AD-A173 477 SPECIAL METHODS FOR PROBLEMS WHOSE OSCILLATORY SOLUTTiW -71-71S DAMPEDCU) NAVAL POSTGRADUATE SCHOOL MONTEREY CAB NETA SEP 86 NP5-53-86-Bi2

UNCLASSIFIED FiG 12/1 U

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-CROCOPY RESOLUTION TEST CHARTNATIONAt1 RUREAU OF STANDARDS 1963-A_

NPS-53-86-012

NAVAL POSTGRADUATE SCHOOLMonterey, California

DTICS--ECrID27 1287

SPECIAL METHODS FOR PROBLEMS WHOSE

OSCILLATORY SOLUTION IS DAMPED

by

Beny Neta

LJ September 1986

Technical Report For Period

April 1986 - September 1986

Approved for public release; distribution unlimited

Prepared for: Naval Postgraduate SchoolMonterey, CA 93943

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.

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NAVAL POSTGRADUATE SCHOOLMONTEREY CALIFORNIA 93943

R. C. AUSTIN D. A. SCHRADYRear Admiral, U. S. Navy ProvostSuperintendent

,Reproduction of all or part of this report is authorized.

This report was prepared by:

BENY NETAAssociate Professor ofMathematics

Reviewed by: Released by:

airman KNEALE T. MARSHALLDepartment of Mathematics Dean of Information

and Policy Sciences

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Special Methods for Problems Whose Technical Report

Oscillatory Solution is Damped 04/86 - 09/86S. PERFORMING ORG. REPORT NUMBER

7. AUTHOR(*) S. CONTRACT OR GRANT NUMBIER()

Beny Neta

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Naval Postgraduate SchoolMonterey, CA 93943

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September 19861. NUMBER OF PAGES

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Approved for public release; distribution unlimited

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18. SUPPLEMENTARY NOTES

19. KEY WORDS (Continue on reverie side it necesary n Identify by block nmb)

ordinary differential equations, exponentially-fitted methods, shallow waterequations, damped oscillations

St

20. ABSTRACT (Continue a reveree aide If neeaary and IdentIfy Z; block .mber)

This paper introduces methods tailored especially for problems whosesolution behave like , where X is complex. The shallow water

equations with topography admit such solution.This paper complements the results of Pratt and others on exponential-

fitted methods and those of Gautschi, Neta, van der Houwen and others ontrigonometrically-fitted methods.

DO I* 1473 EDITION oP INOV 6 IS OBSOLETE UNCLASSIFIEDS,'N 0102- LF- 014- 6601 SECURITY CLASSFICATION OF THIS PAGE ( en Data Entered)

ABSTRACT

This paper introduces methods tailored especially for problems

whose solution behaves like e', where X is complex. The shallow

water equations with topography admit such solution.

This paper complements the results of Pratt and others on

exponential-fitted methods and those of Gautschi, Neta, van der

Houwen and others on trigonometrically-fitted methods.

1. Introduction

In this paper we consider linear multistep methods

k k7h b f(xn-i_,v I ) ' k 1, n > k-l (I)

,-- n+l-Z n+- n

for integrating the initial value problem

(x) = f(x,y(x)) , :(x 0 ) = r (2)

This linear multisteD method is characterized by the polvnom.ials

k k-xn k k-:-) = C' a ' (-) = b " (3)

The main assumption of this paper is that it is a priori known

that the solution is approximately of the for.m

m i ) t.. .... ...........

v(x) -. c0 + c.e (4)

2 A-V r...........................................

where A. w. + ii~j, and the frequencies w. are in a given interval

The special case where X. = with w0 given was considered

first by Gautschi [8]. His approach was the following. Let:

*(z) = )(ez) - zc(ez) (5)

then the local truncation error of (1) is given by Lambert [111

Tk = z(h -E) y(tn) (6)

Insertina (4) in (6) yields

m iX .tTn+k - :(O)c 0 - c.4(ih), )e . = jW0 (7)j-l 3 w

The coefficients bi are chosen in such a way that

S(ihjw 0) = 0 , j = 0,1,...,q , (8)

for the largest value of q possible. q is then called the trigo-

nometric order of the method. Gautschi has chosen a. such that theI

methods are of Adams and Stormer type. However, these methods are

sensitive.to changes in the frequency w0 . Neta and Ford [13]

developed Nystrom and generalized Milne-Simpson type methods. These

methods showed less sensitivity to perturbation in w 0 but require

the eigenvalues of the Jacobian to be purely imaainary. Neta [14]

has developed families of backward differentiation methods that

overcome the above-mentioned restriction. Salzer [17] has developed

3

* - . .,-JJ N -

M

predictor-corrector methods based on trigonometric polynomials.

See also Steifel and Bettis [181 and Bettis [3]. Van der Houwen

and Sommeijer [101 have developed an alternative approach. The

conditions (8) were replaced by

(i) = 0

(9)

*(ihX ( j ) ) = 0 , j = l, 2 ,...,q

where the ,(j ) are appropriately chosen points in the interval

An advantage of this so-called minimax approach over the fitting

approach is the increased accuracy in cases where no accurate esti-

mate of w0 is available or when the frequency is varying in time.

The cther special case considered in the literature is where

= i... Probably the first article on the subject is due to Brock3 j

and Murray [5]. They discuss the use of exponential sums in the

integration of a system of first order ordinary differential equations.

Dennis [7] also suggested special methods for problems whose solution

is exponential. He suggested a transformation of variables. More

recently, Carroll [6] has developed exponentially fitted one-step

methods for the scalar Riccati equation. For the general first

order system of equations, Pratt [16] suggests methods based on the

three parameter exponential function

I(x) = A + Bezx • (10)

4

The parameters A, B are given in terms of values of y and f.

Several possibilities for z are given based on re.ults of Brandon

[2] and Babcock et al. 11].

Lyche [12] analyzes multistep methods which exactly integratew x

the set ,'xme n }, where wn is real or imaginary.

In this article we developed various methods fitting exponentials

and methods obtained via the minimax approach.

2. Construction of Methods

2.1 Fitting Methods

In this subsection we discuss various fitting methods. To this

end, we separate ±(ihj\) = 0, j = 1,2,...,q, into real and imaginary

parts. This yields the following equations relating the coefficients

k k -. (k-.)a-e cos j,.(k-,) be [j,- cosjx,(k-Z)

0 -0

- jv sinjv (k-Z)] 0

k k-, k (k-))a e sin jv(k-Z) - 7 b e3 k [j, sin j' (k-;)

,-,=0 ' .0

+ jv, cosjv(k- 0] = 0

where = w + i1, u = -h;, v = hw, j = 1,2,...,a.

For explicit methods, b = 0. For Adams type methods a 0 = 1, a1 = -1 ,

ai = 0 for i = 2,...,k. For Nlystrom or generalized Milne-Simpson

methods a0 = 1, a2 = -1 and other a. = 0.

5

J''

k = 1 Imp~licit

Adams

b ve-u + sin': v cosv

(i+%)") sin -v(12)

b vew - u siflv - vcosv

0( 2 +v 2) sin v

1-Cos \,

For =0, the coefficients become b 0 sin v - = bwhich aaree

with Gautschi [8] if the coefficients are expanded in Taylor series

with respect to v

k = 2 Exolicit

Adams

b, ( sin 2v -v cos2v,)e~ + v cosv - ~sin%,1 2-.2 sin v

(13)

b~ COS',- sinv,).e 2- ye2 2 2 ) sin

N'.' r on

b (0. sin2v - vcos2,v)e'-(2 +v2 )sin ~

(14)

b (v cosv u ~ sinv)e + .,e-2v 2 2si

ssin

For 0, the coefficients become b 2 sn b 2 0hc

agree with Neta [14].

6

k = 2 Implicit

In this case, one obtains a one-parameter family of (Adams,

generalized Milne-Simpson) methods of trigonometric order 1. The

free parameter can be used to increase the algebraic order of the

method as in 113].

Backward Differentiation

e~ua 2 b 2'

e cos 2v + a1 e cos + 2 - b 0e 2( cos2v - v sin2",) = 00i

e 2 sin 2%; + a1 e sin x. - b0 e 2 (u sin2v + v cos2x) = 0 , (15)

1 + a1 + a2 =0

This system can be solved by mACSYtA (Project .,IAC's SYmbolic KAnipu-

lation system written in LISP and used for performing symbolic as

well as numerical mathematical manipulation [4]) or by REDUCE [9].

The solution is

a e -211 sin2 -.., cos2v,.a1 =

-e (v cos- + y sin,) + . sin2- + v cos2v

(16)

a2= -- a,

b -e 2 L sin I) + e sin2'- - sin

0 -e 2 ,(v cos') + . sinv) + e'(- sin2v + j cos2,)

For 4 = 0, the coefficients agree with those given in [14].

7

k = 3 Explicit

Again here, one obtains a one-parameter family of methods of

trigonometric order 1. In order to get methods of trigonometric

order 2, one has to construct a 3 step implicit method of Adams or

generalized Milne-Simpson type. In order to increase the trigo-

nometric order without going to a higher step number, one can con-

struct linear multistep methods for which the coefficients a; are

also functions of , w. Some examples are given in the next

subsection.

2.2 Generalized Fittina Methods

In this section, we construct some linear mulstistep methods

of the form

k ka-(0 y' . =n+l-Z h b b4X)f , k > 1, n > k-l (17)

-0o k-0

*

Since a- are functions of . one has more free parameters for his

disposal which can be used to obtain higher trigonometric order

methods with relatively lower step number.

k = 2 Imolicit

In this case, one has to solve the following linear system of

five eauations for the parameters al, a2 , b0 , bl, b2 to obtain a

method of trigonometric order 2.

8

+4a1 +a 2 = 0

*2cos 2, +a e' cos v + a2 - b 0 e 2 (w~ cos2v. -,, sin2v)

-b 1e L( . cosV\ -v sin v) - Lb 2 = 0

e ,, sin 2,. +a1 e 1 sinv - b 0e 2 j(i sin2v, -+,o cos2v)

-b 1 e (N sin,, +v\. cos'v) -\b 2 = 0 ,(3

e cos't. +a e 20Cos 21+a 2-b 0e 4,"C 2u cos 4v - 2v sin 4% )

-b 1e 2,-(2', cos2v. -2v sin2.)-2, b2 =0

* 4si4* -j-ae 2 u sin2 v -b 0e 4 1(2,. sin4v + 2 v cos4\,)

-b 1e 2,_(2' sin2v. +2v cos2v,)-2vb 2 =0.

The svste:7 was solved by REDUCE 19]. The expressions for the

coefficients are complicated but REDUCE produces an output in the

form of- Fortr-an statements that can be incorporated in a comouter

prcgra:7 :cr numerical exoeriments with such a method.

..3 V.nimax: :Methods

in th-is section we discuss minimax methods, i.e., methods

obtained bv satisfying conditions (9). These conditions can be

written in terms of a,, b, as follows:

k (j)( )b-e (k-i-l'"

k (j) ( ) ( )( 9

j = 1,2r ... q

9

k k i ))I a Ze ~~~ sink-v

k (k-) ~~ () (j) Ci) (j),b Zb e V ~ cos(k- )v -+LI sin(k-Z~)v , , (20)

Awhere :Cj) = h , v i) -~)= wj

ih ()are the zeros of the function 'Cih,) such that it has

a small maximum norm in the rectangle we < w < wu, < < ~

To obtain the best approximation in this case is certainly

not easy; but we will assume that one can write (i -

,w )as a product of 2 one variable functions. Thus

and w(i) can be taken as Chebyshev's points on the corres-

pondina interval, i.e.

(j) =1 1 )cs21-1-(- +- ) +- )Co

2t 2 u 2qj

(21;

w (i) =-(w, +w ) + -(w -w )cos2-

j = 1,2,... ,q

For this choice of points, one can evaluate the coefficients

a1 ,. by solving

(0) = 0

(22)

(j), ,w (j ) = 0 , j = 1,2 p, ... ,q

We call such methods product rninimax (PM 2

10

The number of free parameters for implicit methods 2k+l and

the number of equations is 2q+l, thus, the trigonometric order

q is equal to the step number k.

k = 1 Implicit

In this case

+'Pu)

w = i(w2 +W) (23)

a=1 -a1

and the system of equations can be solved for b0, b This

yields the coefficients given by (12) where

-h (1) (24)

-h- , v = hw (24)

Thus, the product minimax method would suggest using the center

of the rectangle [,'u]x[wfWu] as X0 . To obtain a product

minimax method of trigonometric order 2, one has to solve a

system of 5 equations similar to equation (18) with the unknowns

b0 , bI, b2 , a1 , a2. The difference is that in the last 2 equa-

tions one should replace 2.i by (2) and 2v by v In the

second and third equations of (18), the p, v, should be replaced

by (i) V respectively. The resulting system can be solved

by MACSYMA [4] or REDUCE [9].

In the next section we implement two methods of trigonometric

order 1 and 2 and see how the product minimax methods compare

with fitting methods.

l1

3. Numerical Example

Both systems (18) and (19)-(21) were solved by REDUCE which

produced a FORTRAN subroutine for the evaluation of the coeffi-

cients. This subroutine is called only once during the integration.

The methods were compared for the solution of the initial

value problem

• i(l +i)z =0 , 0 <t < 4

(25)z(0) = 1

whose exact solution

if(l+i)tz(t) = e , (26)

thus

i (1 + i) , = - , w = (27)2 2' 227

In order to avoid complex arithmetic, we rewrite the differential

equation as a system of equations for the real and imaginary

part of z = u + iv.

u + '(u +v) = 02#

0 <t <4

v - 21U-v) = 0 , (28)

u(0) = 1

v(0) = 0

12

-ji

The system is solved by fitting methods of trigonometric order

1 and 2 with h = .01 and various values of 0 and w. in Table

1 we list the Euclidean norm of the error at t = 4. It is

clear that the method is not sensitive to perturbations in the

values of 4 and w.

1 ferrorM first order second order

0 0 .3678(-12) .4433(-12)

0 .1 .4482(-7) .3508(-11).i 0 .4346(-7) .2544(-11)

.i1.1 .6342(-7) .4421(-11)

0 .2 .9241(-7) .8126(-11)

.2 0 .8706(-7) .5095(-11)

Table 1

Using the product minimax methods of trigonometric order

1 and 2 with h = .01 and various squares centered at 1P = - 7/2,

w = r/2, the error is much laraer but again is insensitive

to small perturbations in the length of the sides of the

squares. In Table 2 we list the Euclidean norm of the error

at t = 4.

13

error

length of side first order second order

.4 .3678(-12) .1469(-6)

.8 .3678 (-12) .2348 (-5)

1.2 .3678(-12) .1186(-4)

1.6 .3678(-12) .3728(-4)

2 .3678(-12) .9001(-4)

Table 2

Note that the perturbations in the product minimax methods are

larger than those allowed in the fitting methods. It is

possible that the larger errors in the product minimax methods

are due to the assumption that € can be written as a product

of 2 one variable functions. Also note that for the first

order method, one always gets a good result since PM2 always

uses the center of the square.

Acknowledgment

The author would like to thank the NPS Foundation Research

Program for its support of this research.

14

REFERENCES

1. Babcock, P. D., Stutzman, L. F., Brandon, D. M., Improve-ments in a single-step integration algorithm, Simulation28, 1 (1979), 1-10.

2. Brandon, D. M., A new single-step integration algorithmwith A-stability and improved accuracy, Simulation 23,1 (1974), 17.

3. Bettis, D. G., Numerical integration of products of Fourierand ordinary polynomials, Numer. Math., 14 (1970), 421-434.

4. Bogen, R. et al., MACSYMA Reference Manual, MIT Press,Cambridge, MA, 1975.

5. Brock, P., Murray, F. J., The use of exponential sums instep by step integration, Math. Tables Aids Comput. 6(1952), 63-78.

6. Carroll, J., Exponentially fitted one-step methods for thenumerical solution of the scalar Riccati Equation, IMA J.Numer. Anal., to appear.

7. Dennis, S. C. R., The numerical integration of ordinarydifferential equations possessing exponential typesolutions, Proc. Cambridge Phil. Soc. 65 (1960), 240-246.

8. Gautschi, W., Numerical integration of ordinary differen-tial eauations based on triconometric polynomials,Numer. Math., 3 (1961), 381-397.

9. Hearn, A. C. (ed.), REDUCE user's manual, Rand Corporation,Santa Monica, CA, 1983.

10. van der Houwen, P. J., Sommeijer, B. P., Linear multistepmethods with minimized truncation error for periodicinitial value problem, IMA J. Numer. Anal., 4 (1984),479-489.

11. Lambert, J. D., Computational Methods in Ordinary Differ-ential Equations, Wiley, London, 1977.

12. Lyche, T., Chebyshevian multistep methods for ordinarydifferential equations, Numer. Math., 19 (1972), 65-75.

13. Neta, B., Ford, C. H., Families of methods for ordinarydifferential equations based on trigonometric polynomials,J. Comp. Appl. Math., 10 (1984), 33-38.

14. Neta, B., Families of backward differentiation methodsbased on trigonometric polynomials, Intern. J. ComputerMath., to appear 1986.

15

11R

15. Neta, B., Williams, R. T., Hinsman, D. E., Studies in ashallow water fluid model with topography, Proc. EleventhIMACS World Congress, Oslo, Norway, August 5-9, 1985(R. Vichnevetsky, ed.).

16. Pratt, D. T., Exponential-fitted methods for solvingstiff systems of ordinary differential equations, Proc.Eleventh IrIACS World Congress, Oslo, Norway, August 5-9,1985 (R. Vichnevetsky, ed.).

17. Salzer, H. E., Trigonometric interpolation and predictor-corrector formulas for numerical integration, ZAMM, 42(1962), 403-412.

18. Steifel, E., Bettis, D. G., Stabilization of Cowell'smethod, Numer. Math., 13 (1969), 154-175.

16

APPENDIX

Here we show that the shallow water equations with

topography have a solution of the form e , where A is complex.

This system of equations consists of three equations with

three forecast variables u, v and ¢. The equations are:

u _u _ ¢ 0(A.I)-+ u -+ v fv + -s-

+ U Lv + c v + +A1

V- + u - + v ^4 + fu +--= 0 , (A.2)Ct ax dy

+ ( I -+- u- v(¢-zB)I = 0 , (A.3)

where ¢ = gh is the geopotential height (h = height of free

surface), *B is the bottom topography (assumed to be independent

of time), u, v are the components of the wind velocity in the

x, y direction, respectively, and f is the Coriolis parameter.

Linearizing the equations by letting

u = U + u' V = V + V' ell

where U, V are the constant mean flow and is independent of

time. Assuming that U, V are related to via the geostrophic

relations

1 V 1 (A.4)f jy 5 x

one obtains the linear system (after dropping the primes):

17

3u + U aU+ V u f +a 0 (A. 5)at ax ay ax

+ U V + V v + fu + LL= 0 ,(A. 6)

3+ U 2 + V I + I(')+ I V)= U3B+ V a('B (A. 7)5a a y dxD ax 3

where 'B'=

If the flow is assumed to be along the topography as in

[153, then the right hand side of (A.7) is zero. In such a

case, one can write the solution in the form

u u 0e i x+y -Ct)

v v 0 ei(Ex +ny -1-t) (A.8)

0 %e

where

(A.9)

n =v -ie

In order for (A.8) to be a solution for (A.5)-(A.7), one must

have

X= -0 + U + nV (A.10)

satisfying

18

2 --- - -.( n -

ix [(ix ) 2 i( ry+ - f[-i, f - i(irny +ay ay

+ (ir'Y + 2-)(-inf + 2)=0 (~lax(A11

The real and imaginary parts of

r = -a + 1-it + vV

(A.12)

'N. = - U - ev

satisfy the following system of equations (after dropping

nonlinear terms in )

f+r(2 +n 2)H + A. (r 2- + -Y

(A.13)

+ ! ?.F2+(" +r T- 2 0r x

In general, ',is complex and, thus, the shallow water equations

have a solution in the class of problems to be discussed here.

19

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