TH(s n u m e r i c a l s o l u t i o n o f p a r t i a l d i f f e r e n t i a l EQUATION WITH THE TAU METHOD
by
K.S. PUN, BSc, ARCS October 1984.
A thesis submitted for the degree of doctor of Philosophy
of the University of London and for the Diploma of Membership of Imperial College.
Mathematics Department, Imperial College, London. S.W.7.
L o v M y i n n g M e m o r y o f M o t h e r *
ACKNOWLEDGEMENTS
I would like to express my deep gratitude to my supervisor
Dr. Eduarado L. Ortiz whose encouragement, guidance, and expert advice had lead me through many difficult areas in my research.
I also owe much to Prof. H. Samara, who has unselfishly helped me through the initial stages of my work.
All my friends and colleagues in the Mathematics Department, to whom I
owe so much, will be remembered deeply. I am grateful to
Drs. P. Onumanyi, K.M. Liu, A.M.E. El Misiery, N. Lambrou,
S. Namasivayam, and J. Loines for many inspiring discussions. A special mention for Dr. T.W. Ng for reading part of this thesis and making a number of useful suggestions.
I also wish to thank BP Exploration Company Limited, for my valuable
three years seconded to Imperial College of Science and Technology. I
am particularly indebted to Dr. G. Rowen, Consultant to BP International
PLC, who had made this possible. My gratitude extends to all my
colleagues in BP who, at one time or another, had given me some of their
valuable time. I am grateful to Dr. A.K. Parrott, Senior Reservoir
Engineer, who has sacrificed many hours of his precious time, reviewing and discussing my research. I am also thankful to Dr. J.H. Divall, Head of Development, for his many encouragements. A word of thanks also goes to Dr. J.W. Buckee, Manager Reservoir Engineering, who despite of his busy schedule had taken time to see me on matters related to my work. To all those concerned in BP, particularly, Dr. A.K. Parrott and Ms
Annette Parascandolo; I am thankful for their help in arranging my
and Leuven (1984). The financial support of BP Exploration Company Limited to attend these meetings is also gratefully acknowledged.
I am also most indebted to my father, whose love and encouragements have been a constant source of moral support in my academic effort.
My sincere gratitude extends also to my siblings, their kindness and
understanding has helped me in many ways. However, for my own
indulgence, I have often neglected their welfare and happiness.
I am thankful to my most beloved friend, Miss Shirley S.L. Cheng, without her love, care, and understanding, I could never have finished
this thesis.
Last but not least, I acknowledge my thanks to Prof. N. Papamichael, Brunei Univ., for sending me his unpublished results on a singular BVP
in Chapter Five. Finally I would like to thank Miss Julie D. Suett of
the Word Processing Centre in BP, for her careful and excellent typing of this thesis.
Notations a a row vector X a column vector T X Transpose of x. tA,ij A'1
i - j th element of the matrix A inverse of A
Tm<*> Chebyshev polynomial degree m
T computing time in cps
MAE Maximum absolute error
even (odd ) indicates an even (odd) function with respect to the
interchange of x and y.
J L the class of linear differential operator with
variable polynomial coefficients
D a domain in R x R
9D the boundary of D
V order of ODE
A Laplacian operator
A2 Biharmonic operator.
Un normal derivative of U = U (x,y)
m,n degree in x, degree in y of the tau approximation.
M,N equals m + 1, n + 1 respectively
Nseg Number of segments
Tstep step length
tau approximation (unless otherwise stated) Chebyshev tau
approximation.
NBI510H-II:198
4
The numerical treatment of Partial differential equation with the Tau method will be considered: a brief discussion on ODE serves as a convenient platform to introduce the subject.
This work is mostly computational and emphasizes the practical and application aspects of the operational approach to the Tau method as proposed by Ortiz and Samara (1984).
A wide range of problems from PDE were solved successfully. Not only is the viability of this method established but also its potential as a practical tool in scientific and engineering applications..
The problems considered are: biharmonic equation, partial differential eigenvalue problems, boundary singularities in Laplace’s equation in rectangular and L-shape domains, some examples of nonlinear partial differential equations, and also the well researched Burgers' equation where the special techniques of segmentation and step-by-step are used. The accuracy obtained in a number of problems is quite remarkable in comparison with numerical results reported in the current and recent literature.
Publications
(1) K.M. Liu, E.L. Ortiz, K.S. Pun, Numerical solution of Steklov's partial differential eigenvalue problems with the Tau method, the third international conference in computional and asymptotic methods for boundary and interior layers, J.J.H. Miller ed., Boole Press, Dublin, pp 244-249, June 1984.
(2) E.L. Ortiz and K.S. Pun, Numerical solution of nonlinear partial
differential equation with the Tau method. Accepted for
publication in the proceedings of international congress on computational and applied mathematics, Leuven, July 1984 (to appear in March 1985 issue).
Chapter One INTRODUCTION
Chapter Two OPERATIONAL APPROACH TO THE TAU METHOD
2.1 Linear and nonlinear ODEs
2.2 A discription of the software
2.3 Extension to PDEs
2.4 Examples
Chapter Three : DIRECT SOLUTION OF BIHARMONIC PROBLEMS
3.1 Introduction
3.2 Special Trial Function: Symmetry
3.3 Boundary Conditions
3.4 Numerical Results
Chapter Four PARTIAL DIFFERENTIAL EIGENVALUE PROBLEMS
4.1 Introduction
4.2 Eigenvalue Problems
4.3 Numerical
Chapter Five BOUNDARY SINGULARITIES IN THE NUMERICAL SOLUTION OF
LAPLACE'S EQUATION
3.1 Introduction
5.3
Re-entrant Corners in L-shape domains
Singularities: Motz's Problem
5.4
Chapter Six : NONLINEAR PDEs
6.1 Preliminaries
6.2 Treatment of Nonlinearity
6.3 Nonhomogeneous Differential Equations
6.4 Numerical Examples
6.5 Burgers’ Equation
R e h 6 £ e N C £ S
A, B, C
ppeNT>!X
CHAPTER ONE INTRODUCTION
Chebyshev (Tshebysheff) polynomials, named after the great Russian
mathematician were known for approximately a century. But it was not
until 1938 when Lanczos proposed the Tau method and discussed the applications of Chebyshev polynomials in approximating solution of
linear differential equations. Since the Tau method is directly
applicable only to differential equation with polynomial coefficients, Lanczos (1938, 57) proposed a generalization of the Tau method which he called the "method of selected points". Later, the names of "Chebyshev Collocation", "Orthogonal Collocation”, and "pseudo-Spectral method" were also used (see respectively Wright 1964, Finlayson 1972, Orszag
1971).
In Clenshaw's method, the solution of a linear differential equation is assumed to be an infinite Chebyshev series with free coefficients
00
b- (i.e. y b. T.(x)). Such a series is then substituted into linear
i . _ l l
i=0
differential equation, and by performing somewhat laborious algebraic manipulations, an infinite system of linear equations in terms of the
adjustible coefficients b^ is obtained. For a finite Chebyshev series
approximation therefore, this infinite system is truncated, giving a finite one for determining the coefficients of the approximate Chebyshev solution of the differential equation.
Alternatively, in the Lanczos' Tau method, one first assumes the approximate solution of the differential equation to be a finite polynomial (say aQ + a^x + . . . + am xm , for some m), then substitute in to the differential equation giving an 'overdetermined' system of linear
1-2
equations for the m + 1 unknown coefficients. A pertipation term is ( r
added to the right hand side of the differential equation so that it has
an exact polynominal solution. In order to minimise such a pectination, jr
a factor or a linear combination of Chebyshev polynomials with free coefficients are used. Furthermore, the degree of overdetermination can be easily established in advance by simple enumeration, the required
number of x terms are introduced to make the linear system
"determinate” , namely one which uniquely determine the unknowns
aQ, a ^ . . ., am , x^, x^, . . . etc. E.L. Ortiz has studied the Tau method extensively and contributed much to the understanding of this
method in the literature. Recently Ortiz and Samara (1981) have
proposed an operational approach to the Tau method. They discussed this new approach in Chebyshev series, although other orthogonal polynomial basis could be used with equal ease.
This approach to the Tau method is an interesting one, since it enables the tau approximations to be obtained in an algorithmic manner which is
particularly suited to be programmed onto a computer. Furthermore when
other polynomial basis is used instead of Chebyshev, the algorithm remains unchanged and the computer program requires no alterations except one parameter which selects the orthogonal polynomial basis the user has requested (see 2.2).
The computational simplicity achieved in the operational approach to the Tau method enables this technique to be extended for the approximate solution of PDE with little difficulty (see Ortiz and Samar^ 1984), and in fact this will receive^' most of our attention.
From the early 1970's onwards, Orszag produced a series of papers with frequent reference to the so-called Spectral and Pseudospectral
methods. These nomenclatures have their origin in the Meteorology and Fluid Dynamics literature where approximation of the flow equations are obtained by Galerkin's method using truncated Fourier series or surface harmonic series, see Orszag (1971).
Later, Gottlieb and Orszag (1977) published a monograph entitled,
"Numerical Analysis of Spectral Method/: Theory and Applications.",
where the meaning of spectral method is one which approximate the solution of a differential equation in terms of a series of known smooth
functions. Such a wide concept covers a great deal, in fact all the
above mentioned methods are in this category. In practice, however,
truncated Fourier series, or orthogonal polynomial expansions are
frequently used. Furthermore the pseudospectral method is basically
same as the Lanczos' method of selected points applied to these series (see Orszag 1971, Gottlieb and Orszag 1977).
Although there is no unified definition and theory of Chebyshev methods
in the literature^ Mason (1970) attempts to clarify and extend the
theory of orthogonal polynomial approximation in the L^, L
2
, Lw ,norms. He defines two types of approximations: (S) and (I). The
former is defined by some unique series expansion criteria, whilst the
latter is defined by some unique interpolation criteria. These two
types of approximation in fact correspond to the spectral and pseudospectral technique respectively.
The operational approach to the Tau method is based on the innovative
use of a pair of very simple and sparse matrices. These can be
ultilized systematically to represent differentiations and products with respect to the independent variables, thereby, simplifies considerably the process of obtaining a tau approximation.
1-4
More recently, Ortiz and Samara (1984) has extended the idea in Ortiz &
Samara (1981) to the numerical solution of PDEs. However, a more
comprehensive numerical experiment is required to assess its usefulness in application. Indeed, it is one of the aims of this thesis to provide such needed experiments, and more importantly to develop this technique further into a useful tool in solving linear and non-linear PDEs.
Encouraging and interesting results were obtained. Particularly, the
accuracy and the ease of which they are obtained is quite remarkable.
This is a beginning rather than an end. Having established the said
technique as a viable approach to the numerical solution of PDEs, and considering its close connection with other spectral techniques, we could draw upon past theory and analysis (from previous authors) and to
take advantage of the appeal/ing simplicity and computational
convenience proposed in the operational approach to the Tau method to treat boldly a wider class of problems in PDEs.
12
OPERATIONAL APPROACH TO THE TAU METHOD
An operational approach to the Tau Method
2.1 Linear and nonlinear ODEs
2.2 Software: ODETAU
2.3 Extension to PDEs
2-1
A very interesting approach to approximate solution of differential equation with the Tau method was proposed by Ortiz and Samara (1980, 81,
84). It is referred to as 'operational' by the original authors in the
sense that it makes it possible to transform a given differential problem into an algebraic one.
The approach to the Tau method can achieve simplification of the usual laborious algebraic manipulation associated with series replacement methods. As a result, the Tau method can be extended to the solution of partial differential equations7 ^Without the aid of any discretization in time or in space variables.
We first discuss the operational approach to the Tau method for
numerical solution of ODE. We restrict our attention to the basic
concepts, and some notations will be introduced, more details can be
found in the original paper of Ortiz and Samara (1981). We then show
how it can be extended to the numerical treatment of PDEs. The
notations in this thesis will follow closely to that of Ortiz and Samara (1981).
2.1 Linear and nonlinear ODEs
Let JL be the class of linear differential operators with
polynomial coefficients. This assumption is not too
'restrictive', because nonlinear problems can be approximated by a
that non-polynomial coefficients can be approximated conveniently
by polynomials. In some cases, the approximate polynomial can be
obtained by using the Tau method itself, or alternatively other
sequence of problems defined and we note
methods such as interpolating at Chebyshev zeros or extrema can be used. These aspects will be discussed further in Chapter Six.
Let L zJi be written as:
V i cd I r
L = l P (x) ^ , Pi(x) = l P XJ/ (2.1) X
i=0 dx j=0
where v is the order of L, and oti is the degree of the polynomial coefficients P^Cx).
oo
Consider now a power series cj>(x) = \ a . X1 . We can re-write it
i=0 1
in matrix form:
(j>(x) = a x (2.2)
where a = (£q> • • • ), and
2 T
x = (1, X, X , . . .) are now infinite row and column
matrices respectively.
Remark: A polynomial degree n can be written also in the form
(2.2) provided that a^ = 0 for i > n-H ,
Differentiation and product
As pointed out by Ortiz and Samara (1981), the effect of differentiation of (2.2) is equivalent of post-multiplication of a
by a sparse matrix n. This matrix q is zero everywhere except in
the first subdiagonal where it consists of a sequence of natural numbers. Similarly, multiplication of (2.2) by the independent
2-3
variable x is equivalent to a post-multiplication of a by another sparse matrix p which consists of unity on the first upper- diagonal and zero elsewhere. Thus we have
(x) = a n x , and x<J>(x) = a p x (2.3) where ' 0 0 1 1 0 0 1 2 0 , u = 0 1 3 • • • • • •
-We note that both n and p are infinite matrices. The expressions for repeated differentiation and multiplication with x of <KX ) are obtained similarly by repetitive post-multiplication of a by n and p
respectively. More explicitly, we have
<j/r ^(x) = a r)r x , and xS <j) (s) = a pS x (2.4)
Furthermore, combination of multiplication and differentiation can be obtained easily, i.e.
vs . (r ) , . r s ,,X <J> ( x ) = a r i p _x. L e ^ , then L <J> (x) = a II x_» where v
n =
l n1 p.(p) . i=0 (2.5)L
C
XSince we have from (2.1),
L<Kx)
v ai * d^
( £ \ P. . x~* r ) <KX ); and using (2.5) we obtain
i=0 j =0 J dX1
L<f>(x) v ai .
I l
P±i (| n1 yJ x) i=0 j=0 3 v ai S ( I I i=0 j=0 P. . ij i n hence L(})(x) = a II x (2.6)Orthogonal polynomial basis
Instead of representing a function in power series, it is often advantageous in practice to express a function in terms of
orthogonal polynomials (see Lanczos 1957). Notably, the well
known Che^yshev polynomials have been given considerable attention j b
in numerical analysis (see Fox and Parker 1968).
Suppose we now wptie / oj'/if-z
00
<£(x) = \ a. y. (x), where V. (x) are orthogonal polynomials degree / Or^-^r ^
. n i l 1 ' C £-<•-£ J
1=0
of i. In practice, Chebyshev or Legendre polynomial basis are
often used. Let a = (a , a^, a^, • • •), T
and v_ = (Vq, v^, v^ . . .) , and $(x) be written as
cf>(x) = a v_ (2.7)
In fact v can be written as _v = V jc, where V is an invertible
lower triangular matrix (see Ortiz and Samara 1981). It is easy
to show that,
A
2-5
A
-A _iwhere II = V II V
Indeed (2.8) is obtained by a similarity transformation applied to
(2.6).
Consider the linear ordinary differential equation
L<j> = f(x) a < x < b (2.9a)
(tyj, 4>) = s.d j = 1 (1) v (2.9b)
where L eX. > and (bj, <j>) are linear functionals acting on $(x)
which are the initial, boundary, or mixed conditions. These
conditions are often called supplementary conditions in the
literature. Ortiz and Samara (1982) shows that (2.9b) can be
written as
a B = s
(
2.
1 0)
where s = (s s ^ • •
that a I^j = sj • Writing
(2.8) & (2.9a) give
.) , and B = [B ^ : B^: 00 f(x) as f(x) = £ f.tf.(x) j=o J 3 . . . ]such f v, then a n = f . (2.11)
Combine (2.10) and (2.11), we have
A
a [B : E] = (s : f) or compactly, a G = S (2.12) 18 nTo obtain a finite approximation in orthogonal polynomial basis, we choose a suitable m and truncate (2.12) to form,
a
=m G = m
s ,
=m (2. 13)where a
=m = (o0> a. , a_, . . ., a ), and 1 z m S
=m = (sL, ^2> • • •» ^2* * * *’ £m-v and
G is m the leading (m+1) x (m+1) sub-matrix of G.
Remark: We note that v equations in (2.13) derived from
supplementary conditions, and the remaining (m+l-v) equations are
obtained from the differential equation (2.9a). Some numerical
examples of linear, stiff, and nonlinear ODEs can be found in (Ortiz and Samara 1980, 81).
2-7
2.2 Software
A computer program (ODETAU) in standard FORTRAN was written and
documented with easy-to-follow instructions for the user. It is
designed to require a minimum of effort from the user and to be computationally efficient.
In Appendix A, a flow-chart of ODETAU is presented, followed by a description of all the common blocks (each with a separate
parameter list). All the routines in the program will be listed
and discussed. Precise instructions will be given as to which of
the parameters are required to be set. The usage of the program
is then exemplified with both a linear, and a nonlinear problem from ODE. The listing of ODETAU will also be given at the end of Appendix A.
20
2.3 Extention to PDE
We now discuss operational approach to the Tau method extended to
numerical solution of PDE by Ortiz and Samara (1984). This two
dimensional formulation of the Tau method, much like its one
dimensional predecpdsor in Ortiz and Samara (1981), enables one to ^
obtain a tau approximation through the use of a systematic and computationally simple procedure. It was shown in 2.1 that when
<j)(x) = a x_> we have
r _ T
xS — — 4)(x ) = a nr pS x = x (pr yS ) a (2.14)
dxr T
where a_ = a. Now suppose we write a function in two variables
00 00 iKx,y) =
l l
a X 1 yj (2.15) i=0 j =0 J as a bilinear form, <Kx,y) = xT a y (2.16) T 2 where sc = ( l , x , x , . . .), T 2 y - (i, y, y » • • • ),and [a]^ = aij for i,J' = * * •
We note also that the partial differential operator:
m n x y .r+t 3xr Sy6 can be factorise as ~t .r
< n d ' v ^ m d>v
ly — rJ lx — J. 9y 9x (2.17)2-9
or f m 8 l x — J ly — J-' v f n d r : ^n 9£ (2.18)
3x 3y
Now applying (2.17) or (2.18) to (2.16) we have,
m n 8 x y r+t 8xr Zy1 T r nuT . t n N = x (n M ) a(p y ) y (2.19)
Let ii = Ux_, and v = Vy be two orthogonal polynomial basis defined by lower triangular matrices U and V respectively. We write
T T
^(u,v) = u Av = x a y (2.20)
T -1 -1
where A = (U ) a V , which is the expansion (2.16) in the (U,V)
basis. Then the corresponding expression for (2.19) in (U,V)
basis is given by m n 8 x y r+t 3xr ay" t|;(u,v) = u^ A Q v
(
2.
2 1)
where P = U nr ym U 1, and Q = V n1 p" V \ T -1 -1since by substituting u_ = Ux, v_ = Vy, and A = (U ) a V into
(2.19), we have T , r nuT . t n N x (n y ) a(n y ) y_ T . -l.T . r itkT T , t tu „-l ,, , . = _u (U ) (n y ) U A V (n y ) v v_ which is precisely (2.21). 22
The differential operator
Let oi, be the class of linear partial differential operator L in
two variables x, y with bivariate polynomial coefficients (see Ortiz and Samara 1984). Let L be written as
L (
2
.22
) where 3^ constants mi ni x y ri+ti „ n „ ti 3x 3y T- with mi, ni, ri, ti e Z and b,- are
1 1
Remark: £ denotes a finite sum. i
Then mu = max (ri), and nu = max (ti) are defined as the order of the differential operator L in x and y respectively. From (2.21) and (2.22), we see that L exact i n g on ij>(u,v) can be expressed as,
L ^(u,v) = u d(A) v (2.23) where d ( A ) = T b. P. A Q . 7 1 1 1 1 „ TT ri mi TT-1 ~ „ ti ni „-l P_^ = U n y U , Qi = V n y V
Therefore for a nonhomogeneous PDE,
L ip(u,v) = f(u,v) = u F v,
i^s an open rectangular domain D, we have A
2-11
Remark: (2.24) is analogous to the ODE case where we
/\
have a II = f . Also since f(u,v) is known explicitly, the matrix F is in principle known.
The Supplementary Conditions
At least as long as they are linear, (which we always assume) the supplementary conditions of a partial differential equation can be defined by linear differential operators of the form (2.22) with an additional constraint restricting them onto the specified
section of the boundary 9D. So they can be treated similarly as
in the differential equation. Assuming D is a rectangular domain with sides parallel to the coordinate axis, then the supplementary
conditions are defined on x = xp > an^ y = Yq can be written as
L(x ) ip(u,v) = h(v);
and L(y ) ^(u,v) = g(u) ;
where L(x ) = P I 9. J — * rj +tj j 3 ' 3xrj 3yC j ’ and L(y ) = I e, i 9 ri+ti i . ri , ti* l 9x 9y
with 0 , constants and i, j, ri, ti, rj, tj £ Z
T T
Remark: g(u) = u_ and h(v) = h_ v_.
24
Then we can write:
L(x ) i|j (u,v) = ( I <$. A Q . ) v = h v, and
P j J 3
T T T
L(y ) i|>(u,v) = u ( £ Pi A y . ) = u g, where
q i 6. = 0 . (U nrJ X ), Q. = V nL PJ V \ and -J J v “ P ; J t J v_1
li
= q (v
rf1 yj,
Pi = u n ri w
1u s
i.e. £ S. A Q, = h and . — J J “ J (2.25) £ P. A y. = g. r i —i — i (2.26)For p = 1, . . } mu we have from (2.25)
LX (A) = H, (2.27)
and similarly for q = 1, . . ., nu we have from (2.26)
LY (A) = G; (2.28)
where the systems in (2.27) and (2.28) are matrices with mu rows, and infinite number of columns, and infinite number of rows and nu number of columns respectively.
Assembly of the equations
2-13
degree tau approximation, we require mu equations to satisfy the supplementary conditions and that the remaining (m + 1 - mu) are related to the differential equation.
Likewise, for a (nr degree in x, nc degree in y) tau
approximation in two dimensions. We have by direct analogy mu(j^)
equations from (2.27), nu(yl) equations from (2.28), and
(m + 1 - mu) (n + 1 - nu) equations from the differential
equation. However as Gotlieb and Orszag (1977) have pointed out, not all the equation from (2.27) and (2.28) are linearly
independent. In general, there exists (my()(nu) linear relations
among them. For example in the Chebyshev tau solution to the
Poisson’s equation, Gottlieb and Orszag (1977) has shown that there exits four linear relations among the boundary conditions. Furthermore in Chapter Three where biharmonic equations are considered, we show there are sixteen linear relations in the boundary conditions.
/hr | /iritf
/
u
In general therefore, we have (mu)j^
4
y((nu) - (mu)(nu) and(m + 1 - mu)(n + 1 - nu) linearly independent equations from the
supplementary conditions and the differential equation
respectively for the determination of the (m + l)(n + 1)
coefficients of the tau approximation (i.e. j > 0 < i < m , 0 < j < n).
26
2.4 Numerical Examples
The first example, we consider is a ^ oisson’s problem in Saint Venant fs torsion problem for a prismatic bar of cross section
2
D = [-1, 1] with homogeneous boundary conditions, namely
Au = -2 in D, (2.29a)
u= 0 on 3D, (2.29b)
and the exact solution is given in Southwell (1946):
u(x,y) = — 7? I it n=l ,3,5, n-1 ( - D cosh (n -^y) cosh (n — ■) ^mrx cos J (2.30)
This example was also considered by Ortiz and Samara (1984), but, we now assume the symmetry in this problem and solve only for the
2
solution in D/4 = [0,1] , namely
Au = -2 in D/4
u = 0 on x = 1, and y = 1
u = 0 n on x = 0, and y = 0.y
An alternative approach to take account of symmetries will be discussed in Chapter Three.
2-15
The second example we consider is the heat equation,
|H. = in D = [0,1] x [0,0.2]
8x
u(x,0)
=
sin it x, 0<
x<
1 u(0,t) = u(l,t) = 0 0 < t < 0.2The solution is symmetric about x = j, so we solve only for D/2 = [0, j] x [0, 0.2].
Numerical results for these two examples using double Chebyshev and Legendre basis are given in Table 2.1 and 2.2, the CPU times on a CDC Cyber 174 will also be shown.
28
m = n
MAX ABS ERROR Time
CPS Chebyshev Legendre 4 .7755-02 .3184-02 .405 6 .9943-03 .5144-03 .908 8 .1516-03 .5004-04 2.514 10 .8614-04 .1450-04 6.739 TABLE 2.1
Heat equation
m = n
MAX ABS ERROR Time
CPS Chebyshev Legendre 4 .1932-02 .5592-03 .488 6 .1485-04 .2236-05 2.208 8 .7111-07 .1728-07 8.621 TABLE 2.2 30 I
DIRECT SOLUTION OF BIHARMONIC PROBLEMS
3.1 Introduction
3.2 Special Trial Function: Symmetry
3.2.1 Choice of trial solution
3.2.2 Symmetry 3.2.3 Trial functions 3.3 Boundary Conditions 3.3.1 Poisson problem 3.3.2 Biharmonic problem 3.4 Numerical Results
3.4.1 Rectangular Plate Problems
3-1
3.1 Introduction
Partial differential equations involving the biharmonic operator
are of interest in several branches of Mathematics. Well known
examples are in linear elasticity for the determination of the tranverse displacement of middle surface of a homogeneous elastic plate, and in fluid mechanics for the determination of the stream function governing the flow of viscous fluid.
Enough situations in the literature concerning the numerical solution of biharmonic equations assure us of the interest it receives from engineers, applied mathematicians, and numerical analysts for theoretical as well as for practical reasons.
A complete survey of numerical methods for biharmonic problems will be outside the scope of this disertation. Our attention will however be centred on the numerical solution of biharmonic
problems with the Tau method. Numerical Comparisons will also be
made with recent results in the literature.
Ortiz and Samara (1981, 84) had proposed what is now known as the 'Operational Approach to the Tau method' for the numerical
solution of ordinary differential equations, and partial
differential equations respectively. Although closely related,
the latter will receive most of our interest.
In Ortiz & Samara (1984), the basics of the operational approach to the Tau method for PDEs was discussed in general terms and examplified with second order PDEs. The simplicity of this method
to obtain tau approximations is very appealling, and offers an extremely interesting alternative approach for the numerical solution of PDEs. Nevertheless, much more numerical experience is required to assess its full potential in application. Indeed, the contribution of this chapter is to solve the biharmonic equation which is fourth order and has two conditions imposed at each point
on the bundary. We appear to be the first to solve directly the
biharmonic problem with the Tau method in two dimensions.
El Misiery and Ortiz (1984) has proposed what they call the Tau Lines Method which is essentially the Method of Lines combined with an extension of Ortiz’ Recursive Formulation of the Tau method for a system of ODEs (see Grisci and Russo (1983), and Freilich and Ortiz (1982)) to solve the biharmonic problems. Their results compare favourably with those of Amara and
Destuynder (1981). We note that the problem they considered
possesses an exact solution of degree 4 in x and y. It is well
known, at least in the ODE case (see Ortiz 1969), that the Tau
method will recover the exact polynomial solution. Indeed, our
computation also verifies this for the two dimensional formulation of the Tau method. This renders our comparison with their results pointless.
First, we will consider two classical examples taken from Timoshenko and Weinowski-Krieger (1959) which describe the lateral displacement of a homogeneous plate under a uniform load. Secondly, we will consider a number of examples which had been
studied by other authors in the recent literature. Some of these
3-3
with our method. Although it is not necessary to choose 'special* trial solution, clearly it would be more efficient if we do it
when symmetry is present. More details will be discussed in
3.2. Incidentally, special trial solution chosen to have the
required symmetries also enables us to obtain very accurate estimations of eigenvalue problems (Chapter Four).
Finally, our numerical results show that the choice of evaluation
schemes can be critical. On the one hand, we have Horner's
algorithm, and on the other hand, we have Nested algorithm due to
Clenshaw (1952). The unwary user might be confronted with
numerical difficulties (lost of significant figures) if the /j
appropriate choice of evaluation scheme is not made correctly.
3.2 Special Trial Function: Symmetry
The treatment of symmetry is suff^iciently important to warrant
some detail discussion. The operational approach to the Tau
method involves the expansion of the solution to a differential equation in an orthogonal basis of polynomials, the coefficients of which are determined by essentially a weighted-residual projection (see Appendix B).
In fact, Gottlieb and Orszag (1977) have exhibited such a
projection explicitly, and also pointed out an important
difference between the Tau technique and its Galerkin counterpart in their treatment the boundary conditions (see Appendix B). The equivalence of the ’infinite' and ’finite’ formulation of the Tau method was also mentioned (see also Fox and Parker (1968), and
Ortiz and Samara (198^)). A general discussion of weighted
residual methods (WRM) is found in Ames (1965, 72) and Finlayson (1972).
Choice of Trial Solution
In the family of Weighted Residual Methods, the choice of the
trial solution is of utmost importance. Without due care, we
could forsake the principal attraction of these type of methods, namely, the possibility of obtaining good approximations with a limited number of adjustible parameters.
Crandal (1956) emphasizes this and urges us to give due
considerations to symmetry or any other special characteristics of the solution which may be known.
3-5
In Chapter Five, the special characteristics of the boundary singularities will be considered. Here, we discuss the choice of trial solutions to account for symmetry.
Symmetry
For the sake of simplicity, let us consider a general function in
one variable u(x) defined on [-1, 1]. When u(x)=u(-x), we say
that it is even; when u(x)=-u(x) we say it is odd. We note also, any function defined on [-1,1] can always be split into a sum of an even, and an odd function in the following way: we defined
g(x) = ( u (x ) + u(-x) )/2 h(x) = ( u (x ) - u(-x) )/2
where g(x) is even, and h(x) is odd. Then we have
u (x ) = g(x) + h(x) (3.1)
In many physical problems, the inherent symmetry limits us to one expansion or the other.
Typically, we write our Tau approximant
m
u (x) = l a 6. (x) (3.2)
i=0
where <f^(x) are orthogonal polynomials defined on [-1,1]. It
well known that <J>^(x) is even when i is even, and odd when i odd. So we can write (/) in the following way
is is
V
a
36
m um (x) - I at i=even m » (x) + l a j=odd J i. (x) J (3.3)
suppose the approximation we seek is even, then the odd component of (3) vanishes indentically. If the Tau method is applied to the trial solution (^), the computed solution indeed tells us the same information, but at the expense of a great deal of computational
effort. Most of which is wasted in inverting a linear system of
equations where approximately half of the unknowns are zeros.
In fact, if the trial solution of the form
m
u (x) = l a <j>. (x) (3.4)
m i=even. 1 1
was used, we only need to perform about 1/8 of the computation
required previously in (3.2). To show this, we note that in
gaussian elimination (see Fox (1979)), the number of operations require to solve the linear system
A x = b of the order N is as follows:
N reciprocals (3.5)
N multiplications
additions
(3.6)
(3.7)
Since almost half of the (n+1) coefficients are zero in (3.2) substituting N/2 in the place of N in (3.5)-(3.7), we see that
3-7
indeed the work require is reduced approximately to 1/8 of the previous amount. We note that storage requirement of the matrix A is also quartered.
Symmetry in Two Dimensions
Occurence of symmetry in two dimensions can be found in many
boundary value problems for PDEs. For example, torsion of a
prismatic bar of rectangular section, bending of plates, modes of vibration in membranes and plates, and also in buckling problems
(see Chapter Four).
Trial Functions
In higher dimensions, it is ESSENTIAL to take symmetry into
account in the choice of trial solutions. This is because the
number of unknowns increases dramatically. For example, a tau
approximation degree ten of the form (3.2) has Eleven unknowns, whilst the number of unknowns in its counterpart in two dimensions has risen to One Hundred and Twenty-One.
Since storage requirement and computational work is a direct function of the number of unknowns, we therefore cannot afford negligence. Especially considering the ease of which this can be done effectively with a double expansion of orthogonal polynomials of Chebyshev or Legendre type.
Consider a general function u(x,y) defined on the square [-1,1] x [-1,1], we call u(x,y) even-even, odd-odd, or even-odd
(or odd-even) respectively as u(x,y) is even in x and even in y, odd in x and odd in y, or even in x and odd in y (or odd in x and even in y). Let <j>, (.) be orthogonal polynomial degree k defined on [-1,1]. Each of the above symmetry functions can be written as
00 00
u(x,y) = l l a (x,y) (3.8)
i=0 j =0 J J
For even-even symmetry, we have
(x,y) = <j>2i (x) <|>2 (y) (3.9)
and for odd-odd symmetry, we have
(x,y) = 4>2i+1(x ) ^2j+l(y^ (3.10)
and for even-odd symmetry, we have
^ (x,y) = (j>2i (x) <t>2j+i(y) (3.11)
Further symmetry are possible in a square, they are even* or odd*
with respect to the interchange of x and y. In these cases, we
can exploit the symmetries further. For example, when u(x,y) is
even-even and also even* with respect to the interchange of x and y, then it can be written as
oo i
u(x,y) = 1 ( 1 a t|>, . (x,y)) (3.12)
1=0 j =0 3 J
3-9
and similarly for the other cases.
We now make use of these symmetry considerations in our numerical solution of the biharmonic problems in 3.4.
When used to solve biharmonic problems, all numerical methods require some ’special' treatment on the boundary. This is because
there are two conditions imposed. For example, in the direction
solution of the biharmonic problem using the classical 13-points finite difference scheme, we would normally need to introduce ’fictitious' values outside the domain in order to preserve the regular structure of our coefficient matrices. Other schemes are
possible (e.g. Stephenson 1984), but we foresake the said
structure above.
It is important to note that the Tau method (and similar method) can accept without difficulty any linear conditions even if they are of a very ’mixed' type (see Fox and Parker 1968). We have no exceptions in the solution of the biharmonic problem with the Tau method.
3.3.1 Poisson’s Problem
Gottlieb and Orszag (1977) have already mentioned that in the Chebyshev tau solution of the Poisson's problem, not all the equations derived from the boundary conditions are
linearly independent. In fact, as they had pointed out,
there are four linear relations.
3.3.2 Bihannonic Problems 3-3 The Boundary Condition
3-11
problems. Indeed, sixteen linear relations in the
boundary conditions can be found. Let us illustrate this
by considering the biharmonic problem on the square [-1,1] x [-1,1],
A2u = f(x,y)
with the boundary conditions,
u = u = 0 n
We seek our Chebyshev tau approximation of the form
m n
Um n (x>y) = \ \ aij V (X) Tj (y) (3<13)
1=0 j=0 J J
Let us denote the Chebyshev expansion coefficients of
(3p+qu/3xP 3yq ) by a<P ’q ) ,
then for the differential equation we have
a (4,0) + 2 a (2,2) + a (0,4)
ij ij ij f . .ij (3.14)
0 < i < m-4 ; 0 < j < n-4
Whilst the boundary condition u=0 gives
m
l (+)1 a . . = 0 0 < j < m (3.15)
i=0 1J
42
n
£ (±)^ a.. = 0 0 < i < n (3.16)
j=0 1J
and the condition un=0 gives
m I (i)1 a (0’ = 0 0 < j < m (3.17) 1=0 1J m I (t)1 *<?• = 0 0 < i < n (3.18) j=0 1J
The sixteen linear relation among (3. 15)— (3.18) are:
m n I y (-)1 (±)J a . . = 0 i .e . u=0 at (±,±1) i=0 n i o ij m I 1=0 n l j=0 (i)1 (±)j II o 0 i.e . ux =0 at (+1,+1) m I n I (i)1 (±)j = 0 i.e . u =0 at (+1, + 1) 1=0 j-0 ij y m l n
y (i)1 (±)j {a?0^ - afl,0)j = 0
i=0 (_i. II c o 1 1J
i.e. ux = uy at (+1,+1) (3.19)
Thus, (^.14)— (3.19) gives (m+1) (n+1) equations for the unknowns a^^ (0 < i < m, 0 < j < n) •
3-13
3.4 Numerical Results
3.4.1 Plate Problems
We consider the biharmonic equation
A u = f(x,y) (3.20)
in the unit square; with two types of boundary
conditions. In the first place, we have
u, un (3.21)
prescribed on the boundary, this is known in the
literature as the First Biharmonic Problem. In the second place, the conditions
u, nn (3.22)
prescribed on the boundary is known as the Second Biharmonic Problem. In the bending theory of thin plates, the unknowns functions in (3.20)-(3.22) is the transverse displacement of a homogeneous plate under lateral loading.
Rectangular Plate Problems
Consider (3.20) above, with the boundary conditions
u nn 0 (3.23)
and
u = un = 0 (3.24)
which corresponds to a plate with all edges simply-
supported, and all edges clamped (or built-in)
respectively.
Simply-supported Edges
The exact solution of (3.20) and (3.23) is a special case
of the solution given by Navier in 1920. Its derivation
is simple which involves expanding the transverse
displacement function u(x,y), and the lateral load
function f(x,y) as a double Sine Fourier Series
respectively as
00 00
u(x,y) = l l b sin(iirx) sin(jxy) (3.25)
i=l j-1 30
and
O O 00
f(x,y) = l l f 'sin(iirx) sin(jTry) (3.26)
3-15
We then apply the biharmonic operator to (3.25) and
equating the Fourier coefficients with (3.26). The
unknowns a^j ' s can be found to be explicitly in terms of the known f^j's. For more details, we refer our reader to the standard text books on elasticity (e.g. Mansfield 1964, Wang 1953).
In the case of a uniform loaded plate, i.e. f(x,y) = l, we have the exact solution given by
u(x,y) = (i|) l l sin(lra).sin(j,y) (3.27)
7T i=odd j =odd (ij)(i + j )
In Table 3.4.1 the maximum absolute error is given for Chebyshev tau approximations of degree in x and degree in
y equals to 6, 8, and 10. The maximum deflection at the
centre of the plate is also tabulated.
In solving (3.20) & (3.23), we have used different trial solution of the form (3.13), (3.9), and (3.12) which respectively corresponds to assuming no symmetry at all,
even-even symmetry, and even-even (even*) symmetry.
Comparisons are made in Table 3.4.2 for the numbers of unknowns (NN), and the computer time (T) taken on the CDC CYBER 174 computer.
We see that both NN and T are reduced substantially when symmetry is taken into account.
Evaluation Schemes
Horner’s Algorithm to evaluate a polynomial of the form
m
p ( x ) =
l
c
X r ( 3 . 2 8 )r=0
is well known. It performs a nested multiplication
process from a ’backward’ recurrence,
q r ( x ) = x q r + 1 ( x ) + c r ; q m + 1 ( x ) = 0 ( 3 . 2 9 )
where q (x) = p(x) .
Clenshaw’s Algorithm first proposed in 1952 is capable of evaluating any sum of following form
m
P ( x ) =
l a
<f> ( x ) ( 3 . 3 0 )r=0
where <j> (x) (not necessarily polynomials) satisfies a linear occurence relation
<}>r+i ( x ) + a . <J>r ( x ) + 3 r ( x ) <J>r - 1 ( x ) = 0
( 3 . 3 1 )
In this case, the sum p(x) is given by
p ( x ) = bQ( x ) <j>o ( x ) + b j ( x ) {4)^( x ) + oco ( x ) 4>q( x ) }
3-17
where
br W + ar^x ) b r+ i (x) + Br+l(x) b r+2(x) = ar ; (3.33)
bm+l (x) = W
x) = 0
More details are found in Clenshaw (1952), Fox and Parker (1968) and Ortiz (1977).
Since all orthogonal polynomials have a linear recurrence relation of the form (3.31), the Clenshaw's algorithm is therefore particularly useful to us. This is because our operational approach to the Tau method works directly^/the
orthogonal polynomial basis. Although we have discussed
evaluation schemes in univariate case, multi-variate expansions can be evaluated in much the same way by step- by-step reduction of the variables.
A situation which ^/unfavourable to the Horner's Algorithm
was pointed out in Fox and Parker (1968). This is when
the coefficients in (3.28) are large (in modulus) even when p(x) is small. We have an analogy of this in the tau
approximation of (3.20) and (3.23). For example, with
m=n=12, some coefficients of (X,Y) basis are of the order Q
of 10 whilst the maximum absolute value of u(x,y) is
approximately equal to 0.00406. Incidently, all the
coefficients of the Chebyshev (U,V) basis has an absolute
value less than 10 In Table 3.4.3, we see clearly
evaluation using the Horner's scheme suffers a lost accuracy (in single precision on a CDC machine).
Degree of Approximation: m=n=12
Evaluation Scheme Max Abs Error
Horner's in (X,Y) .4913-05
Clenshaw's in (U,V) .6859-07
TABLE 3.4.3
The Clenshaw's scheme is used, from now on, throughout this chapter.
Clamped Edges
The exact solution of the simply-supported plate under a
lateral load was comparatively easy. This can be
attributed to the fact that elementary components of the deflection exist which satisfy the boundary conditions 3.23 and the differential equation
3-19
For other boundary conditions, no such elementary
component exists, and the solution to these problems are much more difficult.
In the case of a uniformly loaded plate with clamped edges, (3.20) and (3.24), with f(x,y) = 1, the most flexible ’exact' method is due to Timoshenko (1938). This takes the solution of the simply-supported plate (3.27) as a starting point, then superpose on the deflection of such a plate by some suitably chosen moment/the edges until the boundary condition un = 0 is satisfied.
For a unit square plate, Timoshenko and Woinowsky-Kriger
(1959), p 197, give the max deflection to be
umax = 0.00126. This provides a useful basis for
comparing our accuracy. The advantages of taking special
trial solutions were studied earlier. In Table 3.4.4, we
report numerical results obtained with due consideration to the even-even (even*) symmetry in this problem.
Other Examples
Further examples that have been studied by Gupta and Manohar (1979), Amara and Destuynder (1981), Stephenson (1984) and other authors, provides a useful basis for
further comparison with results in the recent
literature. We note that some of these examples has exact solutions which are low degree bivariate polynomials. Our
method will retrieve these exact solutions with a
sufficiently high degree approximation. Therefore, these
problems offers us no interesting comparison with other
methods. For the sake of completeness, these examples
will be briefly mentioned.
Example 3.4.3
A^u = f(x,y) in D = (-1,1) x (-1,1)
u = u = 0 on 3D
n
where u(x,y) = [ (1-x)2 (1-y)2 ]2 (3.34)
This problem was considered by Amara and Destuynder
(1981), and El Misiery and Ortiz (1984). The latter
authors produced what they called the Tau Lines Method (semi-discretization with MOL and solve the system of ODEs with the Tau method), and use it to solve example 3.4.3. They report results which compares favourably with a finite element technique propose in Amara and Destuynder (1981).
We note that the exact solution u(x,y) is a bivariate
polynomial of degree four in each of the variables. We
now use the operational approach to the Tau method in two dimensions to solve this example numerically, without any discretization of variables or special trial functions. For degrees of approximation greater or equal four in each of the variables, we recover the exact solution in
3-21
(3.34). Therefore, it would be futile, here, to compare
with other methods as competitors.
Example 3.4.4 2 2 2 2 2 A u = 8 [3y (1-y) + 3 x (1-x) + ( 6x 2 - 6x + 1) (6y 2 - 6y + 1)] in D = (0,1) x (0,1) u = u = 0 on 3D n
where u(x,y) = [x (1-x) y(l-y)]2 (3.35)
This example was studied in Gupta and Manohar (1979), and Stephenson (1984), and also other authors. In fact, it is equivalent to the previous example 3.4.3 with the two
solutions differ only by a multiplicative constant. We
can verify this by simply perform linear coordinate transformation, say, from [-1,1] to [0,1] and substitute into (3.34). The details of which are triv^4l.
Our interest, therefore, is in the remaining examples from
these recent literature. One of these examples has no
symmetry at all, whilst the other has an inherent symmetry property.
Example 3.4.5
A^ u = 0 in (0,1) x (0,1)
2 2 x
with u(x,y) = x + y - x e cos(y)
This is a more interesting example - no symmetry is
present. We follow Stephenson (1984), and consider the
first and second biharmonic problem. The numerical
results and comparisons are given in Table 3.4.5 and Table 3.4.6. As earlier, our computations were performed on CDC CYBER 174/720. Example 3.4.6 2 A A u = (2tt) (4 c o s ( 2t t x) c o s (2iry) - c o s (2irx) - c o s ( 2 i T y ) ) in D = (0,1) x (0,1) with u ( x , y ) = (l - c o s ( 2 n x ) ) (l - c o s ( 2 i r y ) )
Stephenson (1984) had used a new method to solve this for
the first and second biharmonic problem. Our results are
obtained giving consideration to the symmetries in this
example. Our accuracy far exceeds that of Gupta and
Manohar (1979) and Stephenson (1984). As we can see from Table 3.4.7 and Table 3.4.8.
m = degree in x , n = degree in y Exact Maximum deflection = 0.00 406 235
m = n Max Deflection Max Abs Error
6 0.00 403 579 .5043-04
8 0.00 406 361 .2740-05
10 0.00 406 264 .3133-06
TABLE 3.4.1
Special Symmetry Trial Functions
No symmetry Even-even Even-even (even*)
m = n NN T(cps) NN T(cps) NN T(cps)
6 49 1.041 16 .322 10 .300
8 81 2.924 25 .408 15 .330
10 121 7.831 36 .598 21 .424
m=n umax NN T(cps) 6 0.00 122 482 10 .312 8 0.00 126 957 15 .347 10 0.00 126 681 21 .421 12 0.00 126 580 28 .547 14 0.00 126 545 36 .732 16 0.00 126 534 45 1.040 18 0.00 126 531 55 1.492 20 0.00 126 530 66 2.730 TABLE 3.4.4
Max. Abs. ERROR
h
n
Gupta & Ma^ohar (1979)
Stephenson (1984) Chebyshev-Tau Approx. Time
(CPS)
h 2nd Order 4th Order m-n Operational Approach
1/5 .6990-03 1 4 2916-04 EXACT 7 .2238-05 .3100 1/10 .8774-04 1 8 .7645-05 EXACT 8 .4668-06 .519 1/20 .4768-05 1 16 * * 9 .2612-06* .890
* _ results affected by /found-of f error / r
EXACT - if Abs. Max Error < 1.-07.
h Stephenson (1984) Chebyshev-Tau Approx. Time (CPS)
2nd Order 4th Order m=n Operational Approach
1 3 .1012-03 .1863-05 7 .4465-05 .307 1 6 .2818-04 .1863-05 8 .8259-06 .523 1 .6735-05 * 9 .4922-05* .899 12 TABLE 3.4.6
Max. Abs. ERROR in u
h
a
Gupta & Ma^ohar Stephenson (1984) Chebyshev--Tau Approx. Time
(1979) (CPS)
h 2nd Order 4th Order m=n NN max abs error
1 20 .1981 (-1) 1 4 .4372 .4080-1 8 15 .1242-01 .098 1 8 .1035 .2618-2 10 21 .1488-03 .105 1 16 .2594-1 .1458-3 12 28 .1885-05 .127 14 36 .1595-07 .162 16 45 .1214-09 .217 TABLE 3-4.7
h
Stephenson (1984) Chebyshev-Tau Approx.
0(h2) 0(h4) m=n NN Max Abs Error Time (CPS)
1 3 .3519 + 01 .3292 + 00 8 15 .8670 - 02 .110 1 6 .9600 + 00 .1979 - 01 10 21 .2065 - 03 .119 1 12 .2241 + 00 .1179 - 02 12 28 .1332 - 05 .144 14 36 .2084 - 07 .175 16 45 .1208 - 09 .230 TABLE 3.4.8
CHAPTER FOUR
PARTIAL DIFFERENTIAL EIGENVALUE PROBLEMS
4.1 Introduction
4.2 Eigenvalue Problems
4.3 Numerical Results
4,1 Introduction
Quite frequently physical problems lead to differential eigenvalue
problems. Crandall (1956) gives an informative survey of exact
and approximate eigenvalue methods. In particular he offers
examples on vibrations and buckling in systems with a finite degree and also in continous systems.
Weinstein and Stenger (1972) in discussing intermediate problems for eigenvalues had selected a list of problems which have an interest in application to engineering, quantum physics and applied mathematics.
In this chapter, we shall consider only linear eigenvalue
problems. Nonlinear eigenvalue problems with a finite power in
the eigenvalue are shown to be equivalent to a linear one in Liu
and Ortiz (1983). The general problem, see Ames (1977), is to
find one or more eigenvalues and the corresponding eigenfunctions, such that the differential equation and the boundary conditions (both in general contain the eigenvalue parameter) are satisfied
on a given domain D. In most of the problems one encounters, a-ee-
t-hoae the eigenvalues appears only in the differential equation; for example, in the vibration or buckling of homogeneous membranes
and plates, see Weinstein and Stenger (1972) and Crandall
(1966). However, eigenvalues appears only in the boundary
conditions do occur. For example in fluid sloshing problems,
small fluid oscillations in a tank, and in the vibration of an elastic membrane with masses concentrated on the boundary. Also, this type of problem is of paramount importance for the method of
4-2
a priori inequalities for the numerical solutions of partial differential equations, see Payne (1967), Kuttler and Sigillito
(1968), and Ames (1977).
The former class of problems is generally known as Eigenvalue Problems, whilst for the latter, the special name of Stekloff
Eigenvalue Problem is reserved. Numerical results for both of
these classes will be reported - Our selection of examples is not intended to be exhaustive, but we shall consider a sufficiently wide range of problems to demonstrate the applicability of the Tau
method for the numerical solution of partial differential
eigenvalue problem.
4.2 Eigenvalue Problems
Ortiz and Samara (1981) proposed a so-called "Operational Approach to the Tau Method" for the numerical solution of ordinary differential equations. This technique has been successfully used for the numerical treatment of eigenvalue problems defined by ordinary differential equations in Ortiz and Samara (1983), Liu
and Ortiz (1982, 1983), and Liu (1984). The accuracy they have
obtained is quite remarkable if compared with results in the current literature.
Ortiz and Samara (1984) extend the ideas reported in Ortiz and
Samara (198'l) to the numerical treatment of partial differential
equations with variable bivariate polynomial coefficients.
Although it is algebraically more involved, this method mirrors the ODE case in Ortiz and Samara (1981).
Consider the general eigenvalue problem
L 2m(u) + A F 2n(u) = 0 (4.1)
in a domain D, and the boundary conditions
B (u) + A C (u) = 0 i = 1, 2, . . . (4.2)
The operators are linear homogeneous differential
operators of order 2m, 2n, p, q respectively, such that m,n and
p, q ^ 2m. The latter two operators and have an additional
4-4
In this chapter, eigenvalue problems defined by partial
differential operators will receive most of our attention. In
fact, Liu (1984) uses the method of lines to treat partial differential eigenvalue problems with Tau lines, a hydrid version of the Tau method, see El Misiery (1984), and Liu (1984).
Here, we will follow closely Ortiz and Samara (1984), and earlier Chapters of this thesis to treat partial differential eigenvalue
problems. Let be the class of linear differential operators
with bivariate polynomial coefficients, and consider L £ / so that it can be written as
L =
l
(x,y)i
where (x,y) = l l a ^ xU yV , and
u v
9ri + Zi +
9. = ---- :---r , with i, u, v, r i , ti e Z
l . vn ti
3 X 9y
and £ denotes a finite sum.
Then the partial differential equations
L u (x,y) = f (x,y) in D (4.4a)
(u) = (x,y) on 3D (4.4b)
where D is a rectangular domain in R x R, can be approximated using the Operational Approach to the Tau Method via the use of two very simple matrices p and n as the basis of constructing our tau approximation.
For the linear PDE in (4.4), we have the algebraic problem A x = b, where A and _b are obtained from the operators L
and , and the functions f(x,y) and G ^ ^ y ) respectively.
Clearly, differential eigenvalue problem (4.1)— (4.2) can be treated in a similar way, see Ortiz and Samara (1983), Liu and
Ortiz (1982, 1983) and Liu et al (1984). Instead of leading to a
system of linear algebraic equations, we have the generalised eigenproblem
(A + XB) x = 0 (4.5)
where A and B are respectively obtained from the operators L
2
m andB^, and F
2
n and . A standard subroutine (e.g. Nag F02BJF, MK 6)is then used for computing the eigenvalues of (4.5).
Generally, the smallest eigenvalue or the largest in absolute
value are of maximum interest. Indeed the latter can be
transformed into the former in a trivial way. We will take into
account of symmetry properties in examples 4.3, 4.4, 4.6 and
4.7. As we shall see, this would substantially reduce the order
of our matrices NN, thus making possible a very accurate estimation of the required eigenvalue, namely the smallest non
zero eigenvalue. The symmetry in example 4.1-4.3 was not assumed
and the first few eigenvalues reported show that accurate estimations of the higher eigenvalues are also possible with the Tau method.
4-6
4.3 Numerical Results
We now consider several examples of eigenvalue problems defined on a square domain D. The exact eigenvalues of our example of second order can be easily found using the separation of variables in
Cartesian coordinates. Since the biharmonic operator is not
separable in Cartesian coordinates, the exact values of our fourth order examples can not be found in this way. At present, we rely on numerical procedures to approximate these eigenvalues.
It should be remarked that Kuttler and Sigillito (1968) had obtained relationships connecting the first non-zero eigenvalues of the membrane problems (example 4.1, 4.2), and the Stekloff
eigenvalue problems (example 4.5-4.7). These eigenvalues are of
interest because they are the optimal constants in the method of a priori inequalities which has applications in bounding solution of elliptic and parabolic PDEs, e.g. Kuttler and Sigillito (1968).
Example 4.1 (fixed membrane)
Au + Xu = 0 in D = (0, 1) x (0, 1)
u = 0 on 3D
Example 4.2 (free membrane)
Au + pu = 0 in D = (0, l ) x ( 0 , 1)
u = 0 on 3D
It is well known that these two problems has exact solutions, namely
Without taking into account of symmetry, we have obtained tau
approximations of these eigenvalues with the eigenfunction
approximated with degrees 6, 8 and 10. Numerical results will be
given in table 4.1 and 4.2, the order of the matrix pencil (A + XB) denoted by NN will also be given.
Symmetry Classes
We call a function defined on D to be even-even, odd-odd, even-odd (or odd-even) as u is respectively even in both x and y, odd in both x and y, even in x and odd in y (or odd in x and even in y). Every eigenfunction of the problems considered can be assumed
to belong to one of these above mentioned symmetry classes.
Functions of each of these classes can be expressed as a double Chebyshev polynomial expansion in x and in y with even or odd
orders according to the symmetry present. For example, without
$
losjt of generality, let D = [-1, 1] x [-1, 1] and suppose u is even-even, then it can be written as
h, k = 0, 1, 2 Jt, m = 1, 2, 3
00 00
u(x,y) = l (4.6)
4-8
Furthermore, if D is a square, the even-even symmetry class can be separate into two sub-classes whose intersection contains only the zero function. These two sub-classes are those even with respect to the interchange of x and y, and those odd with respect to the
interchange of x and y. Then these can be written as (4.7) and
(4.8) respectively.
oo i
u(x,y) = l ( l b <{> (x,y)) (4.7)
i=0 j =0 J 3
where (x,y) = (x) (y) + T 2i (y) (x);
oo i-1
and u(x,y) = \ \ c . . . (x,y)) (4.8)
i=0 j =0 J J
whe
