Semidiscretisations of u tt = u xx and Rational ...rosie/mypapers/2157834.pdf · Williamson, [8]-[10], [12] (see [6] for a full list of references). In this paper we begin an investigation

Semidiscretisations of utt = uxx and Rational Approximations to (ln z)2Author(s): R. A. Renaut-WilliamsonSource: SIAM Journal on Numerical Analysis, Vol. 26, No. 2 (Apr., 1989), pp. 320-337Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2157834 .

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SIAM J. NUMER. ANAL. 1 1989 Society for Industrial and Applied Mathematics Vol. 26, No. 2, pp. 320-337, April 1989 003

SEMIDISCRETISATIONS OF u, =u AND RATIONAL APPROXIMATIONS TO (In z)2*

R. A. RENAUT-WILLIAMSONt

Abstract. Semidiscretisations of the second-order spatial derivative for the numerical solution of u,, = Cx are analysed. It is demonstrated that the stability and order of accuracy of numerical schemes are determined by properties of rational approximations to zL(ln z)2. By using order star techniques, the maximal order of accuracy of such approximations is determined to be p ' 2(r + R). This order is achieved by Pade schemes, which are also proven to be stable. It is noted that the results are also valid for the equation u, = u.,.

Key words. time stepping, finite difference schemes for hyperbolic equations, stability, accuracy barriers, Pade schemes, wave equation, (ln z)2

AMS(MOS) subject classifications. 65M10, 65M20

1. Introduction. We return here to a theme widely investigated in the last few years: the determination of the maximal order of accuracy of numerical schemes for the solution of partial differential equations. First-order hyperbolic equations, as modeled by the linear test equation u, = u,, u = u (x, t), have been studied in a succession of works initiated by Iserles [3] and continued by Iserles, Jeltsch, Strang, Strack, and Williamson, [8]-[10], [12] (see [6] for a full list of references). In this paper we begin an investigation of numerical schemes for the solution of the wave equation

(1.1) Utt = uXX u = u(x, t),

which is the simplest model of a second-order hyperbolic equation. Semidiscretisations (SD) for the solution of this equation will be considered, the discussion of full discretisations is left to a later paper [13].

We assume that the same SD finite difference scheme is used to solve (1.1) at all grid points away from the boundary:

s d2U () 1 (1.2)E bj dm( )=,) >m+j(t)

(1.2) ~~~~j=-R d2j-

Here the coefficients aj and bj are constants, Ax is the grid size in the x-direction, and Um+j(t) approximates the exact solution u((m +j)Ax, t). For this numerical scheme the characteristic functions are

s s

r(z):= Y. ajz3, s(z):= iz j=-r j=-R

so that, with E being the shift operator

Ef(x) := f(x + Ax),

equation (1.2) is a system of equations each with form

(1.3) s(E) d2 U

-= (t )2 r(E) U(x, t)

* Received by the editors August 31, 1987; accepted for publication (in revised form) February 12, 1988. t Templergraben 55, D-5100 Aachen, Federal Republic of Germany. Present address, Department of

Mathematics, Arizona State University, Tempe, Arizona 85287-1804. The work of this author was supported by a postdoctoral fellowship from the Royal Society.

320



SEMIDISCRETISATIONS FOR u,, = u 321

where U(x, t):= Um(t), mAx-Ax/2'x-'mAx+Ax/2. Then, assuming that the implicit part s(E) is invertible, we say that the SD (1.2) has order of accuracy p when for the solution u (x, t),

a2 u(x, t) r(E) 0p+2 2 (U(X, t) = C(AX)p p+2 U + O(Ax)P+. ax (A)sE) O

Now with the differential operator D:= d/ dx, and using the usual operator relationship E = exD, we see that this equation can be written as

(1.4) ((In E)2- R(E))u(x, t) = C (Ax) p+2 a p+2 + O(AX)p+3,

where R(E) = r(E)/s(E). Consequently, the SD has order of accuracy p with error constant C if and only if

(1.5) R(z) = (In z)2 + C(Z-_ 1)p+2 + ?(|Z - 1 1p+3).

Therefore we have an approximation-theoretic problem: the determination of the theoretical order of accuracy of rational approximations to (ln Z)2 in the neighbourhood of z = 1.

The approximation R(z) to (ln Z)2 is a consequence of the second order of the spatial derivative. Consider, for example, the derivation given by Williamson [12] for the equation u, = Lu where L is a general linear spatial differential operator. Clearly for u, = uxx, the equation of heat conduction, approximations to (ln Z)2 must also be investigated. Iserles [5], in his study of full discretisations for the heat equation, looked at the diagonal rational approximations to (ln Z)2, i.e., R = S = r = s. From the definition of order p (1.5) we see, from a count of the number of free coefficients, that maximal order p = 4r - 1 might be expected. However, the form of F(z) = (ln z)2 = F(1/z) enables p _ 4r, and Iserles demonstrates that not only can this bound be achieved, but also that those schemes that achieve the bound are stable.

Naturally we are interested not only in determining p, or a bound on it, but also in the stability of schemes that achieve this bound. Performing the usual Fourier analysis, that is transforming (1.3), we see that the transformed equation can be solved for the Fourier transform function, or equivalently (1.2) for Um(t) if and only if the function s(z) is nonzero for z lying on the unit disc, i.e.,

(1.6) s(z) $ O for lzl = 1.

Further, von Neumann stability, that is global convergence of U(x, t), follows when

(1.7a) Re R(e ')_O -1r _ 0 < 7r,

(1.7b) Im R(e"') = O,05 0 < vr.

Condition (1.6) is the invertibility condition for numerical schemes (1.2). By the application of the Wiener-Hopf theory [8], we see that this is equivalent to the condition that s(z) should be centered: s(z) must have R zeros inside the unit circle and S zeros outside the unit circle.

However, we note from (1.7b) that the requirement Im R(e"') = 0, 0 c [-rT, IT)

means that our numerical scheme should have R =S, r = s with aj = a_j and bj = b_j if it is to be stable. Thus, with this choice, invertibility is just the requirement of analyticity on the unit circle for the function R(z) = R(1/z).

Furthermore, we note in passing that our stability analysis is based on the approximations of maximal order: Pade approximations. For such approximations to



322 R. A. RENAUT-WILLIAMSON

(In Z)2, it can be seen that the linear system of equations determining the coefficients aj and bj always has a solution, R(z) = R(l/z). By Pade theory [1] this must be the unique solution of the system. Consequently, it is not unreasonable to restrict the attention to central differencing methods satisfying R(z) = R(l/z). Additionally, stability results for these Pade approximations are also valid for the heat equation.

Henceforth we only consider the rational approximation problem described by the order condition (1.5) and the stability conditions (1.7). Naturally the analyticity of s(z) for lzl = 1 is also required. As mentioned, Iserles [5] has already considered this problem for the restricted class of diagonal approximations. He employed an order star analysis, which we show in ? 2 to be insufficient for a complete analysis of this wider class of approximations. However, it is extendable to all methods, allowing a proof of the correct order bound. We do not pursue this analysis too far; rather we introduce another order star that enables proof of the stability result.

We now briefly outline the rest of the paper. In ? 2 this new order star will be introduced, and from its geometric properties the theoretical bound on accuracy

(1.8) p 2(r + R)

will be proved. The order star of Iserles will also be briefly described [5]. In ? 4 we study Pade approximations to (ln Z)2. These are shown to attain the bound (1.8), and for polynomial approximation the defining equations can be solved exactly, giving the coefficients of the explicit schemes. Furthermore, consideration of both order stars leads to the proof of our main theorem: All Pade schemes are stable and invertible.

2. Order star theory. In his study of the equation u, = uXX, Iserles considered an order star of the second kind for (ln Z)2 [5]. For the rational function

=M jJ R(z) =N v2N b Djz'

~j=N, j

where min {M1, N1} =0, the function u1(z) is defined by

(2.1) aj1z):= R(ez)- z 2 ze C.

Then the level sets

A1 = (z C clC: Re o-1(z) > 0),

(2.2) DI = (z E clC: Re o-1(z) < 0),

a1 = (z C clC: Re 1(z) = 0)

describe the order star of the second kind of (ln Z)2. This order star is similar to that used by Iserles and Williamson in their study of u, = u, [9] and has similar geometric properties, which depend on the order of the approximation of R(z) to (ln Z)2 and on the location of the poles and zeros of R(z) (see [5] for a precise description of o-1, and [3] and [11] for a more general explanation of order stars).

Examples of this order star are given in Figs. 1-5. Also, for comparison, the same rational approximations are used for the inverse order star, which will be introduced shortly. A full explanation of the geometric properties for this other order star will also be given. Many of the properties of the original order star can be confirmed by analysis of the corresponding inverse order star.

For the rational approximations we have

Ml=(R-r)+, N2 = 2(r-R)+,

M2=(R-r)++2r, N2 =2R +(r -R)+.



SEMIDISCRETISATIONS FOR u,, = u,, 323

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1.;...:.~~~~~~~~~...... .. . .. .... . . 1 1~~~~~~~~~~~.....

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........... X. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ -



SEMIDISCRETISATIONS FOR u,, 3= u,25

' --- AS + ~~~~~~~~~~~~~~~l :3-4x

'~~~~~~~~~~~~

;: U~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~:

o: bs}~~~~~~~~




...........

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.......... ~ ~ ~ ~ ~ ~

.. . .......~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ . ...............

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V 0~~~~~~~~~~~~~....... .X

.......... ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ...... ............ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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SEMIDISCRETISATIONS FOR u, = u, 327

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We can substitute these values into the bound on p as derived by Iserles [5, Lemma 3]:

p-2 max {M2, N2} - (N1 - M1)+ - (M2 - N2)+ + 2,

which gives

(2.3) p 2R + 2 max {r, R}.

This gives the bound p 2(r + R) for the more explicit methods, those with r ' R, and as we will see later, it is achieved by Pade approximations and is thus correct. In fact by counting the available degrees of freedom and together with R(z) = R(1/z), we have p - 2(r + R) for Pade approximations, and thus the bound p _ 4R for the more implicit methods R> r seems somewhat generous. Furthermore, this bound would suggest that the more implicit methods would be better, which is not the case with simple examples, and contrary to experience with the first-order equation [9].

Further investigation of the order star, as in [9], confirms that the bound is too generous. the bound becomes restricted when the position of the zeros of the function R(z) is considered as well as that of the poles.

Instead of describing this analysis we introduce our new order star-the inverse order star. For the function

(2.4) o2(z) := R(ez) Z-2, z E C

the level sets

A2={Z CIC: Re o-2(z)>O},

(2.5) D2 = {z E clC: Re 0-2(Z) < },

d2 = {Z c clC: Re -2(Z) = O}

describe the order star for (ln z)-2. Examples are plotted in Figs. 1-5. From Figs. 1-5 the symmetry in the real and imaginary axis of both order stars

due to R(z) = R(1/z) and R having real coefficients is apparent. Also the reverse role that the poles and zeros play in each order star should be observed. Furthermore, the second order star has only 2(p -2) sectors adjoining the origin as compared with 2(p +2).

We point out that the idea of considering an inverse order star has already been used by Iserles and Norsett in their proof of the Dahlquist barrier [7]. As there, our function 0u2(Z) is essentially analytic [4], and therefore its geometric properties are described by suitable adaptation of the usual lemmas, which now follow without comment.

LEMMA 2.1 (Order). The method (1.2) is of order p if and only if p-2 sectors of A2 and p -2 sectors of D2 approach the origin each with asymptotic angle ii/(p -2).

LEMMA 2.2. Each pole of R(ez) of multiplicity m and each point of interpolation of R(ez) to Z2 order m + 4 is approached asymptotically by m sectors of A2 and D2 each of angle r/inm.

Consideration of o-(z) in the neighbourhood of the appropriate point proves the above lemmas.

LEMMA 2.3 (Essential singularity). There exists a number xo> 0 such that the sets {z E C: jRezj Z-xo, Im zl-},

(i) If R > r, are each composed of 2(R - r) + 1 distinct intervals of A2 and D2; (ii) If r > R, or r = R and bR/ar < O, each lie in D2; (iii) If r=R and bR/ar>0, each lie in A2. Note that by assumption bR 0 .-



SEMIDISCRETISATIONS FOR U,, = u, 329

LEMMA 2.4 (Monotonicity). The imaginary part of o-2 decreases strictly monotonically along any part of the positively oriented boundary of an A2-region, and it increases strictly monotonically along any part of the positively oriented boundary of a D2-region.

This follows by applying the Cauchy-Riemann equations (cf. [9, Lemma 2.5]). This lemma is of crucial importance as it dictates the location of poles of the

function 0u2(Z). It is instructive to consider the following sets as defined by

I:={zcC: JImzIV},

I+:={zc I: Re z>O},

I-:= {z E I: Re z < O}.

The order bound is dominated by the part of the order star in I (cf. [5]). Additionally, for a complete understanding of the order star we require "loops," a portion of a together with a part of the boundary of I, AI, is called a loop if it is an oriented bounded simple curve [7], [9].

The next two lemmas, essentially corollaries to Lemma 2.4, determine the relative position of the poles of o-2(z), all of which lie along a2. Further, they demonstrate that all sectors of A2 and D2 that reach the origin from outside I have poles on their boundaries, either within I or on R ? iir. The only exception is a D2 region bisected by the imaginary axis.

LEMMA 2.5 (Interlace). Singularities and points of interpolation of R(ez) to Z2 (zeros of o-2(z)) interlace along 02 (see [4, Prop. 8]).

This means that if y is a loop reaching the origin with y n ai = 0, then it has exactly one pole of Cr2 along its boundary unless there is a further zero on it. For loops including a portion of the boundary of I we have the following lemma.

LEMMA 2.6. Let y be a loop reaching the origin such that y n {R ? ir} ? 0 and set

-X <A_= min {x c R: x i ir E y},

A+= max{xc R: x+iiTe y}<oo.

Then if O<A <A+<oo, i.e, in I+, and y is a portion of the boundary of D2, positively oriented, then either A + iir is a pole of o-2(z) or there is a pole of 0-2(z) along y from A + iT to the origin. Similarly if y is a portion of aA2 then either A+ + iw is a pole of o-2(z) or there is a pole of a2(z) along the positively oriented boundary of y from the origin to A++ ir.

In I, -,o < A_<A+<O, the roles of A and A+ are exchanged. Proof. We observe that

2xir Im cr2(x + ir)= (. + ) x E R.

Also Im o?2 is zero at the origin and therefore the lemma follows immediately as a corollary to the monotonicity property. O

In our count of the maximal number of sectors of either A2 or D2, which may reach the origin, we require that a maximal number of poles of C2(z) are "efficient." A simple pole is considered efficient if we have the following:

(i) It lies on R ? ir; (ii) It belongs to loops all of which approach the origin; (iii) There are no extra poles along these loops.

Thus a loop lying inside I cannot have an efficient pole on its boundary since such a pole contributes to one sector reaching the origin, while an efficient pole, by periodicity,




contributes to two sectors there. Observe that a nonsimple pole in I may also contribute to more sectors reaching the origin but is not efficient.

Further, by Lemma 2.6, it is clear that in I' an efficient pole lies between an A2-region to the left and a D2-region to the right; this is reversed in 1.

We now observe that the continuity of the function R(-ex)-l away from its poles means that zeros and efficient poles interlace along R + ir so that if insufficient zeros lie on this line then not all poles may be efficient.

LEMMA 2.7 (Efficiency). The number of efficient poles Nin any interval (xl + i-r, x2 + iT), xI, x2>0 is bounded by

(2.6) N _ min {Z + 1, P},

where P and Z are the number of poles and least number of zeros of R(ez)-' along R + iir in the given interval, respectively. If X1 < 0 < X2 then

N min{Z,P} if ireD2,

N 'min {Z, P - 2} if i7r E A2 .

Proof The proof is the same as that for Lemma 2.6 [9]. a A consequence of this lemma, along with Lemma 2.6, is that any sector that

becomes unbounded completely outside I can be neglected in our count because it must have a pole on its boundary, which is then taken care of by Lemma 2.7. Furthermore, Lemma 2.6 means that the periodic repetitions of poles of o2(z) can be ignored.

The above lemmas give a sufficient description of the inverse order star and enable determination of the bound on p. For the moment we ignore the influence of stability and impose maximally efficient configurations when counting the number of sectors adjoining the origin. Further, although we already have the bound p _ 2(r+ R) for r ' R it is instructive to present the alternative proof using o-2(z).

THEOREM A. The order p of the rational function R(z) as an approximation to (ln z)2 at z = 1 is bounded by p 2(r + R).

Proof. We bound from above the number of sectors of A2 adjoining the origin. By the order property, Lemma 2.1, this is equal to p -2.

Let Q be the number of unbounded A2-regions in I, N the number of efficient poles lying along R + iir, and K the number of inefficient poles in the closure of I.

Then

(2.8) p-2_ Q+2N+K,

since each efficient pole contributes to two sectors and each inefficient pole to only one at the origin. Observe that the poles of o(2(z) are the zeros of R(ez) except for the double zero at z = 0. Therefore

K _2(r-1)-N.

In addition, R(ez)-l has at most 2R zeros along R+ ir so that by (2.7)

N_min{2R, 2(r-1)}

and by Lemma 2.3

Q?2(R-r)+2 forR>r,

=0 for R>r.



SEMIDISCRETISATIONS FOR u, = 331

Insertion of these values into (2.8) gives for R - r

p-2'2(R-r)+2+4(r-1)

=2(R + r)-2 and for R < r

p-2 0 + 4R + 2(r-1 )-2R

=2(R + r) - 2. Immediately we have p ' 2(R + r). 0

It is apparent from Lemma 2.3 that with R = r and bR/ar < 0 this proof gives

p ' 2(R + r) - 2.

This contradicts (2.3), the result of Iserles [5]. Consequently we must have bR/ar > 0 for diagonal Pade approximation.

Before investigating the stability of schemes achieving this bound, if any, the differences in Lemmas 2.1-2.7 for the order star o-i are briefly noted. The index sub2 is exchanged for subl throughout.

(1) Lemma 2.1: p - 2 exchanged for p + 2. (2) Lemmas 2.2, 2.4, 2.5: unchanged. (3) Lemma 2.3: for r? R the line segments lie in Dl, For r? R each segment is

composed of 2(r - R) + 1 distinct intervals of Al and Dl. (4) Lemma 2.6: Im o-,(x+ ir-) = -2xii means that efficient poles in I+(I-) lie

between an Al region to the right (left) and a D1 region to the left (right). (5) Lemma 2.7: For X1X2 > 0 N _ min {Z + 1, P}.

For x1<0<x2 N_ min (Z+2, P). This short summary concerning o-,(z) is useful in the following section and can

be confirmed by examining Figs. 1-5.

3. Pade schemes and stability. In this section we prove the stability of those difference schemes that achieve the maximal order given by Theorem A. These schemes of maximal order are called Pade schemes, and their coefficients are derived from the appropriate Pade approximation to z (ln z)2, L = r - R.

In the investigation of the conservation equation, u, = u, ([9], [10]), a bias to Pade schemes that use the much-favoured idea of upwinding was demonstrated. Here our schemes have already been designed to be symmetric, which supports the equal flow of information in both directions for the wave equation. Thus we could only expect to demonstrate a bias towards implicit or explicit schemes. In fact we prove that all schemes are stable. This is encouraging because it implies no advantage in using implicit schemes over explicit schemes for high accuracy; the former of course are more complicated computationally. On the other hand, the solution of the explicit system of equations might yield a severely restricted timestep, thus increasing its cost.

The explicit schemes can be derived without any problem. We set z= e' in the order condition (1.4):

r

R(z) = aO+ Y aj(zi+ z-i) = (ln Z)2+ C(Z _1)2r?2+ O(jz-_ 12r+4) j=l

Differentiating repeatedly and setting z = 1, we obtain the system of equations r

(3.1) Y ajj 2= k , 1<k r j=1

with ao = -2 j> ajQ.




THEOREM B. The explicit schemes of maximal orderfor the solution of utt = usx are given by

aj=4( 1) 1( -j)_(r+j)!j21

rl

aO- 2 2 j=IJ

Proof. Observe that the system of equations (3.1) is Vandermonde (cf. Golub [2]). 0

An immediate corollary of the following analysis is that these schemes are stable. We now return to the more general case: r ' 1, R > 0. Section 2 stated that a count

of the free coefficients available together with R(z) = R(1/z) implies the lower bound

p _2(r+R)

for Pade schemes. Combining this with Theorem A shows that the Pade schemes under consideration belong to 2 x 2 blocks in the Pade table and have order 2(r+ R).

Returning to our investigation of stability, we remind the reader of the von Neumann stability condition and give it in a more directly applicable form.

LEMMA 3.1. An invertible scheme is von Neumann stable if and only if the approximation R (z) has no zeros of odd order on the unit circle.

Proof. Recall from (1.7) the von Neumann stability condition

Re R(e6) 0, -v '-0 < v.

Note also the order condition

R (z) = (In z)2+ C(Z_l )p+2+ ?((|Z-_llp+4))

where the trailing term is of order p+4 since R(z) = R(1/z). Thus

R(e i) = C02? C(_ )(p+2)12op+2+ o(0)P+4,

and R (e < 0 for small 0. Obviously R(e'6) only changes sign at a pole or zero of odd order, and the former

is excluded by invertibility. Therefore an invertible scheme that is von Neumann stable has no zeros of odd order on the unit circle. The proof in the opposite direction is immediate. 0

We can interpret this lemma for both order stars. COROLLARY 3.1 (Inverse order star). The SD scheme (1.2) is stable and invertible

if and only if we have thefollowing: (i) There exist no zeros of 1/R(ez) on the imaginary axis. (ii) No poles of o-2(z) on the imaginary axis have odd order. COROLLARY 3.2 (Order star). The SD scheme (1.2) is stable and invertible if and

only if we have thefollowing: (i) ou1(z) is analytic on the imaginary axis. (ii) No zeros of R (ez) on the imaginary axis are odd. Consequently, our proof of stability requires the confirmation of Corollaries 3.1

and 3.2 for the maximal order schemes. Thus we should investigate the character of the order stars along the imaginary axis. By symmetry it is clear that this axis can only contain points of a and no portion of it. For efficiency these points must correspond to poles of the order star.



SEMIDISCRETISATIONS FOR u,, = u,. 333

In our proof of Theorem A the efficient configurations were not considered in detail; the mere possibility of their existence is sufficient for the order bound. Efficiency in each order star leads to the following lemmas.

LEMMA 3.2 (Order star). (a) For r '-R the Pade' schemes are invertible, and at least 2(R - 1) zeros R(z) lie on the negative real axis.

(b) For R > r all the zeros and at least 2r poles of R(z) lie on the negative real axis. Proof. We look at the number of sectors of A1 adjoining the origin. For a Pade

scheme this is 2(r + R + 1) by the order condition. As for the inverse order star, Lemma 2.6 implies that there can be no contributions from outside I without a pole lying on the boundary of the region either on R + ir- or inside L The only exception is an Al-region adjoining the origin and being bisected by the imaginary axis. Further, all periodic repetititions of poles can be ignored.

Let Q be the number of unbounded regions adjoining the origin, N the number of efficient poles, and K the number of inefficient poles. Then as in (2.8) in Theorem A,

p + 2 Q + N + K + 2.

This time the additional two sectors come from the possible exceptional A, region, which is bisected by the imaginary axis.

(a) r' R. Clearly Q_2(r-R)+2. However, if an unbounded Al-region lies on either side of R + i-n, then by Lemma 2.6 it has an inefficient pole lying on its boundary. As r ' R there are sufficient zeros for efficiency of all poles, and thus

Q+K -2(r-R)+2R- N=2r-N.

Hence

2(r+ R + 1) _ 2r- N+2N+2 = 2r+ N+2,

and N = 2R is the only solution. Further if an efficient pole were to lie at ir- then it would be a double pole because

R(z) = R(1/z). But consistency (1.5) then implies that the imaginary axis bisects a DI region and

so the additional two sectors from the "free" Al region are lost. Therefore the scheme is invertible. Furthermore, continuity of R(-ex) away from

the poles implies at least 2(R - 1) zeros of R(ez) lie on R+ ii.

(b) R > r. By Lemma 2.3 Q = 0, and by Lemma 2.7 not all poles may be efficient:

N _ min - {Z + 2, P}.

Thus

2(r+ R + 1) = p +2 ' 2N+2R - N+2- r =N+2R+2

implies N = 2r.

Consequently, Z+2=P=2r

and part (b) follows. 0 Note in passing that part (a) immediately proves stability and invertibility for

r = R. In the nondiagonal cases there are either free poles or free zeros available, which may destroy analyticity or stability, respectively.




LEMMA 3.3 (Inverse Order Star). For r > R the Pade schemes are invertible and at least 2R zeros at R(z) lie on the negative real axis.

Proof. We proceed as in Lemma 3.2 with Q, N, and K as in Theorem A. From Lemma 2.3(ii) there are no unbounded regions reaching the origin; hence Q = 0. By Lemma 2.7, N is maximal for an imaginary axis bisecting a D,-region. Therefore

N'min {Z, P}.

Now

p-2=2(r+R+1) _22N+2(r-1)-N

- N+2(r- 1)

implies

N =2R,

and we must have a D,-region bisected by the imaginary axis. Consequently, there is no zero of 1/R(ez) at ir- and

Z=P=2R.

Observe that for r > R part (a) of Lemma 3.2 has been refined, and an additional two zeros of R(z) lie on the negative real axis.

LEMMA 3.4. There is at most one pole of 0>2(z) along the imaginary axis inside I. Proof. We assume the opposite and obtain a contradiction. Periodicity implies that all poles are repeated with period 2irii, and thus by

symmetry the existence of poles at i01, i02, 0< 01 < 02< 'IT implies poles at ((2iT- -1)i, (2 T - 02) i, Xi < 2ir - 02 < 2ir- - 01 _ 2Xi-. We only consider the strip of C with 0 < Im z < 2ii, with the argument there being sufficient for the lemma.

Assume two poles exist at i01, i02; then the following situations in Fig. 6 are possible.

Note of course that more complicated situations are possible, but they will also be refuted by our argument. Further, the opposite shadings cannot occur as the order condition (1.5) implies that iOl must be a pole with 1/R(e0i1) tending to -00. Also, as 01 and 02 are simple, the segments of the imaginary axis with 01 < 0 < 02, 2ir- - 02 < 0 < 2ir - 01 must be in A2-

For the situation (a) to occur there are either poles at A_ and A+, or along aA2 between the origin and A and A+, by Lemma 2.6. But then Lemma 2.5 requires a zero of o-2(z) to lie between the additional pole and that at 2Xi- - 01. Similarly in (c) there must be a zero on the boundary of the "blob," one on either side of the imaginary axis.

Alternatively (b) occurs, but then the pole at 2irri has multiplicity greater than two. This is inconsistent with the original approximation, which has a zero of order 2 at the origin.

However if (a) or (c) occurs, then the periodicity of the poles implies that we must have zeros in every strip of width 2irri. These zeros are not periodic;

R(ez+ -i)-(z+21ri)2= R(ez) z2+(4 r2_-41riz)

so that R(z) must interpolate (ln Z)2 infinitely often. Again this is not possible. 0 Applying Lemmas 3.2-3.4 proves our main theorem. THEOREM C. All Pade schemes for u,, = u,, are stable and invertible. Proof. We consider five cases. Although (b) and (c) are also proved by (e), this

is instructive as (b) is immediate and (c) explains the later corollary. (a) r = R: immediate from Lemma 3.2.



SEMIDISCRETISATIONS FOR u,, = ux. 335

0 40 tl_j

~,i q

l0




(b) r = R + 1: immediate from Lemma 3.3. (c) r = R + 2: clearly by Lemma 3.3 there are just two free poles in OU2 that must

lie either on the imaginary or real axis. Consider a pole at ifl, 0 < 01 < Vr. Obviously in oa it is a zero and lies in A1. All poles in ol are efficient, and so this zero lies in an A1-region that neither can surround a D1-region nor be bounded within I, and consequently reaches both the origin and R ? iGr. Thus R(e'o) + 02> 0 near 0 = 0, and by the order condition

R(e'0) = _02+ C(_l)r+R+1o2(r+R+l)+ Q(0)2(r+R+2)

we have C(_l)r+R+l > 0. Thus C < 0, and again by the order condition for small real x we must have x0 E A2 for sufficiently small real xo. But in Oa2 there are no unbounded A2-regions in I for r> R. Hence this xo lies in a bounded A2-region that has a pole on its boundary. However, there are no further available poles.

Consequently, no pole lies at iW1, and the remaining poles lie on the real axis. (d) R > r: By Lemma 3.2(b) all zeros of R(z) are negative real. Therefore if R(z)

is analytic for z = e"' it is also von Neumann stable. Assume R(z) has a pole on the unit circle; then in OU2 it is a zero that lies on the imaginary axis in A2. As in (c) the nonavailability of poles of Cr2 means that this A2-region adjoins both the origin and reaches R ? iir. Therefore by Lemma 2.6 it has 2 poles on its boundary, and by Lemma 2.7 order cannot be achieved. Thus R(z) is analytic; z = e"'.

(e) r> R: R(z) is invertible by Lemma 3.2(a). Further, by Lemma 3.4, no 2 poles Of o-2(Z) can lie on z = iG, 0 > 0. If a single pole lies on iO in o,2(z), then it lies on the boundary of a D2-region adjoining the origin. This region is surrounded by an A2-region reaching R + iT, which by Lemma 2.6 has poles at, at best, A_ and A,. The pole on iO is therefore completely redundant, contributing to no A2-sectors reaching the origin, and order is not achieved. Therefore R(z) has no zeros of odd order on the unit circle. 0

We observe that a further consequence of efficiency is that the imaginary axis always lies in D2(A1). The order condition then implies the sign of the error constant.

COROLLARY 3.3. The error constant C of the Pade scheme satisfies

(-l )r+ R +1C > fs.

Acknowledgments. I would very much like to thank Professor R. Jeltsch for the original suggestion that led to this paper and for his continual encouragement during this investigation. Also I am extremely grateful for many helpful communications from and discussions with Dr. A. Iserles.

REFERENCES

[1] G. A. BAKER, Essentials of Pade Approximants, Academic Press, New York, 1975. [2] G. H. GOLUB AND G. F. VAN LOAN, Matrix Computations, The Johns Hopkins University Press,

Baltimore, MD, 1983. [3] A. ISERLES, Order stars and a saturation theorem for first order hyperbolics, IMA J. Numer. Analysis,

2 (1981), pp. 49-61. [4] , Order stars, approximations and finite differences 1, The general theory of order stars, SIAM J.

Math. Anal., 16 (1985), pp. 559-576. [5] , Order stars and stability barriers, in Proc. Conference on Numerical Analysis, Dundee 1985,

D. F. Griffiths and G. A. Watson, eds., Longman Scientific, London, 1986. [6] , Order stars, approximations and finite differences III. Finite differences for u, = wu'X, SIAM J.

Math. Anal., 16 (1985), pp. 1020-1033. [7] A. ISERLES AND S. P. N0RSETT, A proof of the first Dahlquist barrier by order stars, BIT, 24 (1984),

pp. 529-537.



SEMIDISCRETISATIONS FOR u,, = u, 337

[8] A. ISERLES AND G. STRANG, The optimal accuracy of difference schemes, Trans. Amer. Math. Soc., 277 (1983), pp. 779-803.

[9] A. ISERLES AND R. A. WILLIAMSON, Stability and accuracy of semi-discretisedfinite difference methods, IMA J. Numer. Anal., 4 (1984), pp. 289-307.

[10] R. JELTSCH AND K. G. STRACK, Accuracy boundsfor semi-discretisations of hyperbolicproblems, Math. Comp., 45 (1985), pp. 365-376.

[11] G. WANNER, E. HAIRER, AND S. P. N0RSETT, Order stars and stability theorems, BIT, 18 (1978), pp. 475-489.

[12] R. A. WILLIAMSON, Pade approximations in the numerical solution of hyperbolic differential equations, in Pade Approximations and Its Applications, Bad Honeff, 1983, H. Werner and H. J. Bunger, eds., Springer-Verlag, Berlin, New York, 1984.

[13] R. A. RENAUT-WILLIAMSON, Full discretisations of u, = u,x. and rational approximations to cosh ,uz, SIAM J. Numer. Anal., this issue (1989), pp. 338-347.



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Semidiscretisations of u tt = u xx and Rational ...rosie/mypapers/2157834.pdf · Williamson, [8]-[10], [12] (see [6] for a full list of references). In this paper we begin an investigation