Table Of ContentLecture Notes ni
Mathematics
A collection of informal reports and seminars
Edited yb .A Dold, Heidelberg and .B Eckmann, Z0rich
109
Conference
no the Numerical Solution
of Differential Equations
Held ni Dundee/Scotland, 23-2?, June 1969
Edited by
.J .IL Morris, University of Dundee, Dundee/Scotland
Springer-Verlag
Berlin-Heidelberg • New York 1969
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© by Springer-Verlag Berlin Heidelberg .9691 Library of Congress Catalog Card Number 77-101372 Printed in Germany.
Title No. 5623
Contents
J. Albrecht: Generalisation of an Inclusion Theorem of L.Collatz I
E.G. DtJakonov: On Certain Iterative Methods for Solving Nonlinear
Difference Equations ............................ 7
B. Noble: Instability when Solving Volterra Integral Equations of
the Second Kind by Multistep Methods .................. 23
M. Urabe: Numerical Solution of Boundary Value Problems in
Chebyshev Series ...................................... 40
E. Vitasek: The Numerical Stability in Solution of Differential
Equations ........................................... 87
O. Widlund: On the Effects of Scaling of the Peaceman-Rachford
Method .............................................. 113
J.C° Butcher: The Effective Order of Runge-Kutta Methods ........ 133
G.J. Cooper: Error Bounds for Some Single Step Methods .......... 140
O. Dahl: Approximation of Nonlinesr Operators ................... 148
K. Graf Finck v. Finckenstein: On the Numerical Treatment of Hyper-
bolic Differential Equations with Constant Coefficients .. 154
R. Gorenflo: Monotonic Difference Schemes for Weakly Coupled Systems
of Parabolic Differential Eouations ................ 160
A.R° Gourlay: The Numerical Solution of Evolutionary Partisl
Differential Equations ............................ 168
W.R. Hcdgkins: A Method for the Numerical Integration of Non-Linear
Ordinary Differential Equations .................. 472
M°Lal and P.Gillard: Numerical Solution of Two Differential-
Difference Equations of Analytic Theory of Numbers ........ 179
W.Liniger: Global Accuracy and A-Stability of One- and Two-Step
Integration Formulae ................................. 188
T. Lyche: Optimal Order Multistep Methods with an Arbitrary
Number of Nonsteppoints ............................... 194
S. McKee: Alternating Direction Methods for Parabolic Equations in
Two and Three Space Dimensions ........................ 200
K.O.Mead and L.M.Delves: On the Convergence Rates of Variational
Methods ................................................ 207
S.P. Norsett: An A-Stable Modification of the Adams-Bashforth
Methods .......................................... 214
P. Piotrowski: Stability, Consistency and Convergence of Variable
K-Step Methods ................................... 221
A. Prothero: Local-Error Estimates for Variable-Step Runge-Kutta
Methods ............................................ 228
IV
E.L. Rubin: Time-Dependent Techniques for the Solution of Viscous,
Heat Conducting, Chemically Reacting, Radiating
Discontinuous Flows ................................. 2~
J. Skappel: Attempts to Optimize the Structure of an OdE Program 243
M.N. Spijker: Round-off Error in the Numerical Solution of Second
Order Differential Equations ...................... 249
H.J. Stetter: Stability Properties of the Extrapolation Method .. 255
J.H. Verner: Implicit Methods for Implicit Differential Equations 261
W. Werner: Solution of Elliptic Eigenvalue Problems by Calculating
a "Separable" Solution of a Dynamic Problem ......... 267
List of Contributors
Invited Papers
Albrecht, J., Technische Universitit, 1000 Berlin / Germany
D'Jakonov, E.G., University of Moscow, Moscow / USSR
Noble, B., University of Wisconsin, Madison, ~I / USA
Urabe, M., University of-Kyoto, Kyoto /Japan
Vitasek, E., Ceskoslovenska Akademie Ved, Matematick~ Ustsb,
Prague I / Czechoslovakia
Widlund, 0., Courant Institute of Mathematical Sciences, University
of New York, New York / USA
Submitted Papers
Butcher, J.C., Dept. of Mathematics, University of Auckland,
Auckland / New Zealend
Cooper, G.J., Dept. of Computer Science, University of Edinburgh,
Edinburgh / Scotlsnd
Dahl, O., Dept. of Mathematics, University of Oslo, B]in~rn,
0slo 3 / Norway
Finck .v Finckenstein, K. Graf~ Institut fir Plasmaphysik GmbH,
8046 Garching 6 / Germany
Gorenflo, R., Institut riff Plasmaphysik GmbH, 8046 Garching / 6 Germany
Gourlay, A.R., Dept. of Mathematics, University of Dun4ee, Dundee ~eotland
Hodgkins, W.R., The English Electric Co.Ltd., Nelson Research
Laboratories. Beaconhill, Stafford
Lal, M. and Gillard, P., Dept. of Mathematics, Memorial University of
Newfoundland, St. John's, Newfoundland / Canada
Liniger, W., IBM Watson Research Center, Yorktown Heights, N.Y. / US~
Lyche, T.~ Dept. of Methematics, University of Oslo, Blindern,
Oslo / 3 Norway
McKee, S., Dept. of Mathematics~ University of Dundee~Dundee / Scotland
Mead, K.O. sod Delves, L.M., School of Mathematical and Physical Sciences,
University of Sussex, Brighton, Sussex / England
Norsett, S.P., Dept. of Mathematics, University of 0slo, Blindern,
Oslo 3 / Norway
Piotrowski, P., Institut fir Plasmaphysik GmbH, 80~6 Garching 6 /Germany
Prothero, A., Shell Research Ltd.,Thornton Research Centre,Chester/England
Rubin, E., Dent. of Aerospace Engineering and Applied Mechanics. Poly-
technic Institute of Brooklyn, Farmingda±e, N.Y. / USA
VI
Skappel, ,.J Mathematical Analysis Unit, The Boeing Company,
Commercial Airplane Division, Seattle, NA / USA
Spljker, M.N., Centraal Reken-Institut, Rijksuniverslteit te Leiden,
Leiden / Netherlands
Stetter, H., Institut fur Numerische Mathematik, Technische Hochschule,
Wien / Austria |
Verner, J.H., Dept. of Computer Science, Edinburgh 8 / Scotland
Werner, N., Mathematisches Institut der Technischen Hochschule,
8000 MGnchen 12 / Germany
-I-
Generalisation ' of an Inclusion Theorem of L.COLLATZ
J. Albrecht
For the eigenvalues of a selfad~olnt, definite eigenvalue problem with a
differential equation
m
[ Mu(=) )1-( ~ ))~c)~/,)x/,p(( ;)~(
~=o
L ))x().(u)=(%( ;)
Nu(x) = , (-i) ~ (v m > n >~ 0 ]
L
O=M
and with m (geometric or dynamic) boundary conditions I) at each of the two
boundary points x = o, x = b
)a()P(u o r _ _ o )
=
)a(o~ u )~()P( - )aCuoM = )a(lOCulaCoh(X - ~u(~) ) I
)
u (p)(b)=o r__o ) (p = ,O ...,m-l)
~pCblu (p) (h) . ~u(h) = X(hoCblu(-Pl(h) . ~uCh) )
m
MpuCx) =+P~'.~= 1 (-1)~-(P+l) (P~(x)u(W)(x)) (W-(p+I)) for p = O, ..., m-1
n )1÷p(-v)1-( q( ))x()~(u)x( ))1+p(-~(
for p = 0,.o.,n-i
5u(~) ~ = -_ i+
for p = n,...,m-i
o
the following inclusion theorem holds:
)l
)~(ph = o I
for p = n + i, ..., n - i
)b(ph = o )
.I ~finitions
I.i. Inner pro~cts )2 and RAYLEI@H'S quotient
]~,~[
= p.(x)~(")(~) ~(")(x) ~a +
~=0
-2-
m-I m-I
Z g"(a)~(")(a) ~(") )a( + Z g"(b)~(")(b) ~(") )b(
//=0 ~=0
b n
b,¢t =/~ %(x)~(~)(x) $ (~) (x) ~ .
M=O
h~(b), )~( )b( ~ )b( )~(
=O
R (~) = ¢,~
I.i. Iteration )~
Mvl(x) : NVo(X) in a-~ X ~ b
)~()P( (P)(~) o
vl = ,~ o =
or gp(a)Vl(P)(m) - ) MpVl(a = hp(a)Vo(P)(a) - Nvo(a )
, &=0,...,m-l)
h(~)(b) (P)(b) -_ o i
= V 0
' ' )b()':( ~ N~o(b)
f~or 1,1 -= 0p eoep D.
)2 Asioms of inner products: .i (~,@) = (~,~)
.2 (~,,) >I o ~ for
= 0 for ~ = e
.3 (~*,¢) = ~(~,~)
.: Restriction o~ one step without loss of generality
9~ Assumutions
2.1. %(x) > o ina.<x.<b
for 7; -- 0 9 eoo~ .21
~(~) ~ o, ~(b) >. o
2.2. o < ¢~v(x) < ina~x~.b for ~, -- 0 9 eJej .31
-3-
.5 Theorem
Min ~(x))~x~ xaM ( ~,= %(~))
M~ ( s
,O
V = ...p n a~x.<b v=0,...,n a-<x~b
Adjusting to special cases the first assumption may be simplified in several
ways. For example, the Inclusion Theorem of L. COLLATE for the "single-term"
class holds [i]:
2.* snoitpmJuss~
( %(:~) - o ina~x~b for v = 0, ..., n-i
I ),(,~,.~ -_ o, h,,(b)
~.1,2 ( = 0
( %(~)
> 0 Ins.< x~b
(
( i~(~) ~ o, h(b)
0
2.2* 0 < • (x) < ina~x~b
n
3.* ,~sore~
Min ~ (x) ~ x ~=~ )x(
n s n
b<.x~a b~x<.a
~. Proof 4)
With th@ abbreviations
Min ( ~Ln *v(x) ) Max ~= %(x) )
L =
I = v=O,...~n b<~x~a v=O,...,n ( a~x~b
~(x) ~ %(x) . %(x) ina=x.<b !
) for v=O,...,n
hv(a) .= hv(a) . @v(a) , hv(b) .= hu(b)'*w(b)
2
etc.
m%d with the estimations
1[¢,¢I <- [¢,¢I ~ ,T I~,¢[
4 -
we apply the Comparison Theorem (which is derived from themin~-~! properties 5)
of the RAYLEIGH quotient) to the following Eigenvalue problems
~) ., llI
cou~
)5
h. = R(h) : Min R(v) ; 2 : k R(U2) = Min R(v) ; . . .
veV vcV
vlu 1
V: Set of the admissible functions (satisfying the essential boundary
conditions)
~i(x) = )~(iu~.~!lx fma.<x~b
l )
I u '(a) : 0 or
i p( = ,o )i-=,...
I )
uI(P )(b) = 0 r_r_o )
kll (hp(b)ul(P)(b) + ~ui(b))
( ,~
y<
(~igenvalues ~ = kl, -~ , eigerd'unctiens ui,~(x ) (~ = 1,2,...)
I
Muii(x) = Xllf~%ll(x) Ina~x~b
Ul! (P)(~) : o or
)p(
: Xii(~o(a)u!i (p)(a) - f~uzi(a))
(p=O,...,m-1
: ~i(Zp(b EII~) 'p( )b(J + }~z(b))
[g,(b)uii (0)(b) + ~Uil(b )
eige~otion8 ~i,~(~) = h(x)
