 . 6 0 4 7 3 e - 0 2 2 , 5 7 2 4 0 e - 0 2 2 . 5 9 7 4 0 c - 0 2 3 . 1 1 7 0 5 c - 0 1 2.83636e-02 TH5 133197e-05 1 . 9 9 8 6 5 e - 0 5 9 . 2 1 1 6 2 e - 0 6 2 . 3 7 4 8 4 c - 0 4 4 . 4 2 4 8 5 e - 0 4 2.05748e-04 THL5 1 . 3 3 1 7 4 e - 0 5 2 . 0 0 1 5 3 c - 0 5 9 . 2 3 7 2 8 e - 0 6 2 . 3 7 4 7 2 e - 0 4 4 . 4 2 5 1 9 e - 0 4 2.05752c-04 THD5 6 . 6 1 7 7 2 e - 0 3 6 . 2 4 0 6 2 e - 0 3 6 . 6 1 1 0 7 e - 0 3 9.84898e-03 65901 le-03 9.80937c-I)3 THTI5 6 . 9 2 1 9 9 e - 0 3 6 . 2 7 9 3 5 c - 0 3 5 , 9 2 6 6 8 e - 0 3 1 .15609c -02 6 . 5 7 8 9 4 c - 0 3 6.57894¢-03 THT25 ~ .30515e-02 126901 e - 0 2 1.30448e-02 1.36311 c-02 1.31578e-1)2 1.35822c-02 THT35 6 . 3 4 1 7 3 e - 0 3 6 . 7 0 3 2 3 e - 0 3 6 . 3 4 8 3 6 e - 0 3 6 . 5 7 8 9 4 e - 0 3 9 . 9 7 7 3 2 e - 0 3 6.58390c-03 THT45 6 3 4 1 7 3 e - ( 1 3 6 . 7 0 3 2 3 e - 0 3 6 . 3 4 8 3 6 e - 0 3 6 , 5 7 8 9 4 e - 0 3 9.97732e-113 6,58390e-03 THT55 2 . 0 5 7 5 3 e - 0 4 2 , 3 8 8 4 7 e - 0 4 7 . 8 5 8 1 0 e - 0 4 2 . 0 4 8 5 8 e - 0 3 5 , 3 2 4 3 5 c - 0 3 1.01773¢-02 THT65 1 ,30515e. .02 1.26901e-112 1.3(1448e-02 1.36311 e q 1 2 1.31578e-112 1.35822e-02

!inearized methods is O(h) even for O = 0.5. Table 2 also indicates that, for problem S2, the Tit5 and THL5 methods are more accurate than the THD$ and THTK5 ones for K = 1, 2; however, for problem Sl, the THT10 and THT20 algorithms are more accurate than the TH0, THL0 and THD0 ones.

The results presented in Table 2 also indicate that, for problem Sl, the THTI0 and THT25 algorithms are more accurate than the THD0 and THT20 ones, and the THD5 and THTI5 ones, respectively; however, for problem $2, the THTI5 method is more accurate than the THD5 and THT25 ones.

Table 4 Errors [or problem 54

$4 Emcan ( I ) Eme,m (2 } E'. . . . . . (3) Ema,~ ( I ) Emax ( 2 ) Ern,lx (3)

EM 1.03847e-03 9 . 8 6 5 2 0 c - 0 4 3 . 0 7 7 3 8 e - 0 4 3 .52236e413 2.65852c-03 9.50811 e-04 RK 7.51718e.119 7.13498c-(1q 2.24260e-09 2.52571 c-08 1 , 8 7 8 7 9 e - 0 ~ 6.44214¢-09 TH0 9.40056c -02 7.29128e 4)2 2 23113e-02 2.44842e4) I 2 . 3 4 6 7 8 e - 0 1 6.60277e-(12 THL0 9 . 4 1 7 3 7 e - 0 2 7 . 3 0 2 9 3 c - 0 2 2 . 2 2 3 2 7 e - 0 2 2 . 4 5 9 4 3 e - 0 1 2 . 3 4 6 3 9 e - 0 1 6.60788e-02 THD0 t . 2 1 6 7 2 e - 0 1 8.26745c-02 3.25311 e-02 3 . 4 4 2 4 6 e - 0 1 3.14179e-01 I. 181121 e-0 I THTI0 1,911 i 4 ¢ - 0 1 2 . 0 6 0 8 4 c - 0 1 6 , 2 5 4 3 1 e 4 ) 2 8 . 5 0 8 3 5 c - 0 1 7 . 9 7 7 9 6 c - 0 1 2.68555c-01 131-117.0 2 , 1 2 6 3 5 c . 0 1 1 .78912c -01 6 . 8 9 9 2 1 c - 0 2 9.80736c-11t 6 . 3 7 3 7 4 c - 0 1 3.07585c-01 'I'I.-IT.'~ 2 , 1 5 1 7 8 e -01 1 . 9 8 8 4 0 e - 0 1 6.97200e-02 9.88607e 4) 1 6 . 5 0 3 5 6 c - 0 1 3.09743e-0 I THT40 1 , 9 2 2 2 8 c - 0 t 1 . 86829c -01 5 . 9 7 8 8 8 e - 0 2 8 . 5 4 3 8 8 e - 0 t 7 . 7 1 0 6 3 c - 0 1 2.61934e-01 " rHTS0 2 .1 4 3 4 8 c -01 1 .98134e4 )1 7 . 2 3 5 8 5 e - 0 2 9,86311)c-01 6 , 4 9 5 9 8 c - 0 1 3.16027e-01 THT60 1 .93622c-01 2.08760c-01 611074 le-02 8 . 5 8 4 2 6 c - 0 1 8 , 0 5 0 8 3 c - 0 1 2,62754c-0t TH5 2.08468c-04 1.90948e-04 7.75634c-05 6.66359c-04 5.63963c-0.I 2.43507c-04

Tit1.5 8 , 1 8 3 5 8 c - 0 4 7 . 7 1 9 4 8 e - 0 4 2 . 4 3 7 5 8 c - 0 4 2 . 7 2 0 6 5 c - 0 3 2 . 0 1 4 8 7 c - 0 3 b.98256e-04 THD5 1.21672c 411 8,26745 c-02 3.2 ';31 i e-02 3 .44246e - I ) l 3,14179e-01 I. 181121 e-01 I'I'ITI5 1 ,33030e-01 1 . 2 3 3 6 3 e . 0 1 4 . 0 4 0 8 3 c - 0 2 5 ,79795c -01 4 . 5 9 3 4 8 e - 0 1 1,87987c-01 T1~q[25 1 .39808c-01 t . 1 6 8 1 8 c - 0 1 4 . 6 5 4 0 5 e - 0 2 6 . 1 7 9 6 6 c - o 1 4 . 9 1 4 3 8 c - O l 1.95812e-0 i THT35 I A0763e-01 125775e-0 i 4 . 6 8 5 7 4 e - 0 2 6,21565c-0 t 5.07525 e-01 1.96851 c-0 I THT45 1.334,,t7c-01 1 .14791e -01 3 . 8 7 6 3 0 c - 0 2 5 . 8 1 3 2 3 e - 0 1 4 . 4 4 1 7 0 c 4 ) 1 1,83491c-01 "!"!t"1"55 1.401. ' ;5c-01 1 . 2 5 2 5 1 e - 0 | 4 . 8 8 2 2 7 e . 0 2 6 . 1 9 4 8 6 c - 0 1 5,0615{1c-01 2.01288c-01 THT65 13405)e.01 1 .24476e -01 3 . 8 9 2 5 2 e - 0 2 5 .83569c -01 4 . 6 2 8 2 6 e - 0 1 1,84034c-01

J.I. Ramos. C.M. Garchi.LfpezlComput. Methods Appl. Mech. Engrg. 129 (1996)255-269 263

Table 3 shows that, for problem S3, the THD meth~d is more accurate than the TNT one. Further- more, the results presented in Tables 2, 3 and 4 clearly indicate that the accuracy of partially-linearized triangular methods depends on the order in which the equations are linearized and solved, and on the system of ordinary differential equations to be solved. For example, Table 3 shows that the THTS0 and THT55 schemes are more accurate than the THD0 and THTI0 ones, and the THD5 and THTM5 ones, respectively, for problem $3, where ! = 1--4 and M = 1-4 and 6. For problem 54, Table 4 indicates that the THT techniques are less accurate than the THD one.

Figs. 1--6 show the solution to problem $5 obtained with the fourth-order accurate Runge-Kutta method (top left) and the differences between this solution and those obtained w~,th the THL, THD and THT techniques which are denoted by EL, ED and ET, respectively, for O = 0 and 0.5. Due to the large number of variables in this system of equations, only the partially-linearized technique which results from the partial linearization of the equations in the order given above, i.e. 1 to q, is considered. This partial linearization leads to a lower triangular system.

Figs. 1-6 show that the solution obtained with RK tends to the equilibrium or stationary point of the system as : increases, and that the different components of the system exhibit oscillations of different amplitude and frequency. Figs. 1 and 2 indicate that the behaviour of y3 is a damped oscillation whose

50 t l~ I

40

3o ', I t

20

I x

0 20 40 60 80 100 t

120

1.5

1

0.5

0

-0.5

0 20 40 60 80 100 120 t

1.5

1

0.5

0

-0.5

-1

-1.5

o5 I 0.4 ~L

r t 0,3t ,I,

o lHI' ',

0 ~ "+; . i . , . ~ ~ _ - _ _ ,

0 20 40 60 80 100 120 0 20 40 60 80 t00 120 t t

Fig. I. Solution and local errors for problem 55 with 6) = I). (rlbp left: RK solution: top right: El_, Ix)ttom left: ED, bottom right: ET; Yt: solid line: Y3: dashed line: 3'5: dashed-dotted line.)

264 J.i. Ramos, C M. Garcla-L6pez / Comput. Methods Appl. Mech. Engrg. 129 (I9O6) 255-269

4O

30 I ' ~ ' - 7 - ' t " . , - . . - . . . . . . 4 ~.

> I, I I /'l k- / ~ ~ ' " ~

,Ooi, 0 20 40 60 80 100

t 120

0,01

0.005

0

-0.005

-0.01

,, • , L , . ~ , m

..~ = ..... -0"0150 20 40 60 80 1C)0 120

t

1.5

1

0.5

0

-0.5

-1

-1,5

-2

-2.5

0.5

p- uJ C

-0.5

-1 0 20 40 60 80 100 120 0 20 40 60 80 100 120

t t

Fig. 2. Soluti~m and local errors for problem $5 with 6) = 0.5. (Top left: RK solution: top right: EL, bottom left: ED, bottom right: ET; ,vw: solid line; Y3: dashed line: y~: dashed-dolled line,)

largest amplitude is larger than those of Yl and Ys, and Yl shows oscillations of higher frequency than those of Y3 and Ys. The errors illustrated in Fig. i also show an oscillatory, behaviour whose amplitude and frequency depend on the linearization. For a fully-linearized system, the results presented in Fig. 1 (top right) indicate that the largest amplitude of the errors is associated with Yl which is the component of smallest amplitude. The errors in Y5 are smaller than those of y~ and Y3 because this component has oscillations of lower frequency than the other two shown in Fig. 1.

Fig. 1 also shows that the errors of THD (bottom left) exhibit similar trends to and are of the same order of magnitude as those of THL, but they decay to zero at a smaller rate. The errors of "I'HT (bottom right) are of smaller amplitude than those of THL and THD. The errors presented in Fig. 2 exhibit the same trends for all the linearization techniques presented in this paper, i.e. the errors are damped oscillations, This figure also shows that the errors of THL decrease by about two orders of magnitude as (9 is varied from 0 to 0.5 on account of the decrease in the magnitude of the truncation errors. However, the errors of T I t D and TI lT are of the same order of magnitude for (9 = 0 and 0,5 because these methods are first-order accurate in time regardless of the value of (9.

Figs. 3 and 4 illustrate the behaviour of and errors in Y2, Y8 and Yg, and indicate that the amplitude of Y9 is larger than those of the other two components. The three components presented in these figures

J.I. Ramos, CM. Garc[a-Lfpez /Comput. Methods AppI. Mech. Engrg. 120 (1996)255-269 265

25

t 15

i

t /

t- /

! /

/

5 t0 15 20 t

1 . 5 1 • "'

li

1 j I

0.5 0 i -0.5

~" ... ~-'~" I _ ~ _ _ "" t "~

I

f

~f

0 5 10 15 t

:t' 1. I~

/ i J

0.5 1 o i -0.5

-1

-1.5 i . ~

-2

i"

I ~ I"~ F "~.

I

P

0 , 4 , •

0.3 t t L

0.2 I

0.1 A~

-0.1 ' v ~ " , , /

_0.2! - . .

- 0 3 i

-0,4 0 5 10 15 20 0 5 10 15 20 t t

Fig. 3. Solution and local errors for problem $5 with O = 0. ('R~p left: RK solution: top right: EL. bnttom left: Eli. bottom right: ET: y~: solid line: Ys: dashed line: y~: dashed-dotted line.)

have an oscillatory behaviour, and the frequency of oscillation of Y2 is larger than those of Y8 and Yg. As a consequence, the largest errors are observed in this component for times greater than about 7, whereas those of Ys are of smaller amplitude because of the lower frequency exhibited by this component, even though its amplitude is larger than that of Y2. Fig. 3 also indicates that the errors in Y9 are initially larger than those of the other two components shown in the figure due to the steep time derivative of this component.

Fig. 3 also shows that the errors of T I t D are comparable to those of TILL, but larger than those of TILT. The results presented in Fig. 4 indicate that the errors of T H L are smaller for O = 0.5 than those for t9 = 0, whereas those of T l t D are nearly independent of the value of O. The errors of ~ are about a factor of two larger for 19 = 0.5 than for 19 = 0 at least for t ~< 7. These results are consistent with those shown in Figs. 1 and 2.

The results presented in Figs. 5 and 6 exhibit similar trends to those of previous figures and indicate that the accuracy of T I lL and T I l T increases and decreases, respectively, as O is increased from 0 to 0.5, whereas that of T H D is nearly independent of this parameter. Furthermore, the errors of T l t T are smaller than those of T i l l for O = 0, while the opposite is true for O = 0.5. Although, for $$ and the same value of O, the errors of THT are smaller than those of TI tD, the computational cost of the former is higher than that of the latter; therefore, it may be possible to obtain very accurate numerical

266 J.L Ramos, C.M. Garcia.L@ez/Cmnput. Methods' AppL Meci,. Engrg. 129 (1996) 255-269

25 ~-., . • ~ " ....

I

20 't I

/ >.- // 10 - .

%, . . /

o 0 5 10 15 20

t

0.03

0.02

0.01

0

-0.01

-0.02

-0.03

-0.04

-0.05

J I I

li

i I

I I V

5 10 15 t

2 [ . • ~ .

11 JL J

o.5 . , f i i ~ ,,

-1 !u

-1.5

-2o s lo is 2o t

0.8

0"6 t 0.4

0.2

I - - ,,, 0

-0,2

-0.4

-0.6

-0.8 0

?

il

i t ' , '

l ' ~ ; i, 2,- ; •

5 10 15 20 t

Fig. 4. Solution and local errors for problem $5 with t9 = 0.5. (Top left: RK solution: top right: EL, bottom left: ED. bottom right: ET: Y2: solid line: ys: dashed line: y~: da:~hed.dotted line.)

results with THD by using small time steps at less computational cost than those associated with THL and THT, especially for very large systems of ordinary differential equations.

5. Conclusions

Fully-linearized O-methods for autonomous and non-autonomous, ordinary differential equations have been derived by approximating the non-linear terms by means of the first-degree polynomials which result from Taylor's series expansions. These methods are implicit but result in explicit solutions, and are A- stable, consistent and convergent; however, they may be very demanding in terms of both computer time and 6torage because the matrix that results from the linearization has to be inverted at each time step, and this matrix is, in general, dense. The accuracy of fully-linearized O-methods is comparable to that of the standard, implicit, iterative O-methods, and deteriorates as the value of 19 is decreased from @ = 0.5 for which both ~9- and fully-linearized O-methods are second-order accurate.

For the ordinary differential equations considered in this paper, the accuracy of both O- and fully- linearized O-methods is higher than that of the second-order Runge-Kutta method, but lower than that of the fourth-order Runge-Kutta method.

J.I. Ramos. C.M. Garcia-Lfpez/Comput. Methods Appl. Mech. E.grg. 129 (I996)255-260 267

30

~5

20

>'15

10

5

0 0 20 40 60 80 1 O0 120

t

1,5

I

-!.5 0 20 40 60 80 100 120

t

1

0

-1

-2

o.,]

-30 20 40 60 80 100 120 -0"60 2~0 4'0 60 80 100 120 t

0.2

I

-0.2 II 11

-0.4

t

Fig. 5. Soluliorl and I~vcal err~rs for problem S5 with (-) : O. ('l'~p telt: RK solution: tup ri~,ht: EL. bottom left: ED. bottom right: ET; Y4: solid line: YC dashed line; YT: dashed-dotted line,)

Partially-linearized O-methods based on the partial linearization of non-linear terms have also been developed. These melhods result in diagonal or triangular matrices which may be easily solved by substitution. Their accuracy, however, is lower than that of fully-linearized O-methods.

Since the fully-linearized O-methods developed in this paper require the evaluation of both the Jaco- bian matrix at the previous time step, an automatic control of the time step may be easily incorporated in these methods by simply considering the norm of this Jacobian matrix. Furthermore, since these methods also account for the variations of the right-hand sides of the ordinary differential equations with respect to the independent variables, automatic control strategies of the time step may also be based on the magnitude of these variations even when the norm of the Jaeobian matrix is of order one.

The accuracy of partially-linearized methods depends on both the system of ordinary differential equations to be solved and the order in which the equations are solved. For a system of N ordinary differential equations, there are N! partially-linearized triangular O-methods whose formal accuracy is O(h) even for O = 0.5, i.e. their accuracy is lower than those of the standard and fully-linearized O-methods for the same value of O. Some of the examples presented in this paper indicate, however, that, in some cases, the error. ~f the partially-linearized methods are smaller than those of the standard

268

25

20

>-15

10

5

0 0

J.i. Ramos, C.M. Garcia-L6pez /Comput. Metlupds AppL Mecil. Engrg. 129 (1996)255-269

0.015

j~ 0.01 ~Li ~ 0.005 I /

L i

~ - ~ 1

J ,, ~ ~ - . . . . . -0 .005

~/ " " -0.01

-0,015 20 40 60 80 100 120 0

t 20 40 60 80 100 120

t

2 1

0.5

0 0 . . . .

o l - w w

-1 -0 .5

- 2 -1

- 3 -1 .5 0 20 40 60 80 100 120 0 20 40 60 80 100 120

t t

Fig. 6. Solution and k~al errors for problem 55 with (9 = 0.5, ('lk~p left: RK solution: lop right: El, bottc~m left: ED, bottom right: ET; Y4: solid line: y~: dashed line; YT: dashed-dotted line.)

and fully-linearized ones. Furthermore it has been shown that partial linearization may indicate the dependent variable which has the greatest effect on both the system of equations and its solution.

Acknowledgement

The research reported in this paper was supported by Projects PB91-0767 and PB94-1494 from the D.G.I.C.Y.T. of Spain.

References

[I 1 J.l. Ramos, Numerical methods for onc-dimcnsiona] reaction-diffusion equations arising in combustion theory, in: TC. Chaw[a, ed., Annual Review of Numerical Fluid Mechanics and Heat Transfer. Volume I (Hemisphere Publishing Corporation. New Yerk, 1987) 150-261.

f2] J.l. Ramos. Finite element methods for one-dimensional flame propagation problems, in: T.J. ('hung, ed., Numerical Modeling in Combustion (Hemisphere Publishing Corporation, New Yofk, 1993) 3-131.

[3] T. Mavoungou and Y. Cherruault, Numerical study of Fisher',,; equation by Adomian's methf~l, Math. Comput. Modelling 19 (1994) 89-95.

J.l. Ramo.~, ('.M. Garcht.Lfpez/f'ompnt. Methodx AppL Mech. Engrg. 129 {1996)255-269 269

[4] J.I. Ramies, Modilied cquaiion techniques fi~r reaction-diffusive systems, l'ar! 2: l"imc-linearization and operator-splitting methods, ('ompul. Metht~ds Appl. Mech. Engrg. 64 (1~)87) 221-236.

t5] J.l. Ramos, (In the accuracy ¢,f blo,;k implicit and ¢~pcra',or-splitting algorilhms in contined flame propagation problems, Int. J. ('tmlpul. Math. 211 (19861 299-324.

(61 E,H. 'l~,vizell, Y. Wang, W,(i. Price and E Fakhr. Finile-diffcrence methods for solving |he reaction--difussion equations of a simplt t isolherm:d chemic~al system, Namer. Mclhods l'arlml Diff. Eqns. 10 11994) 435--454.

I7] J.D. Lamherl, Numerical Mctht,ds for [)rdin~lry I)iffercnlial Equations (John Wiley & Sons. New York, 1991). [81 "I:E. Hull, W.H. Enright. I?..M. F¢llen ~nd A.E. Sedgv,,ick, Comparing numerical mctht,ds for ordinary differential equations,

SIAM J. Nu,ncr. Anal. 9 (1972) 603-6,37. 191 ['.M. (}arcia L6pcz, Estudio an:~litico .v numcrico de algunos modclos dc pttblaci6n rcl~scionad0s con ccuaciones diferenciales

ordinarias y ccuacioncs cn deriw~das parcialcs. Tcsis dc Liccnciatura, l.lniversidad dc M~ilaga, 1994. [10] E. Haircr. S.P. Nt~rsctt ~md (;. Wanner. Striving ()rdin:trv Differential Equation.', I. Nonstiff Problems (Springer-Verlag, New

Y~rk. 1987}.