A High Order Difference Method for Differential Equation

7/27/2019 A High Order Difference Method for Differential Equation

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A HIGH ORDER DIFFERENCE METHOD FOR DIFFERENTIAL EQUATIONSRobert E. Lynch and John R. Rice

Division of Mathema tical SciencesPurdue Univer sity, West Laf ayette , IN 47907

CSD-TR 244S e p t e m b e r 1 9 7 7

AbstractThis paper analyzes a high accuracy approximation to the m-th

order linear ordinar y differen tial equation Mu = f. At mesh pointsU is the estima te of u and U satisfi es MU = I f where M Un n nis a linear combina tion of values of U at m+1 stencil points


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A HIGH ORDER DIFFERENCE METHOD FOR DIFFERENTIAL EQUATIONSRobert E. Lynch* and John R. Rice*Division of Mathematical Sciences

Purdue Unviersi ty, West Lafaye tte, IN 47907

1. Introduction. We consider some aspects of a new flexiblefinite difference method which gives high accuracy approximationto solutions u of linear differentia l equati ons Mu = f subjectto rather general initial or boundary condit ion s. The approximati onto u is taken as U defined at mesh points as the solution ofa system of differe nce equations M n U = I n f together with appropriateboundary conditions; n is used to Identify a particular partitionof the domain of u. M n is a diffe rence operator and M nU is


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Theo ry. We have named this method High Order Differenceapproximtion with Identity Expansions which leads to thepronouncible acronym HODIE .

In this paper the application of the HODIE method toordinary differential equation problems is treated . The analysisand results presented here give insight into the more complicated--and more importantapplication of HODIE to the solution ofpartial differential equati ons. Preliminary results about themulti-dimensional applications are given by Lynch and Rice [1975,1977a,1977b]and by Lynch [1977a,1977b] and more detailed analyses will bepresented at a later time . The method was discovered by R.E. Lynchduring a study of methods for approximating solutions of elliptic


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[1960] after one replaces derivatives of f with divided differ ences,the method of obtaining the coefficients of the difference equationis different from that of Mehrste llenverf ahren.

For ordinary differential equat ions, the HODIE method gives thesame difference equations as obtained by Osborn [1967] whogenerali zed the Styirmer-Numerov scheme . Osborn was pessimis ticabout its practicality; he did not prove convergence results.More recently and indep endently , Doedel [1976] presented anessentially equivalent method for the ordinary differentialequatio n case and he proved some resu lts. Doedel also presentsresults about difference schemes which use more than the minmialnumb er, m+ 1, of stencil points for an m-th order ordinary


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error is demonstrated and Gauss-type auxiliary points areintroduced and analyze d. These Gauss-type points are the zerosof polynomials orthogonal with respect to an integral innerproduct with weight function a polynomial B-spline . InSection 5, we extend t he results of Section 4 to the generallinear variable coefficient differential operator with leadinqterm d m / d t m . In Section 6 , we show that the HODIE method givesa stable difference approximation and that the order of thediscretization error is equal to the order of. the truncation error.Section 7 contai ns a comparison of the computatio nal effor tfor the HODIE method and five other method s; this suggests thatthe HODIE method is among the most efficient methods available


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2. Approximation of differential ope rators. We construct andanalyze high accuracy (m+l)-point difference approximation tom-th order different ial equati ons 1f\ [u,f] = 0 subje ct toi/appropriate initial or two-point boundary conditions W [u.c^] = 0,k = 0,...,m-l where(2-la) W[ u, f] (t ) = Mu(t) - f(t), A < t < B,

m-1 ,(2-lb) Mu(t) = D u(t) + I a.(t) d\{t), D = d/dt,i= 0 1(2-1 c) W k [ u , c k ] = M ku(A ) + M ku(B ) - c k , k = 0,...,m-l,

. m-1(2-1d) M u(t ) = I a, .(t) D {t)i= 0 K > 1

For the initial value problem , a k ,.(A) = 0 if i f k, a^ k (A ) = 1,


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The second set of points comprise J distinct auxiliary pointsF k = (i k -j,...,tk j) subject to the restrictions t^ < x k -j < .. .< t ^ j < tfc+m- T h e identity expansion with coefficients0 is

V i c f k , j ' fk . j ; f ( T k , j ) -

For a given f, U is the solution of L [U ,f ], = H 1.1, - I f , = 0n L J k n k n ksubject to appropriate boundary conditions.

The coefficients a,6 of the operators M n and I aredetermined so that the approx imation is exact on an (L+l)-dimensionallinear space S of fun ctio ns. A basis for S is


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Remarks about bases and efficient methods of solving the HODIE equations(2-2) are given in Section 7.

Boundar y conditions for U are obtained in a simila r wa y.The equation ftj [u,c k] = 0 is approx imated with

^ ' aA , k , i Ui + aB , k , i Un-i *" ( BA.k,j fA,k,j + eB , k , j fB,k,j > " c k = 0

whe re the values f. . . and f R . . are taken at auxiliary" , J B T K , Jpoints near t = A and t = B, resp ecti vely . The coeffi cientsa,f3 are dete rmin ed by

m m- 1


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The truncation error is related to the discretization error,define d as the max-n orm of the error e = u - U at mesh po int s.This is because if u e Z , then M e = M u - M U = M u - I ( M u )n n n n n= T u; that is, e satisfies the equatio n M e = T u. Inn ^ n nSection 6 we show that with natrual hypotheses and appropriateboundary condition appr oximati on, a bound on the truncation erroryields a similar bound on the discretization error.

Example s. We consider a few examples for equal spaced2

mesh points with spacing h and the opera tor Mu = D u + a^Du + a^u.It is sufficient to consider t^ = -h , t k + 1 = 0, = h. Forbrevi ty, we use a single subscripted notation for the coefficients


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0 " 1 + ^ ( T j J C T j - h / Z ] + a o ( T j ) C T j - T j h 3 / 2 la i " + W ^ j 3 + a o ( T j ) C h 2 - T ^ lJ 'a2 ' f 1 + a^TjJCTj+h/Z] + j) CTj+Tjh3/2}J 'i f e,j-i 0 6 J f S + * , < V [ 3 V h 2 j + a O ( T j ) C T j - T j h - Z ^0 " Jj.! ' j f ^ j - a 2 + ^ ( T j J C ^ - z x / ] . of Tj JC tJ -T jV ii0 = 6 J , 2 0 T j " 6 T J h 2 + w ^ W * 3 +

and so on.

1= 0 0 =A = 1 0 = = 2 0 =

normalizationl = 3I = 4I = 5


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the auxiliahy points change.' Below O(h^) denotes the truncationerro r with respe ct to the space of functions 1 = offunctions with continuous (p+2)-nd deriva tive.

Example 2-1: For J = 1 and -h = t k < x^ 4 t ^ = h, x-j f 0,the equati on f or = 3 is not satis fied [a-j = ag = 0] and withI n f k = f(xj) we obtain an 0(h) scheme which is exact on P ^ .

Example 2-2: For J = 1 and x-| = t ^ = 0 , the equation forSL = 3 is satis fied, but the one for = 4 is not satis fied, andpwith I n f k = f(x-j) we obtain an 0(h ) scheme which is exact on v

Example 2-3: For J = 2 and -x^ = Xg = h ( l / 6 ) ^ 2 we obtain4


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To use these schemes for the Dirichlet pro blem , one solvesthe system(2-5a) U Q = u(A), U n (B) = u(B)(2-5b) ( U k l - 2 U k + U k + 1 ) / h 2 = 9 k , k = 1,..'. ,n-l

with h = ( B - A ) / n , t k = A + kh and"k " 'n fk-l Ej., ' W W t j )

i

where g differs from example to exampl e. In each cas e, howe ver,the matrix formulation has the same tridiagonal (n-1)- by-(n- l)coefficient matr ix. Once g has been eval uate d, the work tosolve the system is independent of the particular g used. Thu s,


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Example 2-1': 0(h) , g = f(A)Example 2-2': 0( h 2 ) , g = f(A+h/3)Example 2-4': 0( h 4) , g = [9f(A) + 25f(A+2h/5) + 2f(A+h)]/36.Example 2-5': 0( h 6) , g = B ^ A + r ^ + B 2f ( A + T 2) + B ^ f A ^ ) ,

3 1 = 0.4018638275, 6 2 = 0.4584822127, B 3 = 0.1396539598,t 1 = 0.0885879595h, T 2 = 0.4094668644h, T 3 = 0.7876594618h.

Example 2-6': 0 (h 1 0 ) , g = B ^ A + t ^ + ... + B 5f ( A + r 5 ) ,B 1 = 0.1935631805, B 2 = 0.3343492762, B 3 = 0.2927739742,B 4 = 0.1478177401, B 5 = 0.0314958290,


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3. Truncation error for polynomial approxim ation. We onlyconsider approximation away from boundaries and approximation whichis exact on a polynomial space P ^ for some L m. Results forapproximation of boundary conditions are obtained by an easy modification.Results for other spaces, such as those appropriate for approximationnear singular points of differential equa tions , will be presented elsewhere

We use ., j = 0,1,. .., to denote distinct points such thatK Jl k ^ V m a n d s e t = ^ k , 0 w e a l s o s e t(3-1) A F k J = m i n M = 0 j ( - C k > q | .We use the polynomials

(3-2a) w ^ k , j ; t ) = nq = 0 f t " ? k , q ) / ( j + 1 ) ! ' J * 0 ' . 1 - - - ,


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Because T n u k = M n u k - I n[ M u ] k involves derivatives of u onlyup to order m < L, it follows (see, for exampl e, Theorem 2.1 ofde Boor and Lynch [1966]) that for

u e ^ [ t ^ t ^ ] = { v | D L V is absolutely continuous,(3-5)

D ^ v is square integrable on t^ < t < t k + m }we have

fc** T n ( t ) C ^ k . L 8 t ' X k d x(3-6a)

+'k

where


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one has q = 0 on the stencil points and hence M ^ j q = 0;(3-6a) then reduces to

J rt D L + 1 U ( X ) dx,k t = = Tk , j

where

M ( t ) q ( ^ , L U ' x )(3-7b) L L= f a,(t) (( t-x )^V (L- i) ! - I (5 .-x) D V ( ? . ;t )/ L! )

i=0 j=0 K , L J K , LIn (3-7 a), points T k j, x , and those in k L are between t k an dt k + There fore, by (3-4) we can bound the quantity in curly bracketsin (3-7b ) by + K 2 [ h k/ A f k L ] L ) where K ],Kg are constants

L + 1


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Furthermore, set(3-9) H = ma x. n m ( t < + m - t.)/mn j-0,...,n -m J+m jand we have the following.

THEOREM 3-1: Suppose the coefficients a . of M are continuous.Let A = tQ < t^ < ... < t = B , n > m , be a set of mesh points andt^, k = 0,..., n-m, sets of auxiliary points. Suppose that fork = 0,...,n-m there are coefficients a^ j which satisfy(2-2) and (2-3b) for Sg ,.. .,s L, L > m , a basis for P ^ . Thenthere is a constant K which depends only on B-A , the order m ofM , and the coeffic ients a^ such that for any u with continuous


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4. Analysis of the special case M = D m . The main resultsabout the special case M = D m carry over to the general caseof the variabl e coeff icien t operator M in (2-lb). In thissectio n, we consider in detail the special case. To distinguishbetween the two cas es, we use the superscript 0 for quantitieswhich apply to the special c ase , in partic ular, we use a^ , B^ , M^,and for the coeffic ients and the operato rs when M = D m .

In (2-2) set M = D m , replace a,B with a? B? and use thefollowing basis for P L [see (3-2) and (3-3)]:

(4-1 a) s.(t) = A i( t k ;t) , i = 0 m,w(r k ) i_-| it), 1 = rrr+1,... ,L


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(4-2a) a k , 1 / h k " tnii /W(t k; t k + i) J 6 k J = 0 , 1 = 0 m ,(4-2b) ^ D m w ( C k j m + J l _ 2 i T k J ) = = 1 . . , L- m+ l ,where . denotes the Kronecker delta function,i

Since the sum of the B's is un ity , (4-2a) shows that the operatorM^ is m! times the usual divided differe nce approxi mation to M = D1":

M S U k = Z " n ak , i u < W ' h k = F. n(4-3) 1 - 0 1 = 0

= m! u[ t k , t k + 1 t k + m ] ,that is, M^u k is the m-th deri vati ve of the unique polynomialin P ^ which Interpolates to the values u U k + 1 -) at t k + 1- i = 0,...,m.

By Taylor's Theo rem, any u in F 1" can be represente d as


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at the stencil points in t k . This B-spli ne satisfies (Curry andSchoenberg [1966])

(4-5a) B m ( t k ;* ) = { > *k < X < W

(4-5b)

k+ mtk + m B m( t k;x) dx = 1.

Therefore, we have

- > H

- cOrnf T


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now show that there exist special sets of auxiliary points whichmake the approximation exact on P ^ for L up to m+2J -l.

Since B m ( t k ; * ) is positive on the range of integration, wecan define the following inner product:

rt,(u,v) = k + m B m { t k ;x) u(x) v(x) dx rh

For fixed m , k, and B (t^;-) let bg , b^,. .. with , b^ in P ^denote the normalized orthogonal polynomials with respect to thisinner product ; we call these the B-spline orthogonal po lynomia ls.Based on the well-known theory of orthogonal polyno mials, b^ hasi dist inct real zeros in t k < t < t k + m > an d, for fixed i, wecall these the B-spline Gauss points.


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THEOR EM 4-1: Let M = D m and let the normalization for HODIEappro ximat ion be (2-3c). For any set of m+1 stencil and J > 0auxiliary points t ^ T ^ , there is a HODIE approximation withcoefficients a. H = a^ ., 0. . = B? whic h is exact on P, forK I J K J K J J K | J Lany L with 0 < L-m < J-l . The operator M n = M n is unique, itis m! times the divided difference operator with respect to thestencil point s. There are sets of J auxiliary points for whicha HODIE approx imatio n is exact for L with J < L-m < 2J- 1.L-m > J- l, then the coefficients of I are unique and aregiven by (4-6). The J auxilia ry points which give exactness on J^j+ m-lare the zeros of the J-th degree B-spline orthogonal polynomial bj


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because of this (or, alterna tively , symme try), the scheme 1sexact on P ^ . Another set of three auxiliary points (Example 2-5)yield s an approximation exact on P ^ .We now derive bounds on the elements of the inverse of the coefficientmatr ix of the system in (4-2b) with L~m+1 = J; these are used in the next sectior

For i = l,...,j conside r the systems

^ Xj,i m w ( W - 2 ' T k , j > = 1 =

Mult iply the Ji-th equation by the constan t (determined below ) n , ' r-1,m+-2and sum with respect to I to obtain

( 4 ' 7 ) Xj , i ^ V l , m + S , - 2 = V l . m + i - 2 -


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Xr,T = V l . m + i - 2 = P r - l ^ f c . O - ' - ^ k . m + l - l ^The points are disti nct, are between and t f c + f n and s k =A = 0,..., m. Hence it follows from (4-4) that

x - = f t k + n l B , , .;x) D m + i _ 1 p ,(x) dx

f ) k + m B m + i - l ^ k , m + i - l ; x ) d 1 " V i < V x > d x ' .kwhere B^.. 7 _ i'>') denotes the polynomial B-spline of degreem+i-2 with joints at f^ l = 0 , . , m + i - l . For the case i = 1,this reduces to x j ] = \ j w l t h \ j given in (4-6). By (4-5) and(3-4), we hav e, ther efore , the following res ult.


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5. Analysis of the variable coefficient case. Let \ anddenote the functions obtained by applying M to the basis element% and w in (4-1 a):(5-la) (t) = M s ^ t ) = M^i (t"k;t), 1 = 0,....m,(5-1 b) ^ ( t ) = M s m + j l(t) = Mw(C k i r n n_-,; t), = l,...,L-m,and set

(5-lc) ^ Q (t ) = = 1.We use A 9 , ^ to denote these functions in the special case M = D m .

The HODIE equations are then


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for i = 0,... ,m we have

(5-3a)= 1

m m+ { h k V l ( T k (Pk i-Y k o) + + V o ( T k i ) n Y k Q} / mK M I q = o K ' J K ' Q K U K , J Q = 0 > Q ? ( I K,q

Fo r i = l,...,L-m+l we have

(5-3b) " { C ^ ^ ^ t - i ' P k j ) + ^ V T k , j > P w ( W r > k , .


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To show that HODIE approx imation s exist for L-m = 2J-1 withspecial auxiliar y point s, we need some preliminary result s.After changing to nondimensional parame ters, the functions ^in (5-1) have the same form as the functions in the next theorem.This theorem shows that the set of functions ij^, I = 0,...,L-mis a Chebyshev set.

THEOREM 5-2: Let K and m denote positive integer s. LetY k , k = 0,...,K+m-l denote distinct points in the unit interv al.Let the functi ons have the form


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p such that V(h) is nonsingular for all h, 0 < h H^. Thenthere are sequences with index i = 1,2, ... , *

H i + 1 = O.-'-K-" 1' w it h m a x J c ^ H . ) ) = 1 ,p"(Hi) = (p 1(H.),...,p K(H.)) i P.(p) = Cj,{H.) * ( H . ; p ) ,

where P^ has zeros at p = P j ^ - ) , j = 1 K. There exist, therefore,convergent subsequences (whose elements we also denote as above) such that

^ ( H . J - c * , PjtH,-) - P], and P. - P*.By continuity and the form of the functions the limitingfunction P* is a polynomial of degree at most K- l. Again by conti nuit y,P*(pj) = 0, j = 1,... ,K. Since max fc|c*| = 1 , P* is not identically


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Let p denote any nondecreasing right continuous function ofbounded vari ation on t^ < t < t k + . Let a = 0 L denot efunctions of a Chebyshev set on this interval. The -th moment q^of the set with respect to the measu re dy is

= f t k + m M X ) d y( x ) , * = 0 , .. . ,L .J . WkFor each meas ure, one gets a set of moments q = {qQ,...,q^) andthe set of all such q is a subse t Q of Euclidi an (L+l)-spacewhich is called the moment space of the Chebyshev set. This momentspace is the smallest cone with vertex at the origin which containsthe curve *(t) = ( Q( t ) . . . ^ ( t ) ) , t R < t < t k + m ; this curveis not in Euclidean L-sp ace . If L = 2J -1 , J > 1, and q e Q is


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then q Q = (q 0,0' q0,l " '' ' q0 , L }' q0,*-l = 1 s 1 n V T h u s 'the principle representation is given with t. ., the zeros of theK J JJ-th degree B-spli ne orthogonal polynomi al and with Bi, * equalK , Jto B k j in (4-6). By uni que nes s, cfg is an interior point of themom ent space Qg and so there is a closed sphere Sg with centerqg in the interior of Qg.

It follows from Theorem 5-2 that if the coefficients a . of Mare contin uous, then the functions in (5-lb) form aChebyshev set for all h k sufficiently smal l. Let Q denote themoment space for this Chebyshev set . The curve (t ) - ( I^Q ,...,^)converges uniformly to the curve ^ ( t ) = (1 s ^ ( t ) . . ,D ms L(t))o n t^ ^ t


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The system (5-2b ) with L-m+1 = J can be writte n in matri x form as(B + B 1 ) b = e v e ^ = (1,0 0)

where B is the matri x in Lerrma 4- 1. With 0 = +


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where the norms are thn vector max-nomi and the matrix row-sum -norm.For all suffic ientl y small h^ there 1s , the ref ore , a constant Kgsuch that( 5-4) \ m \ m < h k ( V A 7 k ) J " ] K 0 m a x j ' 6 k , j l -Lemma 4-1 with i = 1 gives a bound on . which yiel dsK , J

m a x j l B k , j 1 = ( h k / A 7 k ) J ^ ^ r ^ r j ) t 1 + h k ^ k ^ k ^ " 1 K 0 ] 'This gives the following result.

LEMMA 5-1: Under the same hypothe ses as Theor em 5- 3, there is aconsta nt K which is independe nt of h^/Ax ^ such that for all suffic iently


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6. Discretization error for polynomial approximation . He beginby obtaining a bound on the solution of a homogeneous HODIE dififerenceequation problem with values of the first m-1 divided differencesgiven at tg = A. Let V denot e the solution of

M n V k = 0, k = 0,1,...,V [ t 0] , V [ t 0 >tj ], ... , V[t Q,.. . .tm_.j] are giv en,

where M n is from a HODIE approximati on which is exact on Pj withL > m.

For fixed . k, let p denote that unique element in P whichminterpolates to V k , V k + 1 , . . . , V k + m at t k , t k + 1 t k + l f |. Writingp in the Newton form of the interpolation polynom ial, we have


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Set H n = max k Because the auxili ary points are between t kand t k + m ' t h e r e a r e constants K^ which depend on maxi|Ja. U ^ ,but not on the mesh points nor on the auxil iary poi nts , nor nn H rsuch that for H p < 1

< K , n - X j l B ^ j I C l - H * + 1 ) / ( l - H n )

By Lemma -5-1, m a x ^ S ^ | < K R J _ 1 , R = h k / A f k . Consequ ently, if


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We also have

(6-lb) "'= +

( - V V l )1 = 1 k+m-1.Let ||V k|! m, 1 denote

H v k m - i " + M V W I + ... + |v[t k W l ] | .From (6-1) we obtain

ll vk+lllm-i i f 1 + H n K ) H v k Hm-1where


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we haveM n V A - 1 = CH -l ,m V r - ' W l H =

Hence, for k = A.2 .+ 1. . ,n-m,H K(k -) H K(k -) H K(k-Jl)

The soluti on W of the initial value problemM n W k = F k ' 1 = 0 , 1W [ t Q ] , W C t g . t ^ , W [ t 0,... ,t m _ 1 ] given,

is bounded by


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One can choose a set of solutions u ^ , j = 0,...,m-l, whichspan the space of all solutions of Mu - f = 0 and the correspo ndingHODIE approximations U ^ subject to initial conditions convergeto u ^ a s 0 ( H j | "m + 1). These can be used to obtain the unique HODIEapproximation of the solution of (2-1) subject to the general boundaryconditio ns in (2-1) where the HODIE approxi mation sat isfies th egeneral boundary conditions in (2-4). In addition to exist ence,uniqu eness , and smoothness of the solution u of (2-1), one needsthat the boundary conditions in (2-lc) are linearly independent onifthe space of polynomials P m , that is, if }ff [p,0] = 0 , k = 0, ..., m-l,for p in then p = 0. We then have the following re sult

THEOREM 6-1: Suppose the coefficients a. of M in (2-lb)


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andR-, = max. { h . /min - , [ T , - T

are bounded as n >. Suppose that the HODIE approx imatio n isexact on P L with L > m + J -1 and let denot e the HODIEapproximation on the n-th partiti on. Then for all sufficientlysmall H n


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7, Computation analysis . In this secti on, we consider thecomputational aspects of the HODIE meth od. We discuss specific featuresof our implementation and we compare the amount of work, with other availablemethod s. The discussion is restricted to the case of second order equationssubject to Dirichlet boundary conditions for four reasons: it is simpl e,it is the most important case, it is readily generalized, andthere are detailed analyses of other methods available for compari son.

The differential equation problem is

Mu(t) = a 2(t)u"(t) + a 1(t)u'(t) + a Q(t) u(t) = f(t), A < t < B,u(A) and u(B) given,

whe re, for gene rali ty, we have taken the coefficient of u" in M


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There are two distinct parts in an implementation of a specificHODIE approximation. The first part consists in the determinationof the values of the coeffici ents a.. , i = 0, 1,2 , and ,,

K , 1 K , Jj = 1,. .., J, for each k = 0,... ,n-2 and then the determi nation ofthe values I nf k k = 0,.. .,n- 2. The second part is the determinationof the values U^ , k = l,.,., n-2, of the solution of the resulting(n-1)-by-(n-l) tridiagonal system of difference equations .

In the first par t, the system of algebraic equations for thea's and 3's is reducibl e: one solve a J-by-J system for theB's and then a 3-by- 3 system for the a' s; this is done foreach k = 0,..., n-2. This reducibility results in significant savingsof work for the special second order ca se , m = 2, as well as in the


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where the X's indicate nonzero elemen ts. Thi s, of cour se, is veryadvantageous for solving for the B's in the regular case.

We consider the computational effort required first for a uniformpartition: t^ = kh , k = 0,...,n. We measure the effort in terms ofthe number F of function evaluations ( a 2 , a^ , ag , or f) andthe number M of multiplications requir ed. In regard to the non-function-evaluation wo rk, we assume: the total computational effort is proportionalto the number of multipl ication s. Table 7-1 lists the effort requiredfor various part of an implementation of the HODIE scheme.

Computation step J = Regular Case3 5 7 9 Gauss-type Case2 3 4 5


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Regural Case is assumed for estimating the work to solve this matrixequati on. For the Gauss-type Cas e, we have a general (J-l)-by-(J-l) system to solve. Note that we assume that the Gauss-typeauxiliary points have been previously computed or are otherwise known.The right sides of the a-equ ation s are of a special form and the computationis carried out by forming B b ^ ( T , , .) and then combining theseK ,J JT K , Jappropri ately. The solution of the a-equat1ons is trivial and thefinal multiplications occur in solving the large tridiagonal systemplus the evaluation of its right side. In the Regular Case, the functionvalues at the mesh points and the auxiliary points are used more thanonce without recomputatio n.

We now use these work estimates to compar e, roug hly, the work of


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We emphasize that the exact values of these operations countsdepend on small details of the implementation of a particular algorithmand one can trade multiplications for addi tion s, and so on, in some instances.

Order of the method and mesh typeMethod FourthUniform General SixthUniform General EighthUniform General

HODIE, Regular Case 34M+4F 40M+4F 89M+12F 113M+12F 183M+20F 241M+20FHODIE, Gauss-type Case 28M+8F 32M+8F 49M+12F 57M+12F 109M+16F 140M+16FCollocation, piecewiseHermit e r " " 38M+8F 42M+8F 62M+12F 72M+12F 145M+16F 159M+16F

Collocation, splines 24M+4F 56M+4F 37M+ 4F 99M+ 4F 52M+ 4F 152M+ 4FExtrapolation of thetrapezoid rule 32M+8F 32M+8F 70M+16F 70M+16F 165M+32F 165M+32FLeast square s, splines 66M+8F 90M+8F 198M+16F 270M+16F 440M+24F 580M+24F


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extrap olatio n, even for uniform sp acin g, for a problem for which theerror behavior is as in Figure 8-3).

Considerable caution should be taken in attaching importance to thespecific num bers in Table 7- 2. These give only roufrh compa rison s andvarious other considerations can completely override the differencebetween , say , 28 and 35 multiplications per poin t. We can only concludethat the first five methods are generally comparable in work and the lasttwo seem unlikely to be competitive. Collocation with splines seems togain a work advantage as the order increa ses, but it is simultaneouslyincreasingly complicated near the boundaries which may well negate thisadvantage somewhat.

To obtain a realistic evaluation of these metho ds, one needs not


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8. Experimental results. We present support for the following points:(1) The HODIE method converges as predicted by theory; there are nounforeseen numerical complications . (2) There are no unforeseen difficultiesor complexities in implementation. (3) There is a definite pattern in therelationship among the accuracy actually achie ved, the actual computationti me, and the order of the meth od. Specif ically, the higher the desiredaccuracy, the higher should the order of the method be to minimizecomputation ti me. (4) The use of Gauss-type auxiliary points gives the rateof convergence predicted by theory. (5) The use of Gauss-type

2auxiliary point for the operator D improves the rate of convergencefor a general second order operator M over that expected for ageneral set of auxiliary points.


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confidence in the reliability of the HODIE method.The Fortran program we wrote seemed to be as easy to write and to

debug as a.program for any other metho d of solving this class of proble ms.Howe ver, we quickly found that in order to verify the rates of convergencefor very high order HODIE sche mes, we had to use very high precision.In the remainder of this s ecti on, we discuss only a small subset of theexperiments which we performed.

All computation was done on the Purdue University CDC6500 withdouble precision arithmetic which uses values with about 28 decimaldigits . In each experim ent, the domain of the problem was partitionedby an equal-spaced mesh with N subin terval s, so the mesh spacingh was proportional to 1/N.


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conver gence . The central auxiliary point is the central mesh point ofthe three-poi nt difference operator M^ and it is clear from the symnetryof the differential operator that this auxiliary point is a zero ofevery odd-degree generalized B-spline orthogonal polynomial. This(or symmetry) shows that one expects 0( h 6) rather than n( h 5) convergence

(b) There is a set of nine curves in Figure 8-1 which h ave sharpdownward spikes at N = 4,8,16,2 5,32,50, 64,100, and 20 0, respectively.The set of 5 auxili ary points used for each one of these curves is theset of 5 Gauss-type points for that value of N at which the spikeoccurs . One has nine different sets of these Gauss-type points becausetheir locations depends on h = 1/N. The curve with spike at N = 8 istypical and we describe some of its features. Fir st, the spike is


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(d) The last curve is the one for 5 Gauss-t ype points for the2operator M = D . Except for the central auxiliary poi nt, these are

2not the Gauss -type points for the opera tor D - 4. One expects at leastO(h^) rate of convergence; however,' a very consistent 0(h) rate ofconvergence is observe d. As h tends to zer o, the Gauss-type2auxiliary poi nts tend to those of the operato r D , hence one expectsimprovement over an arbitrary set of auxiliary points, even a set whichcontains the central mesh point of the operator

Example 8-2: Typical of a fairly difficu lt problem is one takenfrom Rachford and Wheeler [1974]:

t Q = 0.36388(.01 + 100(t-t Q) g u( t) ] = -2{1 + 100(t-t 0)(tan" 1[100(t-t Q)]


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O ( h ^ ) , respectively. For a general set of seven auxiliary points, oneexpects O(h^ ) rate of convergence; the use of the Gauss-type pointsfor the operator D 2 improves the rate of convergence to O ( h ^ ) .

To compa re effi cie ncy , we note that the Sttfrmer-Numerov s chemewith N = 300 required almost exactl y the same amou nt of comput ation tim eas the seven-poin t scheme with 100 poi nts . The StfJrmer-Numerov schem eachieved a maximum erro of .00026 which is almost exactly 100 timesgreater than the error for the higher order scheme.

Fina lly, we note that the usefulness of extrapolation techniquesis doubt ful for eithe r of these schemes for N less than about 100 .Exampl e S-3 : The final example we discus s is:

u"(t) + sin(t) u'(t) + 4 t 2 u{t) = 2[1 + t sin(t)] cos(t 2), 0 t 5,


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502 fi3-point D Gauss-type method both are 0 ( h ) metho ds, but the

maximum error of the Regular method is about 10 times larger than theGauss-type method for the same execution time.


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51REFERENCES

Birk hoff, G. , and C.R. de Boor, [196 5], Piecewise polynomialinterpolation and appro ximat ion, in Approximation of func tion s.Editor H.L. Garabedian, Elsevier Publishing C o A m s t e r d a m , 164-190.

de Boor, C. , and R.E. Lyn ch, [1966 ], On splines and their minimumprop ertie s, J. Math and Mech 25 953- 970.

de Boo r, C.W. and B. Swar tz, [1973] , Collocation at Gaussian po ints ,SIAM J. Num. Anal., 10 582-606.

Coll atz, L., [I960], The numerical treatement of differential equ ation s,3rd Editi on, Springer-Verlag, Berlin.Curr y, H.B ., and I.J. Schoe nberg , [1966 ], On Polya frequency functions IV .The spline functions and their limi ts, J. Analyse Hat h. 1771-107.


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Lync h, R.E., and J.R. Rice [1975], The HODIE method: A brief introductionwith summary of computational properties, Department of ComputerScience Report 170 , Purdue Universi ty, Nov . 18.Lyn ch, R.E ., and J.R . Rice [1977a], High accuracy finite differenceapproximation to solutions of elliptic partial differentialequa tion s, Department of Computer Science Report CSD-TR 223, PurdueUniversity, Feb. 21.

Lync h, R.E ., and J.R. Rice [1977b],. High accuracy finite differenceapproximation to solutions of elliptic partial differentialequat ions, (complete revision of Report CSD-TR 223) to appear .

Osbo rne , M. R. , Minimizing truncation error in finite difference


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Russ ell, R.D. , and L.F. Sham pine, [197 2], A collocation method forboundary value problems, Numer. Math. 1-28.

Russe ll, R.D ., and J.M . Vara h, [1975], A comparision of globalmethods for linear two-point boundary value prob lems . Mat h. Comp29 1007-1019.


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.00-1

- 5 . 0 0 -

54

D D U - 4 * U = 2 * C 0 S H ( n O N ( 0 , 1 )S O L U T I O N U ( T ) = C 0 S H ( 2 * T - 1 ) - C Q S H ( 1 )

- 1 0 . 0 0 -

enocna : - I S . 0 0U JX< x

o - 2 0 . 0 0 -


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Z.OO-t

.00 -

-e.oo-

- 4 . 0 0 -Q001 -B.OOX(Xx

N u ( - 4 )> * >

STQRHER-NUMEROV


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Figure 8-3 : Illustr ation of the relation ship between work (execution tim e),accuracy ach ieve d, and order of the HODIE method for Exampl e 8-

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A High Order Difference Method for Differential Equation