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TRANSCRIPT
GARPPUBLICATIONS
SERIES
No.10
I
INTERNATIONAL COUNCILOF SCIENTIFIC UNIONS
GARP
JOINT ORGANIZING COMMIJTE-E
WORLD METEOROLOGICALORGANIZATION
GLOBAL ATMOSPHERIC RESEARCH PROGRAMME (GARP)
WMO -ICSU Joint Organizing Committee
"'.. .';:
~- ... ,r-'~ ,
METHODSFOR TH E APPROXI MATE SOLUTION
OF TIME DEPENDENT PROBLEMS
By H. Kreiss and J. Oliger
GARP PUBLICATIONS SERIES No.10
February 1973
03-5053
C 2
© 1973, World MeteOl'ological OrganizationInternational Council of Scientific Unions
Since you are on our regular mai I ing I ist we are, as usual,
sending you a free copy of the latest GARP Publ ication: No. 10 "Methods
for the Approximate Solution of Time Dependent Problems".
Because of the expected wide interest in and demand for this
particular publ ication, a special stronger-backed version has been
prepared and requests for any such additional copies should be sent to
the World Meteorological Organization, C.P. No. I, 121 I Geneva 20,
SWitzerland, who wil I be pleased to supply them at a cost of Sw. Frs. 14.
per copy.
iii
CONTENTS
. . . . . . . . . . . .FOREWORD • • • • • • • •
SUMMARY • • • • • • •
INTRODUCTION • • • •
. . .. . . .
. . . . . . . . . . .. . . . . . . . . .
· . .· . .· . .
Page
vii
xvii
1
1. Difference Approximations for Ordinary DifferentialEquations . • • . . . • . • • • . . • .••.. . . . . 3
2. Some Simple Difference Approximations for OrdinaryDifferential Equations ••••••••••••••• • • • 7
Step Size ••.•••.••....••.•••.•••.
The Importance of the Truncation Error and the StabilityDefinition for Error Estimates • • • • • • • • • • •
4. Some Remarks on the Choice of a Difference Method and the
72
11
12
1416
19
27
3338
42
46
54
59
64
71
7982
86
97
101
103
. . .
. . . . . . .
. . . . .. . . . . . .
Trigonometric Interpolation
The Fourier Method • • • •
The Leap-Frog Scheme • • • • • • • •
Notation and Elementary Theorems • • • • • •
Well-Posed Cauchy Problems • • • • • • •
Stable Difference Approximations for the Cauchy Problem • •
Difference Approximations for Hyperbolic Systems
On the Choice of a Difference Scheme ••••
20. Grids. • • • • . • • • • • • • • • .
21. Discontinuities. • • • • • • •••
REFERENCES • • • • • • • • • • • • •
BIBLIOGRAPHY • • • • • • • • • • • • •
Implicit Difference Methods •••••••••••••
Nonlinear Instability • •• • • • • •••
Initial Boundary-Value Problems for Hyperbolic Equations
Initial Boundary-Value Problems for Parabolic Equations ••
Difference Approximations for the Initial Boundary-ValueProbl~m. Stability Definition • • • • • • • • • • • • • •
Difference Approximations for the Initial Boundary-ValueProblem. Some Stable Methods • • • • • • • • • • • •
19. The Shallow-Water Equations •••
18.
8.
9.10.
ll.
12.
13.14.15.
16.
17.
5.
6.
7.
v
One of the conditions which made the concept of the Global
Atmospheric Research Programme viable was the development of numerical
models with which to attack the problems of atmospheric dynamics and
of weather prediction. In recent years many able minds have been
focussed on this development and progress has been rapid,as is
attested by the rich literature which has grown up in many scientific
journals.
In order to make this work more accessible and understandable
to the community of geophysical fluid dynamics, the Joint Organizing
Committee for GARPdecided to commission two leading scientists,
Professor Heinz Kreiss, University of Uppsala, Sweden, and Dr. Joseph
Oliger, the National Center for Atmospheric Research, USA, to write an
analysis of the numerical techniques involved. The resulting monograph
is an outstanding contribution to the GARP Publications Series and
should be of great value not only to meteorologists but also to
oceanographers.
As well as expressing appreciation to the authors, I wish to
thank Dr. Akira Kasahara who has written a perceptive foreword which
serves as a valuable guide to the monograph. Thanks are also due to
the Meteorological Institute of the University of Stockholm, Sweden,
and the National Center for Atmospheric Research, Boulder, USA, for
their support of this project.
R.W. Stewart,Chairman, Joint Organizing Committee
vii
FOREWORD
This monograph was written by two outstanding numerical
analysts having extensive experience in the actual formulation of
finite-difference schemes for numerical weather prediction. Dynamic
meteorologists and physical oceanographers have developed many use
ful approximation methods for solving time-dependent problems. These
methods were formulated to solve individual problems. Therefore, it
is not su;rprising to find that the numerical scheme which \yorked
well for one problem may not be useful for solving others. This
also applies to selecting grid structures and boundary conditions
for finite-difference schemes.
The theoretical analysis of numerical methods presented in this
monograph will fill the need for systematic treatment in formuLating
finite-difference methods for geophysical fluid dynamics problems.
In my view, this monograph is aimed at serving two objectives. One
is to review fundamentals in the construction of finite-difference
methods for obtaining approximate solutions to time-dependent dif
ferential equations. Special emphasis is placed on treating the
hyperbolic partial differential equations which appear in geophysical
fluid dynamics. The other objective is to bring to the attention of
meteorologists and oceanographers the recently developed material in
the field of numerical analysis relevant to the present problem. The
references to each chapter are useful for interested readers to
further pursue particular topics.
In Chapters 1-5, the concepts of stability and consistency of
difference equations are discussed for ordinary differential equa
tions. When the spacial dependence of atmospheric prediction equa
tions is expressed by the use of orthogonal harmonic functions, such
viii
as trigonometric functions, surface spherical harmonics or even
empirical orthogonal functions, .the prediction equations are reduced
to systems of ordinary differential equations with respect to time
for the amplitudes of the orthogonal functions. (This is called
the spectral method in contrast to the grid point method for direct
approximation of partial differential equations in the physical
domain.) Several basic difference schemes are presented in detail
to demonstrate the nature of stability and discuss error estimates.
The fact that stability and consistency imply convergence also
becomes a useful tool in the analysis of difference equations for
partial differential equations. The selections of a difference
method and the time step, both important subjects in the spectral
method, are diSCURsed from the viewpoint of accuracy of numerical
solutions.
Knowledge of matrix algebra, norms, and difference operators
needed in the subsequent chapters are briefly summarized in Chapter 6.
Proofs of some elementary theorems are left to the reader for
exercise. Perhaps the use of translation operators to define
difference operators may be new to some. The notations introduced
in this chapter are very common in the field of numerical analysis.
The definition of well posedness of the Cauchy, or initial
value, problem for partial differential equations is presented in
Chapter 7. Because the domain of integration spans from _00 to 00
in space, the Cauchy problem is relevant only for the prediction
of global-scale motions. However, any system of prediction equa
tions to be used for short-range forecasting over a limited area
must also be well posed as the Cauchy problem before the boundary
conditions are imposed. The equation of soundwave propagation and
the shallow-water equations are examples of well posed problems.
ix
A gravitational (convective) instability in a hydrostatic system
is an example of an ill posed problem, since the exponential growth
rate is proportional to the wavenumber of horizontal scale of motion.
Various difference approximations for Cauchy problems are
presented in Chapters 8 and 9. An example of an unstable scheme
and a remedy for its instability (both already familiar to many
readers) are instructive introductions to the concept of computational
stability. Just as discussed in Chapter 1, the definitions of
stability and consistency are introduced again for difference equa
tions approximating partial differential equations. The order of
accuracy (ql,q2) for a particular difference scheme defined on
page 30 is an important concept. The Von Neumann condition is also
an important tool to discover a necessary condition for stability.
General necessary and sufficient conditions for a difference approxi
mation to have no growing solutions are very complicated. For a
long-term integration of primitive equations with thermal forcing,
dissipative terms are often requisite to ensure computational
stability. Through the examples of Lax-Wendroff and leap-frog
schemes, the authors discuss how dissipation terms should be incor
porated without reducing, the order of accuracy of the finite
difference scheme.
The choice of difference schemes presented in Chapter 10 is of
particular interest to many readers. The usefulness of the fourth
order scheme over the second-order scheme is clearly demonstrated.
However, it is a significant conclusion that difference methods of
. order greater than six may not have any practical advantage for
geophysical fluid dynamics calculations.
x
In Chapters 11 and 12, the use of trigonometric -functions for
solving partial differential equations is reviewed. The usefulness
of trigonometric interpolation for approximating functions is
demonstrated when the functions are sufficiently smooth. The
operational counts for comparing the Fourier method with the fourth
order scheme and some difficulties connected with the application
of the Fourier method to equations \l7ith variable coefficients are
discussed in detail. The application of the Fourier method in
evaluating the spacial derivatives, which is called the pseudo
spectral method (as compared with the spectral method referred to
earlier), recently became very popular for the time integration of
atmospheric primitive equations owing to the fast Fourier transform
method. The authors analyze the accuracy of the pseudo-spectral
technique. The approximations of spacial derivative terms are
involved in the error estimate and, therefore, the accuracy of the
technique for equations with variable coefficients is difficult to
determine in general. The users of the pseudo-spectral method
must provide their own accuracy estimates by comparing with those
based on grid point methods.
Recently in the field of numerical weather prediction, con
siderable efforts were made to speed up the computer calculations
of prediction equations using implicit difference methods. In
Chapter 13, various aspects of implicit difference methods are
discussed for linear one-dimensional and two-dimensional equations
and a nonlinear equation. The advantage of implicit methods is in
the use of a relatively longer time step than normally allowed by
the linear stability conditions of explicit schemes. However, all
waves traveling faster than the phase velocity determined by the
xi
time step satisfying the explicit stability condition are slowed
down artificially and their contribution to the solution becomes
erroneous. Thus, as long as the contribution from the fast traveling
waves is negligible, the implicit schemes.can be used effectively.
The use of the splitting technique is discussed to speed up the
implicit calculations for the two-dimensional problem.
In Chapter 14, the nature of nonlinear instability, first dis
covered by N. Phillips in his numerical simulation experiments for
the atmospheric general circulation, is discussed based on a simple
nonlinear equation. Instability of the same nature can also appear
with linear equations with sufficiently rough variable coefficients.
The authors recommend the addition of dissipation to remove this
type of instability. A similar procedure has already been practiced
in the field of numerical weather prediction. The use of an energy
conserving scheme, such as the one developed by A. Arakawa, can still
produce instabilities, although the amount of dissipation needed to
control instabilities is much smaller. The authors propose a fourth
order accurate scheme (including a nonlinear dissipation term) to
solve the shallow-water equations.
In Chapters 15 and 16, the authors discuss the initial boundary
value problems for hyperbolic and parabolic partial differential
equations. The atmospheric models for short-range forecasting over
limited areas and ocean circulation models fall into this category
of problems. For the one-dimensional hyperbolic system, the boundary
conditions are specified based on the solutions along the characteristics
. which intersect the boundaries. The authors further discuss how
to specify the boundary conditions for a two-dimensional hyperbolic
xii
equation for a single dependent variable. As the authors demonstrated,
however, this particular approach may fail for the system of two
dimensional hyperbolic equations. It appears that a case study is
needed in certain two-dimensional problems such as the shallow-
water equations.
Various finite-difference schemes for the initial boundary
value problem are presented in Chapters 17 and 18. These sections
are very relevant for the formulation of difference approximations
to solve prediction equations for atmospheric circulation over a
limited domain or for ocean circulations in an enclosed basin. In
the case of atmospheric problems, the need for imposing horizontal
boundaries is largely computational to economize the computations,
since there are no physical horizontal boundaries--in contrast with
the oceanographic problem. In predicting atmospheric motions for a
period of beyond, say, 5 days, the domain of integration must be at
least hemispherical or preferably global. On the other hand, for
short-range forecasts of up to 2 days or so, the use of a limited
area model with fine horizontal (and vertical) grid resolution is
very valuable. Ideally speaking, a great deal of economy in the
computation may be achieved by combining the limited-area model
with the global model. This particular question of combining two
(or more) grid meshes with different resolutions is discussed in
detail in Chapter 20. The foundation of the question of grid
refinement lies in the design of stable difference approximations
for the initial boundary-value problem.
The basic approach to examine the stability condition of dif
fer:~nceequations incorporating suitable boundary conditions (which
xiii
have been developed only in the last few years) is described in
Chapter 17. Examination of the stability condition is demonstrated
through an example of the leap-frog scheme~ Then the authors
generalize the approach in the form of three theorems. There are
a number of references cited in this chapter and interested readers
are urged to explore them.
In Chapter 18, some stable difference methods for the initial
boundary-value problem for hyperbolic equations are discussed. The
purpose of this discussion is to develop stable boundary conditions
to solve the shallow-water equations over a limited domain and
ultimately to solve baroclinic (atmospheric) primitive equations
for a limited area. Specification of these difference boundary
conditions is guided by the theorems described in previous chapters.
Computer demonstrations of these methods and many others are dis
cussed extensively in papers referenced in this chapter. Dif
ference methods for irregular boundaries such as those connected
with the simulation of ocean circulations must be examined by
extensive numerical experiments. However, as the authors point out,
these problems should be considered by locally mapping the irregular
boundary to a tan-gent plane and then introducing a local net with
points on planes normal and tangent to the plane tangent to the
boundary.
The "best" difference approximations to solve the shallow
water equations, from the authors' viewpoint, are presented in
Chapter 19. Both the second- and fourth-order schemes are dis
. cussed with comments on the boundary conditions. It is possible
to devise a semi-implicit scheme for the shallow-water equations
xiv
. to take longer time steps, but caution is necessary in formulating
difference boundary conditions for the implicit schemes.
In Chapter 20, various problems associated with the selection of
grid structure are discussed. Obviously, from the economy of cal
culations we would like to choose a uniform grid mesh with large
grid increments. However, the size of the grid increments is
determined from accuracy as discussed in Chapter 10. rt is then
attractive to explore the pOSSibility of mesh refinement techniques,
Le., the use of different mesh intervals in different parts of the
region. An example of mesh refinement and the method of examining
computational stability are presented. Even if we can formulate
stable difference .schemes for mesh refinement, the refinement
technique produces undesirable side effects which must be clearly
understood to use this technique advantageously. As the authors
discuss, one serious problem is intrinsic in the technique and
unavoidable. Any wave which is poorly represented in a coarser
grid will change phase speed when passing through an interface
into a finer grid. If this wave later passe,s from the fine grid
back into the coarse grid, a serious interaction can result with
that part of the wave which has remained in the coarse net. In
these situations, a refined area must be treated as a separate
initial boundary-value problem.
Many global atmospheric models use finite-difference approxi
mations over grids defined by intersec·tions of latitude and longi
tude circles. These grids must be modified near the poles for use
with explicit methods because the convergence of meridians, and
hence grid points, approaching the poles imposes a very severe
xv
restriction on the maximum allowable time step through the linear
stability criterion. The restriction is usually avoided by increas
ing the longitudinal grid increment in the neighborhood of the poles
and suitably modifying the finite differences used in those regions.
Using the barotropic vorticity equation, the authors demonstrated
that the change in the longitudinal grid increment causes phase
errors in wave propagation. The distortion of wave patterns due
to phase errors creates an erroneous transport of angular momentum
in the meridional direction. One solution the authors suggest is to
use the constant longitudinal increment throughout the latitudes
and apply a smoothing operator in the longitudinal direction to
relax the linear stability condition of a grid with a constant
longitudinal increment.
Finally, in Chapter 21 difference methods for discontinuities
are presented. Atmospheric fronts often associated with extra
tropical cyclones are a manifestation of discontinuities in the
atmospheric flow. If we use nondissipative schemes for flows
with discontinuities, then noise waves are generated at the dis
continuities and travel in a direction opposite to the character
istics. For the system of equations which has no coupling between
the dependent variables corresponding to the ingoing and outgoing
characteristics (such as the shallow-water equation with no effect
of the earth's rotation), the use of a dissipative scheme '\lith a
higher-order dissipation or with restrictive dissipation only in
the neighborhood of the discontinuities may be used effectively.
However, for the system in which the ingoing and outgoing variables
are coupled through the nature of equations, error in the neighbor
hood of the discontinuity can propagate into that region of the
xvi
domain which can be reached by toe characteristics originating
from the discontinuity line. This means that the accuracy of cal
culations is eventually lost. The authors propose two approaches
to deal with accurate calculations of discontinuities. For the
large-scale motion in the atmosphere, the degree of discontinuity
is relatively small and, therefore, a higher-order dissipation or
selective dissipation in difference equations may satisfactorily
accomplish the calctilationof large-scale flow "With fronts. How
ever ,special consideration should be given in dealing with the
problem of predicting medium-scale motions with the intention of
resolving the frontal structure in detaiL
xvii
SUMMARY
Numerical experimentation and modeling have been emphasized and hold a central
position in the plans of GARP. This monograph discusses approximate methods
for problems in dynamic meteorology and oceanography. Since many of the
systems of equations used in these fields have essentially hyperbolic behavior
the emphasis is on the approximate solution of hyperbolic partial differential
equations.
The first six chapters discuss the initial value problem for ordinary
differential equations. The concepts of stability and convergence are
introduced and several methods are examined and analyzed for computational
efficiency.
Chapters seven through fourteen treat the Cauchy, or initial value, problem
for partial differential equations. A parallel treatment of the properties
of the differential equations and of their approximations is carried out.
Several methods are analyzed for computational efficiency.
Chapters fifteen through eighteen discuss the initial boundary-value problem.
An outline of the theory for the differential equations and their approximations
is accompanied by several examples of its application.
In chapter nineteen the shallow-water equations are discussed utilizing the
concepts developed in the earlier chapters.
Chapter twenty includes discussion of finite difference grids and mesh
refinement.
Problems with discontinuous solutions are treated in chapter twenty-one.
The theory developed for the initial boundary-value problem is used in this
treatment.
xviii
RESUME
L'importance de l'experimentation numerique et de l'elaboration
de modeles a ete maintes fois soulignee et cette branche d'activite est
au centre des preoccupations dans les projets du GAHP. La presente mono
graphie analyse les methodes d'approximation qui peuvent etre utilisees
pour resoudre les problemes de la meteorologie et de l'oceanographie dynamique.
Etant donne que la plupart des systemes d'equations utilises avec ces cham~s
ont un caractere essentiellement hyperbolique on s'est surtout int~resse ~
la resolution approchee des equations aux derivees partielles hyperboliques.
Les six premiers chapitres traitent du probleme des valeurs
initiales pour les equations differentielles ordinaires. Les concepts de
stabilite et de convergence y sont exposes et l'on considere plusieurs
methodes visant ~ ameliorer l'efficacite et la rapidite du calcul.
Les chapitres 7 ~ 14 traitent du probleme de Cauchy, c'est ~
dire des valeurs initiales, poUr les equations aux derivees partielles.
On considere de mIme les proprietes des equationsdifferentielleset des
equations approchees correspondantes. Plusieurs methodes sont analysees
du point de vue de l'efficacite du calcul.
Les chapitres 15 ~ 18 abordent le probleme initial des valeurs
aux limites. Un apergu de la theorie pour les equations differentielles
et les equations approchees est complete par plusieurs exemples d'application
pratique de cette theorie.
Au chapitre 19 on considere les equations pour une masse d'eau
peu profonde en faisant appel aux concepts developpes dans les chapitres
precedents.
Le chapitre 20 considere les grilles aux differences finies et
la question des pas fractionnaires.
Le chapitre 21 est consacre aux problemes comportant des
solutions discontinues. A cet effet, il est fait appel ~ la theorie
elaboree pour le probleme initial des valeurs aux limites.
xix
RESUMEU
En los planes del GARP la experimentacion numerica y la
formulacion de modelos han adquirido gran importancia y ocupan un
lugar preeminente dentro de este programa de investigacion. Esta
monograf!aestudia metodos aproximados para la resolucion de los
problemas que se plantean en los campos de la meteorolog1.a dinamica
y de la oceanograf1.a. Como muchos de los sistemas.de ecuaciones
utilizados en estos campos tienen esencialmente caracter hiperbolico,
se presta mayor atencion a la resolucion de ecuaciones de derivadas
parciales hiperbolicas.
Los seis primeros capItulos tratan del problema del valor
inicial para las ecuaciones diferenciales ordinarias. Se introducen
los conceptos de estabilidad y de convergencia y se estudian y
analizan varios metodos, desde el punto de vista de su ·eficacia para
el calculo.
Los capItulos siete a catorce tratan del problema de Cauchy,
o del valor inicial, con respecto alas ecuaciones de derivadas
parciales. Se lleva a cabo un estudio paralelo de las propiedades
de las ecuaciones diferenciales y de sus aproximaciones. Se analizan
varios metodos, desde el punto de vista de su eficacia para el calculo.
Los cap1.tulos quince a dieciocho tratan sobre el problema
del valor lImite inicial. Se hace una exposicion general de la teorIa
en la que se apoyan las ecuaciones diferenciales y sus aproximaciones,
acompanandola con varios ejemplos de su aplicacion.
En el capItulo diecinueve se estudian las ecuaciones
aplicables alas aguas poco profundas,. utilizando los conceptos
desarrollados en los cap1.tulos anteriores.
El capItulo veinte se ocupa de los retIculos de diferencia
finita y del perfeccionamiento de la malla de dichos ret1.culos.
Los problemas con soluciones discontinuas son estudiados en
el capItulo veintiuno. En este estudio se aplica la teorIa desarrollada
para el problema del valor l!mite inicial.
xx
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1
INTRODUCTION
In this monograph we discuss methods for obtaining approximate solutions to time
dependent differential equations. We focus our attention on methods useful for the
time dependent equations encountered in dynamic meteorology and oceanography and, there
fore, our emphasis is on hyperbolic partial differential equations. The development of
these methods has been intimately connected with meteorology and oceanography. The quest
for efficient methods for these problems continues to be an extremely important problem.
The research problems are ever increasing in scope and complexity, and in the past few
years numerical models have become tools for the daily work of many forecasting groups.
Thus, there is the need to be able to handle larger problems and the need to do this work
economically and quickly is greater than ever.
We have developed many concepts and carried out many analyses by studying model equa
tions. We find that most computational difficulties are linear effects and can be studied
in simple situations where a detailed analysis is possible. We stress the importance of
this technique. An adequate, rigorous .analysis is usually practically impossible for the
large nonlinear models; computationa~ difficulties are apt to be wrongly ascribed. This
can easily lead to a large and incorrect folklore which can steer future research in the
wrong direction. There are certainly many examples of this in the past. This is not to
say that there are no pitfalls inherent in this technique. Great care must be used in
selecting model equations and in extrapolating conclusions to more complicated situations.
However, this is an invaluable tool to isolate and analyze phenomena.
We have developed the theory for differential equations and their approximations in
parallel. This is essential for the development of a meaningful theory for the approxi
mating equations. Unfortunately, many of the results needed from the theory of differen
tial equations have been relatively inaccessible, and the parallel development we feel
necessary is often overlooked.
In chapters one through six we introduce the concepts of stability and convergence by
studying difference approximations for ordinary differential equations. Several difference
approximations are analyzed for computational efficiency. These first six chapters are an
introduction to the concepts, methods and point of view of this monograph. This develop
ment is repeated again and again as we move from the easier to the more difficult problems.
In chapters seven through fourteen we treat the Cauchy, or initial value, problem.
This is the relevant theory for global weather forecasting or climatological problems. The
theory for the Cauchy problem for linear equations is essentially complete and various ap
proximations are relatively easy to analyze and compare. In these chapters we progress
from linear equations with constant coefficients to those with variable coefficients and
finally to nonlinear problems. The aim is always to reduce the analysis of more complicated
equations to that of simpler equations insofar as this is possible.
2
In chapters fifteen through eighteen we discuss the initial boundary-value problem.
This is the relevant theory for ocean circulation problems and weather forecasts over
limited sub regions of the globe. This theory is considerably more difficult than the pre
ceding sections and has only become rather complete in the last few years. Again the ap
proach is from simple to more complex problems.
In chapter nineteen we discuss the shallow-water equations utilizing the concepts
developed in the earlier chapters. It is hoped that this chapter provides good examples
of applications of the principles developed here for the meteorological and oceanographical
communities.
Chapter twenty includes discussions of finite difference grids and mesh refinement;
The earlier chapters discuss resolution requirements and it is natural to consider using
a grid which is denser in areas where the solution is less smooth. We discuss the situa
tions where this procedure is advisable and ana1yze this technique using the theory of the
initial boundary-value problem.
Problems with discontinuous solutions are treated in chapter twenty-one. This develop
ment is again via the theory for the initial boundary-value problem.
We have included proofs or outlines of proofs for most of the results but have left
out proofs of several results which would have required the development of complicated
machinery; for these we refer to the appropriate literature. We have included many examples
which we hope will make the theory and its applications clear.
Our bibliography includes general reference works for the various sections and source
works for the material discussed. We have listed general references pertinent to.particu
1ar chapters in a section preceding the bibliography. We make no pretense of completeness
and refer to the bibliographies of our references for more complete lists, in particular
to Richtmyer and Morton (1967) and Thomee (1969).
3
1. DIFFERENCE APPROXIMATIONS FOR ORDINARY DIFFERENTIAL EQUATIONS
Difference methods for solving th~ initial value problem for·partial differential
equations can be considered as the numerical solution of systems of ordinary differential
equations by difference methods. Many of the relevant properties of these systems can be
demonstrated using the scalar equation
Ly = dy/dt - Ay = aeat
, y(O) = Yo (1.1)
with constant coefficients. For simplicity we consider only the nonresonance case a * A.
Then the solution of (1.1) is given by
(1.2)
with
As usual, we call YI(t) the forced solution and YH(t) the transient solution. In many
of the practical applications that we are concerned with, the solutions of (1.1) are uni
formly bounded for all time. Therefore, we assume that
Real A ~ 0 and Real a ~ 0 (1.3)
We want to solve the above problem using a multistep-method. Therefore, we introduce
a time-step k > 0 and define gridpoints tv
and gridfunctions Vv by
t v vk, Vv = v(tv)' v = 0, 1, 2, . . . . (1.4)
We then approximate (1.1) by
p
L YjVV-j - Akj=-l
p
Lj=-l
(1. 5)
where the Yj are constants, the Sj Sj(k) may depend on k, and gv is an approximation of
aeatv• We assume that Y-1* 0 and therefore, that (1.5) can, for sufficiently small k, be
written in the form
p
Lj=O
Thus we can compute Vv for all v ~ p + 1 if the p + 1 initial values
v v .... v0' l' , P (1. 6)
are given. The initial value y(O) = Yo of the differential equation supplies us with one
value. For p > 0 we need special methods for determining the other values. There are
essentially two techniques for doing this:
(1) Use a special one-step method (a method with p = 0) to compute v1
' •
from Vo y(O). In this case one must be careful not to destroy the accuracy.
discuss this point later.
vp
We shall
4
(2) The differential equation gives us
dy/dtlt=o = Ay(O) + a,
and therefore, by Taylor expension,
02y(O) + O(Ay(O) + a) + "2 (A 2y(0) + Aa + aa) +
(1. 7)
(1. 8)
Then we can compute v., j = 0, 1, 2, .. " P by choosing 0 jk, replacing y(o) by v.,J J
and neglecting the o(or+l) terms. We can of course also use (1.7) to construct a special
one step method for determining v , . . " v. To do this we need only to replace 0 by k,6.Vk P ( r+l) ( r+l)y(o) by vv+l' y(O) by vv ' a by ae and neglect the 0 0 = 0 k terms. Thus lye get
vv+l = Vv + k(Avv
+ aeaVk
}+ ~2 (>,2 VV + (A + aa)eaVk
) + .
Equations (1. 7) and (1. 8) have the san,,,, order of accuracy. In both cases we have
However, the best constant c is generally smaller for the method (1.8).
We shall now define stability for the difference approximation. We consider the
homogeneous difference equation (1.5)
p
~j=-l
y.v . - AkJ V-J
p
~j=-l
o (1.9)
for all initial values vo' • and make
Definition 1.1. The difference approximation (1. 5) is stable if there are constants
o and K, independent of k and of the initial values vo' ..
of (1.9) satisfy the estimate
v , such that the solutionsp
(1.10)
for all t = t ~ pk and all sufficiently small k.V
Let y(t) be the solution of the diff~rential equation (1.1) and substitute it into the
difference equation, i.e., consider the truncation error sv:
p
~.j=-l
y.y . - AkJ V-J
p
~j=-l
ksV
(1.11 )
5
Definition 1.2. The approximation (1. 5) is accurate of order q if there is a function
d(t), uniformly bounded on every finite time interval, and a constant c such that for all k
Iksvl = I~yv - kgvl ~ d(tv)kq+
l, t v
IYj - vj l ~ cjkq
, j = 0, 1, 2,
The approximation is said to be consistent if q > O.
vk,
" p.
(1.12)
(1.13)
We shall now state the main theorem in the theory of difference approximation,
Dahlquist (1956).
Theo~em 1.1. Let y be a solution of the differential equation for which (1.12) holds,
and v be a solution of the difference approximation for which the inequalities (1.10) and
(1.13) are fulfilled. Then we get for all t = tv
(1.14)
c =
with
t/!(t,a) _f!eat ~: :: ~ll-eak
max1cjlj
From (1.14) it follows that stability and consistency imply convergence on every finite
t-interval. Note that the estimate (1.14) is not only valid as k + 0 but holds for every
fixed k. This is important because in actual calculations one is not interested in an
asymptotic error estimate but in an estimate for the k one is using in the computation.
We shall now discuss the estimate (1.14) in more detail. It is, in general, not diffi
cult to determine the truncation error, at least when the solution we are looking for is
smooth. Then we develop YV-j in a Taylor series
jkdy/dtl + (jk)2 d2y/dt21 +YV-j = Yv - t=tv
2 t=tv
and the left side of (1.11) in a power series in k, which, if (1.12) holds; starts with a
term of ~rder kq+l • In most applications that we are concerned with, the solution of the
differential equation and its derivatives are uniformly bounded for all time. Therefore,
we can replace d(t) in (1.14) by a constant d. In this case we have uniform convergence
for 0 ~ t < 00 only if a < 0, and the constant involved is small if there is large damping,
i.e., large IReal AI. If 0'=0 then the error will grow linearly with time and after a
sufficiently large time interval no accuracy remains. When a > 0, the approximation is
only useful for relatively small time intervals. Furthermore, decreasing k does not help
very much, except for the case 10'1 ~ const. k. If Real A ~ 0 only those difference approx
imations for which a ~ 0 are efficient. The influence of the errors in the initial values
is given by KkqcJ(p+l)ea(t-pk). Thus it is necessary to devise special methods for cal
culating the initial values which are so accurate that the inequalities (1.13) hold. Fur
thermore, c must be of the same order as d. There is one exception to this. If we are
6
not interested in the transient solution, and a < 0, then this initial error has no influ
ence after a sufficiently long time.
It is very easy to determine
be solved exp1~cit1y by an ansatz
called characteristic equation
the stability properties of equation (1.9) since it can
v =~A.K.V where the K. are the solutions of the so-V ~ J J J
p
L (Yv - Akf3v ) KP-
V0
v=-l
In the next chapter we give a number of examples of this procedure.
7
2. SOME SIMPLE DIFFERENCE APPROXIMATIONS FOR ORDINARY DIFFERENTIAL EQUATIONS
In this chapter we want to discuss four simple difference approximations for integrat
ing the differential equation (1.1), namely:
avkvv+l = (1 + Ak)vV + kae ,
avk(1 - Ak)vv+l = Vv + kae ,
(1 - tAk)VV+l = (1 + tAk)Vv + kaea(v + ~)k,
avkvv+l = vv-l + 2Akvv + 2kae ,
(Eulers method)
(backward method)
(midpoint role)
(Leap-frog method)
(2.1)
(2.2)
(2.3)
(2.4)
The solutions of (2.1)-(2.3) are uniquely determined by the initial condition
For (2.4) we must specify v1
• We use Taylor expansion and obtain
v 1 = Yo + kdy/dtlt=o = (1 + Ak)yo + ak
(2.5)
(2.6)
We assume that lakl ~ 1, i.e., the forcing function is smooth. We now study several rather
common situations.
(1) Let IAkl ~ 1 and the interval of integration be small. By theorem 1.1 this case
is triviaL
(2) Let IAk I ~ 1 and consider a large time intervaL Here the magnitude of the
damping, IReal AI, is very important.
(3) Let IAkl - 0(1). Here two situations have to be considered:
(a) Let -Real A ~ 1, then the transient solution is decaying rapidly and it is
the forced solution that is of interest. This is typical of control problems.
(b) Let IReal Akl ~ 1 (for example Real A = 0, and IImAkl - 0(1)). In this case
the transient solution YH(t) oscillates rapidly. Again one is not interested in
YH(t). One only wants to compute the forced solution.
We now solve the difference equations explicitly and start with the case where the
forcing function is zero, i.e., a = O. The general solutions of the (homogeneous) equa
tions (2.1)-(2.4) are given by:
(1) V 1 + Ak = Ak-~A2k2 + .(2.la)Vv 'lK1 with K1 e
(2) V _1 Ak+!.:A 2k 2 + • .(2.2a)vv '2K2 with K2 (1 - Ak) e 2
(3) V(1 + A~)(l A~r1 = eAk+X2A3k3 +. .
(2.3a)Vv '3K3' with K =3
Equation (2.4) is a two-step method and therefore its general solution can be written as
(2.4a)
8
where Kif and Klfl are the solutions of the characteristic equation
K2 - 1 + 2AkK = 0
i.e. ,
Kif = Ak +~ = eAk-16A3k3 + •• '.
Klfl
= Ak -JA2k 2+1 = _e-Ak+XA3k3 + ••
(')Now we must determine the constants 'j' j = 1, 2, 3, 4 and 'If 1 in such a way that the vv
J
fulfill the initial conditions (2.5) and (2.6) for j 4. From (2.5) we have
'j = Yo' j = 1, 2, 3
'If and 'If 1 are determined by
i.e. ,
We have determined the solutions of the difference equations explicitly, and we want
to compare them with the solution of the differential equation y(t) = yoeAt
Furthermore,
(2.8)
We have convergence for all the methods. For k sufficiently small, the solution obtained
using the mid-point rule converges fastest, followed by the leap-frog scheme and the im
proved Euler method. These are second order methods while the others are only of first
order.
Case 2. The solution of the differential equation is uniformly bounded. Therefore,
a useful method must not have exponentially increasing solutions. (2.8) implies that
I V I IReal AltKlfl - e V, and we therefore cannot use the leap-frog scheme if Real A < 0,
Dahlquist (1956). We will later show how to modify this method so that its solutions are
nonincreasing. Let A Al + A2
, Aj
real; Al ~ O. For Eulers method we have
Therefore,
9
and the method is useless if Al
point rule we always have
IK.I ~ 1, jJ
2, 3 (2.9)
We obtain from (2.7), for t '" tV
I v Atl IYole-IAlltl 1 - e-P-1A2kt I'2K2 - yoe
and/ V At/ IYo/e-/Al/tl 1 - e- 'i(2Aak 2t I'aKa - yoe
o then the error initially grows linearly and reaches its maximum
Case 3. Eulers method certainly cannot be used for kA I < -2 or IkA 21 > 1 because
then IKII > 1. Similarly, the leap-frog scheme is useless for IkAI > 1. For the backward
method we always have
IK21 < 1 with lim IK21IkAl-+-()
o (2.10)
(2.11)but lim K "'-1IkA/-- a
(2)Therefore, v decays very fast and the backward method can be used. Since it is only a
first order method, it should only be used for small time intervals, after which one should
switch to a second order accurate method (at least). For the mid-point rule we have
{
< 1 if Real A < 0
IK a /
1 if Real A '" 0
(a)Therefore, v may decay very slowly. Later we will discuss methods to overcome this
difficulty.
This concludes the discussion of the homogeneous equations (2.1)-(2.4). Let a * O.
We first determine the forced solution, which is of the form
(") avkw J '" pe' V'" 0 1 2 . ',' J' '" 1,2,3,4V j , ", (2.12)
Substituting (2.12) into the difference equations (2.1)-(2.4) yields:
_o-----=k;.:.:a:o.-__ '" _a_ (1eak _ (1 + Ak) a - A
a2
)2 (a - A) k + .... (2 .lb)
kaeak __a_ (1 _ -::--7a::..2_~(1 _ kA)eak _ 1 - a - A 2(a - A) k +
~akkae
.)
( ( 1 ! )a 2 ~ - A=_a_l_~3. + .
a - A 8 a - A
(2.2b)
(2.3b)
2kaeak 2kaeak _ ZAk _ e-ak (2.4b)
10
It is obvious that
Um sup I YI(t ) - wv(j) I = Uml a Pil =·0k+O o<v<oo V k+Q a. - A -
in all four cases. Observe that we have uniform convergence in time. If we exclude a
neighb orhood of the resonance frequency a. = A (I a. - AI ;;. 0 > 0), then the convergence is
also independent of A and depends only on a.k. As expected, the convergence is much faster
in the last two cases, O(k 2 ), than in the first two, O(k).
The complete solution of our problem is of the form
j = 1, 2, 3
for the first three methods and
for the leap-frog scheme. The coefficients are determined such that Vv fulfills the initial
conditions (2.5) and (2.6). This leads to the equations:
T j = Y0 - Pj .' j = 1, 2, 3
Therefore,
Tj Yo -
__a_+O(h), j 1, 2,a. - A
T. Y -__a_+ O(h 2
) , j 3, 4,J 0 a. - A
T'I1O(h 2 )
Since lim p. = a/(a. - A) and the convergence is uniform for all Awith la. - AI > 0, the conk+Q J
vergence behavior of our four methods is completely described by our discussion of the case
a = O. The most important conclusions we can draw are:
(1) If Real A < 0 then we can achieve uniform convergence for all time.
(2) If Real A= 0 then the transient solution will not converge uniformly· in time.
In fact the accuracy will be completely destroyed.
(3) The forced solution always converges uniformly.
11
3. THE IMPORTANCE OF THE TRUNCATION ERROR AND THE STABILITY DEFINITION FOR ERROR ESTIMATES
In the first chapter we derived an error estimate. One might suspect that this error
estimate is only of theoretical interest. However, if IRea1 Akl ~ 1 and we are interested
in the transient solution, then the inequality (1.14) can be used to describe the behavior
of our examples in chapter 2 rather accurately. On the other hand, if IRea1 Akl = 0(1) and
we are only interested in the forced solution (suppressing the transient solution in some
way), then we often obtain a gross overestimation of the error. At any rate, for the com
parison of different methods the truncation error and the stability constants k,cr are the
most important parameters. We shall illustrate this by deriving error estimates for the
backward method and the mid-point rule for the case Real A = O. The truncation errors are
given by
and
Yv+1 - Yv - AkYv+1 - kaeavk
= ~Z (dZYv/dtZ - lAdYv/dt + 0(k 3)
~Z (_ d2Yv/dt2 + aaeatv) + 0(k 3 )
1 ( ) a( v+J-)kYv+1 - Yv - "2 Ak Yv+1 + Yv - ke '2
(3.1)
(3.2)
It follows from (2.2a) and (2.3a) that IKzl ~ 1 and IK 3 1 = 1. Therefore, we can choose
k = 1 and cr = O. Furthermore, both are one-step methods and Vo = y(O). Therefore, c j = 0
in (1.14). First consider the case where a = O. Then Id2Yv/dtZI ~ IAlz and we get from
(1.14)
Iy(t) for the backward method and (3.3)
These estimates are as good as (2.7).
t v for the mid-point rule. (3.4)
Now let a * 0 and assume that we are only interested in the forced solution, i.e., we
have either choosen the initial values properly or have some other method to suppress the
transient. Then instead of (3.3) and (3.4) we obtain, for t = tv'
(3.3a)
and
(3.4a)
If a and A are not too large, or A < a, the estimates (3.3a) and (3.4a) are in reasonable
agreement with (2.2b) and (2.3b) over short time intervals. If A or t is large, (3.3a) and
(3.4a) are quite bad. Still it is rather obvious that the mid-point rule is much more
accurate than Eu1er's method.
a given,;::-L"'" 100'
12
4. SOME REMARKS ON THE CHOICE OF A DIFFERENCE METHOD AND THE STEP-SIZE
Consider the homogeneous differential equation (1.1) with A 21Ti, yo = 1 and a = O.
Let k = IIN, N a natural number, i.e., we have N points per wave length. We want to com
pute N such that the error at time t = j (after j wave lengths of the solution have passed21T"
point) of either the amplitude or the phase is at mostp%, Le. ,lvjN - e 1J IFor the backward method we obtain from (2.7)
i.e. ,
Therefore, if we allow an error of at most 10% we need 200j points per wave length. This
is, for most geophysical applications, unbearable. In fact all first order methods have
similar properties.
-L = ·1 IN _ 21Tij ·1100 3 e
For the mid-point rule we have instead
-.!(21T) 3 j"" =1,::.2--,,--_
N2
and for the leap-frog scheme
i.e •• N (.2 ) 3 10.0 .1T p.12 J
-L = 1IN _ 21Tijl100 If e
~(21T) 3
~ N2 j, i.e., N = (2 ) 3 100.1T p.lZ J
If we allow an error of at most 10% we must use 15 v"'j'" points per wave length an~ Zl v""Jpoints per wave length, respectively. If we decrease the error to 1%, we have only to
multiply the above numbers by VlOtoobtain the appropriate estimates.
These values are often satisfactory. However, one can easily devise methods for
which the number of points necessary is much less. For example, if we use Simpson' s rule
+Ak3 (vV
- l + 4vv + vv+lO
)vv+l = vV_ l
then the truncation error dB given by
(4.1)
For A 21Ti we have
"I 1 .~ ( 21T 5) \lOo:-VjN - y(j) I = NIf.180 (21T)5, Le., N = 21TJ p' 9
Therefore, we need - 4o.5'j~points per wave length if we allow an error of at most 10%.
Observe that Simpson's rule requir~s no more work for every time-step than the mid-point
rule. It only requires storage for one more time-level.
Another implicit method is given by
(1 -i Ak + li (Ak)2)VV+l = (1 + %Ak + li (Ak)2) Vv (4.2)
~In this case we need 3.3·j points per wave length for an error of less than 10%. This
method requires approximately twice as much work for every time-step as Simpson's rule.
13
Thus the usefulness of the method is not obvious. However, we have for all Ak that
/Yv+1' ~ IYvl, i.e., the method is unconditionally stable. This is not true for ~impson's
rule.
We can also construct explicit methods of comparable efficiency, for example,
((Ak) 3)
vv+1 = vv-1 + 2, Ak + 31 vv
This can also be written as a three-step formula
v - v + 2Akv(2)v+1 - v-1 V
(4.3)
(4.4)
1
In this case we need 5.3·j~ points per wave length if we allow an error at most 10%.
(4.3) requires three times as much work for every time-step as the leap-frog scheme. The
superiority is increased ~f we only a110~ an error of 1% because the above numbers need
only to be multiplied by vTO instead of VIO.
Of course one can construct methods which are even more accurate but, as we will see
later, not much is gained. The fourth order methods we have discussed might be inferior
to second order methods using a smaller time-step. This seems reasonable since they are
either implicit or require computing higher powers of complicated operators. Another
alternative is to use higher order uncentered formulas like Adam's methods. Since these
are not centered, the truncation error is not favorab1e and more time-levels must be
stored. For example, if we consider the explicit fourth order accurate Adam's method, we
need 13.4'j!t; points per wave length, if we allow at most 10% error.
Another way to increase the accuracy is to use Richardson extrapolation. This will
be discussed in chapter 10. We will see that nothing is gained at the 10% error level but
that there is an advantage if we allow at most 1% error.
14
5. THE LEAP-FROG SCHEME
In this chapter we want to discuss the properties of the leap-frog scheme in detail.
We again consider the differential equation
dY/dt = Ay, y(O) = Yo' A =. \ +1A 2; AI' 1.. 2 real
and approximate it by
1 + t Ah
1 - .!. Ah Yo2
We know that. the solution of (5.2) can be written in the form
with
(5.1)
(5.2)
(5.3)
and
We assume that we want to integrate (5.3) over a long time interval and that
moderate size, i.e., IAkl ~ 1. Furthermore, we assume that Al Real A < O.
and we cannot use the leap-frog scheme directly. Let y = - Al + iY2' Al and
constant and substitute the new variable y = eyty into (5.1). Then y is the
which we approximate by
Vv+1 = v 1 + 2ki(A + Y )vv- 2 2 V
We obtain an approximation of (5.1) by setting v = eytvV V
+ykLet us now replace e- by 1 ± yk. Then we have
A is of
ThenlK 1>1'+1
Y2 real, be a
solution of
(5.4)
(5.5)
which is second order accurate. Its general solution is again of the form (5.3) but now
To see this we need only substitute
v =V (1-Yk)~
1 + yk Wv
into (5.5) and obtain
wV
15
The same procedure can be applied to the approximation (4.3). Then we have
((A + YYk2
)eYkVv+l e-ykvV_
l+ 2i(A 2 + Y2)k 1 - . 2 6
2Vv
+ykIf we again replace e- by 1 ± yk then
Though this approximation is only second order accurate, it can be useful if y ~ A. A
fourth order accurate approximation is given by
(y2k2 y3 k 3)
1 + yk + 21 + 31 vv+1
k< u*u > 2;;>0.
16
6 . NOTATION AND ELEMENTARY THEOREMS
If x is a real number we denote its absolute value by Ixl. ~f x = a + ib, a and b
real, i 2 = -1, is a complex number we denote its complex conjugate by x = a - ib and its
absolute value by Ixl = (a2 + b2)~.
Now consider vectors:
u C'). v = G)n n
where the U., v., j = 1, 2, .•. n, are complex numbers. The scalar product, < u,v >,J J
and norm, lul, (length of the vector) are defined by
n
< u,V > = LUjVj , lul =j=l
Note that the symbol ,., is used in three different ways. This should not cause any con-
fusion, since the correct interpretation will be clear from the context. The following
results are well known:
If A
Ixul~lxl 'Iul (x complex, u a vector),
lu + vl~lul + Ivl (Triangle inequality),
1< u,v >1~lul'lvl (Cauchy-Schwarz inequality).
(a ij ) is a n x n matrix,
_ (all' .. all1)A - ~2l a 2n
a anl nn
(6.1)
then its adjoint, A*, is given by
_ (~11a2l •.._anl)A* - ~12"" am
- -a a.In nn
A matrix is called Hermitian if A
unit matrix
I
A* and unitary if AA*
(10... 0)o 1 .• 0
001
A*A I, where I denotes the
The eigenvalues, A, and the eigenvectors, u, of a matrix are the nontrivia1 solutions of
Au = Au.
17
The norm of a matrix A is defined by
sup IAul.lul=l
The following results are well known for matrices A,B and vectors u,v.
(AB)* = B*A*,
< u,Av > = <A*u,v >,
1< u,Av >1.;;;;IAI·lullvl·
(6.2)
Let A be hermitian. Its eigenva1ues are real and there is a unitary matrix u which trans
forms A to diagonal form.
U*AU
where the A. are the eigenva1ues of A and the column vectors of u are eigenvectors. ForJ
all matrices A,B and vectors u we have
IAul.;;;;IAI·lul,
I(A + B)ul.;;;;(IAI + IBI)lul, (6.3)
IABul.;;;;IAI·IBI·lul,
and one can show that
IAI 2 = largest eigenva1ue of A*A.
Therefore, IAI = 1 if A is unitary.
If,u,v are vector functions depending on xl' ... , Xs then we define the L2-sca1ar
product, (u,v) and norm, I lul I, respectively, by
+00 +00
(u,v) f .f u*vdx l • .. dXs ' Ilull (u,u)\_00 _00
The following relations are well known
(u,v) = (v,u),
(u,Av) = (A*u,v),
I (u,v) 1.;;;;lul·lvl,
(u,3v/3xj) = - (aU/3Xj ,v).
(6.4)
Let u be a sufficiently smooth function. The Fourier transform u of u is given by
Fu
where w
-8/2 f +00 -i< w x >u(w)" = (21f) e' u(x)dx_00
... , ws
) denotes the (real) dual variable to x and dx
The following well known results are crucial.
Theorem 6.1. (Fourier inversion theorem).
Hu Fu then uCx)
18
Theorem 6.2. (Parseval's relation).
Theorem 6.3. (Multiplication theorem).
We now define several difference operators. We begin with the translation operators
Ej(h). h>O •
. Eju(x) = U(X 1, ••• , x j _1 ' x j + h, x j +1 ' .... , xn}. j =1,2, ••• n.
The translation operator has the following property:
(EiEj)U(X) = Ei(EjU(X)}.
Therefore
We can now define
where r E~ is the unit operator.J
h- 1 (Ej - I).h-1(r _ Ejl),
2h-1(Ej - Ejl) = t~+j + D_j ).
19
7. WELL-POSED CAUCHY PROBLEMS
Consider the Cauchy problem for a linear system of partial differential equations:
au/at = P(x, a/ax)u,
u(x,O) = f, ~<x <00 jj , 1,2, ... , s,
(7.1)
(7.2)
P. (x,a lax) = ~ A (x)a lvI/axv1
J ~_ V 1
Iv I=j
functions depending on x
numbers, Iv I =L V j '
m
p(x,a/ax) = '2)j(x,a/ax),
j=O
where
are vecto.r
Vj natural
(
u (1) (x, t))u = u(x,t) = : ' f
u(n)(x,t)
(Xl' ... ,
(~(1) (X))
= f(x) = :fen) (x)
xs ) and t.
axVss
is a differential operator with smooth matrix coefficients.
(7.1) and (7.2) do not always behave like physical processes. Consider, for example,
the system
(7.3)
with initial values
(7.4)u(x,O)-~f +R iwx A
= (2~) e f(w)dw,
-R
where R is some constant. Then the solution of (7.3), (7.4) can be written in the form
u(x,t) f+R
-k iWXA= (2~) 2 e u(w, t)dw.
-R
(7.5)
Substituting (7.5) into (7.3) gives us, for every frequency w,
du(w,t) ( 0 1)dt = iw -1 0 u(w,t), u(w,o)A
few) • (7.6)
The solution of (7.6) is
(7.7)
where the constants A.J
A.(W) are determined from the equation
J (i(1) (w):)Al(_~) + A2(~) = few) = f t2 )(w) .
(7.7) shows us that the solution of (7.3) and (7.4) can grow like eRt
(7.8)
R is arbitrary.
Therefore, the Cauchy problem (7.3) has solutions which grow arbitrarily fast. Thus,
(7.3) and (7.4) do not behave like any physical system. Instead consider
20
·with the same initial values (7.4). The solutions can again be written in the form (7.5)
where nowu(w, t)
Le. ,
and, therefore,
Parseva1's relation gives us, for every fixed t,
Ilu(x,'>lI' • j"'"'-00
In this case the L2
norm is constant, i.e., if the initial values are small then the solu
tion is small for all time. This is the sort of behavior we expect for a physical system.
We define
Definition 7.1. Consider the problem (7.1), (7.2) for all initial values f with
1 If 1 1<00. The problem is well posed if there are constants K,a (independent of f) such
that for all solutions and all t an estimate
holds.
Ilu(x,t) II~eatllu(x,O)11 (7.10)
(7.11)
Our first example does not fu1fi11 (7.10) because we cannot find an a which holds
uniformly for all solutions. The second example is well posed because (7.10) holds with
K = 1 and a = O.
Now consider the Cauchy problem for systems
m
ou/at = P(%x) u = ~Pj(%x)uj=O
with constant coefficients. It is easy to derive algebraic conditions for this problem
to be well posed. Let
u(x,O) =>....
f(w)dw. (7.12)
The solution is of the form
u(x,t)
-too-"'2! i< w x >....(21f). " e ' u(w,t)dw.
-00
(7.13)
21
Substituting (7.13) into (7.11) gives us, for every frequency,
du~~,t) = P(iw)u(w,t), u(w,o), = f(w).
By theorem 6.3
(7.14)
(. )\is• • • • 1.WS •
The solution of (7.14) is given by
u(w,t)
and, therefore, by theorem 6.1
u(x, t) = 2.)-'" j"'"'e i < W,X >.. (iw)t, (w)dw.
_00
Parseval's relation implies+00
Ilu(x,t) 11 2 =j leP(iW)tf(w) 12dw_00
+00
~ m~xleP(iw)t12 f If(~v) 12dw
_00
We now state
(7.15)
Theorem 7.1. For the system (7.11) the Cauchy problem is well posed if and only if
there are constants K and a such that
I P(hv)tl'""'v atmax e ~e.
w(7.16)
Proof: If (7.16) is fulfilled then it follows from (7.15) that also (7.10) holds,
i.e., (7.16) is a sufficient condition for well posedness. We shall not prove that
(7.16) is also necessary.
We now discuss hyperbolic systems. We begin with
Definition 7.2. The system (7.11) is hyperbolic if m = 1 and for every w there is a
nonsingular matrix T such that
and
T(w)P I (iw)T- I (w) iA .(~~. 0)1. • , A. real
o A Jn
(7.8) is an example of a hyperbolic system because the eigenva1ues of
P(iw) = PI (iw) = iw (~ ~) are Al = iw, A2 = -iw.
Furthermore, P(iw) can be transformed to diagonal form by a unitary matrix, i.e., ITI
IT-II = 1.
22
The linearized shallow water equations are another example:
a (U) (U 0 1) a (U) (V 0 0) a (U) (0 -f O)(U)- a- V = 0 U 0 a- v + 0 v 1 a- v + f 0 0 vt ~ ~ 0 U x ~ 0 ~ v y ~ 000 ~
These equations can easily be transformed to a symmetric system by introducing the new
variable
which gives us
(u 0 ~~) a (U ') (V 0 01? a (U ') (0 -f O)(U ')= 01 U 0 a- v' + 0 V1 ~'2 a- v' + f 0 0 V t •
~'2 0 U x ~t 0 ~'2 V y~, 0 0 0 ~'
For the last equation we have that P(iw) = is(w) where S(w) is a symmetric matrix. Thus
P(iw) can be transformed to diagonal form by a unitary transformation, and the eigen
values of P(iw) are imaginary. Thus the system is hyperbolic.
We now prove
Theorem 7.2. If the system (7.11) is hyperbolic then the Cauchy problem is well
posed.
Proof: Consider the system (7.14). Let T be the transformation of Definition 7.2,
and introduce the new variable
V Tu.
Then we have
and therefore
d IvAI2 = i- < A A > = < dv A >dt dt v,v dt'V
< A dv >v, dt
(7.17)
Using 6.2 we get
< iAv,v > + < v,iAv > = 0,
1< TPoT-1v,v > + < V, TPoT-1v >1<2ITI·IT-11·lpol·lvI2.
(7.17) and definition (7.2) imply
and therefore
lu(w,t)1
23
The last inequality implies
I P(iw)t I"'K2 ate "" le
which concludes the proof.
We now briefly discuss parabolic equations.
Definition 7.3. The system (7.11) is parabolic if m = 2p is an even integer and if
there is a constant 0 > 0 such that the eigenva1ues Kjm)Of Pm(iw) satisfy the inequality
Real K(m)~ - ow2P • (7.18)j
The simplest example of a parabolic equation is the heat equation
aula t = a 2 u/ax2 •
In this case P(iw) = P2
(iw) = _w2 and' /eP(iw)tl = e-wZt~l.
Therefore, the Ca\)chy problem is well posed. The same is true if we add lower order
terms. We consider au/at = aZu/axz + aau/ax + bu a,b complex numbers. Then
P(iw) = _wz + aiw + b
and
and the Cauchy problem is again well posed. In fact, this is always true.
We state without proof
Theorem 7.3. For parabolic systems (7.11) the Cauchy problem is always well posed.
Finally, we state a general negative result.
Theorem 7.4. Assume that there is an eigenva1ue K(m) of P (iw) withj m
Real K(m) > 0j
(7.19)
for some frequency w. Then the Cauchy problem for (7.11) is not well posed.
We can use equation (7.3) as an example. Then
P(iw)
and its eigenva1ues are given by
Al = w, Az = -wo
Therefore, by theorem 7.4, the problem is not well posed.
We have, up to now, only discussed equations with constant coefficients. Now con
sider the equation (7.1) with variable coefficients. We obtain a family of equations
with constant coefficients by freezing the coefficients, i.e., we consider
P(xo,a /ax)u (7.20)
for every fixed x o• The behavior of the equation (7.1) is essentially described by the
behavior of the family (7.20). To make this precise we state
24
Theorem 7.5. If there is a Xo such that (7.19) holds, then the Cauchy problem (7.1)
is not well posed.
Theorem 7.6. If all the systems (7.20) are parabolic and the constant 0 in (7.18)
is independent of x, then the Cauchy problem (7.1) is well posed.
Theorem 7.7. If all the systems (7.20) are hyperbolic and the matrices T w,xo of
definition 7.2 are smooth functions of wand x o' then the Cauchy problem (7.1) is well
posed.
If the coefficients of (7.1) are also smooth functions of t, then analogous results
hold.
It should be pointed out that this reduction to the case of constant coefficients
does not always yield the desired results if the equations are not hyperbolic or para
bolic. Consider, for example, the equation
au/at = ia(x)u + i(da/dx)u , a(x)~a > O.xx x 0
The Cauchy problem is well posed because
If we freeze the coefficients, i.e., consider
au/at = ia(xo)uxx + i(da(Xo)/dX)Ux
then
P(iW,X O) = -ia(xo)wZ
- (da(xo)/dX)W
and eP(iW,Xo)t is obviously not bounded if da(xo)/dX ~ O. Thus, if a(x) ~ constant, then
not all the members of the associated family of Cauchy problems are well posed.
One can also construct examples where the associated family of Cauchy problems is
well posed, but the variable coefficient problem is not.
For nonlinear equations no general global results are known. We state only
Theorem 7.8. Consider a system of the form
aula t
s~
= ) A.(x,t,u)au/ax.L.J J Jj=O
+ B(x,t,u)" (7.21)
where A.(x,t,u) are symmetric matrices depending smoothly on x,t, and u. Then theJ
Cauchy problem is well posed in a sufficiently small interval ~t~T which depends on the
smoothness properties of u.
For special equations there are better results. See O. Ladyzhenskaya (1963) on the
Navier-Stokes equations, and the results of Aronson and Serrin for parabolic equations
(Aronson and Serrin, 1967).
For many problems the estimate (7.10) can be derived directly. We start with
Lemma 7.1. Let A(x, t) be a matrix; then
25
Proof: Partial integration gives us
+00-f u*3A/3x.vdx.J J
_00
and (7.22) follows easily.
We now introduce the following concept.
Definition 7.3. A differential operator P(x,t,%x) is half bounded if there is a
constant a such that for all t and all sufficiently smooth functions w
(w,P(x,t,3/ox)w) + (P(x,t,%x)w,w)~2a(w,w).
Theorem 7.9.
estimate
If the differential operator P is half bounded then we have the
Ilu(x,t) II~eatllu(x,O)11
for the solutions of the corresponding Cauchy problem.
Proof:
i.e. ,
which proves the theorem.
Examples. Let
P(x,t,%x) = A(x,t)3/ox + B(x,t)%y + C (7.23)
where A = A*, B = B* are smooth sYmmetric matrices. Then the operator P is half bounded.
This follows directly from lemma 7.1.
Let
P(x,t,%x) = A(X,t)o2/ox2 + Ba/ox + C,
where A is positive definite, i.e., there is a constant 0 > 0 such that
Then P is half bounded. By lemma 7.1
-(ow/h, (A + A*)3w/ox)
- (w,oA/oxow/'dx) - ('dA/oxow/ox,w)
~ -20 Ilow/oxl12 + 2113A/oxll·llwll·ll ow/axll
26
Therefore
(w,Pw) + (Pw,w)< - ollow/oxll Z + 21IB/I'llwll'll ow/oxll
+ (211CII + o-llloA/oxll) IIwl1 2
Now assume that the Cauchy problem is well posed for the system
ou/o t = P(x, t,o /ox)u.
Then we can solve it for any time interval t1<t<t z provided u(x,t 1) is given. He can'
write
S, the solution operator, has the properties
S(t, t) = I, S(t 3 ,tJ = S(t3,tJS(tz,t 1),
Ils(tZ,t 1) II<r<ea(tz-tJ
(7.24)
v(x,t)
all of which are obvious consequences of well posedness.
Consider the inhomogeneous problem
av/a t Pv + F(x, t)
v(x,O) = f(x).
He want to express its solution using the solution operator S.
Theorem 7.10. The solution of (7.25) can be written
t
S(t,O)f(x) +f S(t,T)F(x,r)dT
oand the following estimate holds:
atIlv(x,t) II<r<eatllfll + Kmax IIF(x,T) II~
D<T<t a
(7.25)
(7.26)
Proof·: Let v be defined by (7.26). Then v(x,O)
(7.25). Formally we get from (7.24)
f(x) and we show that v satisfies
and therefore,
oS(t,T)at Hm
fIt-+O
Hml'It-+O
S(t+~t,t) - S(t,T)fit
S(t+~t,t) - I S(t,T)M
P(X,t,%X)S(t,T) .
t
ov Jat = p(x,t,%x)(S(t,O)f(x) + S(t,T)F(X,T)dT) + F(x,t)
°= P(x,t,3/3x)v + F(x,t).
The estimate (7.27) follows from (7.26) and (7.24).
27
8. STABLE DIFFERENCE APPROXIMATIONS FOR THE CAUCHY PROBLEM
In this chapter we develop a theory for difference approximations of the Cauchy
problem. Let us start with the following simple example:
au/at = au/ox, - 00 < x < 00
(8.1)u(x,O) = f(x)
If
then it is obvious that the solution of (8.1) is given by
/
-+00-!" iw x + t Au(x, t) = (21T) 2 e ( ) f (w)dw
_00
f(x +t).
(8.2)
Let us now approximate (8.1) by a difference equation. We introduce a time step k > 0
and a mesh interval h > 0. Then we approximate (8.1) for x = Xv = vh, v = 0, ±l,
±2, •• , t = t~ = ~k, ~ = 0, 1, 2,. by
v(x,t+k) - v(x,t) v(x+h,t) - v(x-h,t)k = 2h
(8.3)v(x,O) f(x)
which can also be written in the form
v(x,t + k) = (I + kD o) v(x,t), v(x,O) = f(x). (8.4)
We can use (8.3) to compute v(x,t) for all x = xv' t = t~. Hopefully, v converges
to u. Let us compute the solution of (8.3) explicitly. From (8.2) it follows that
Observing that
v(x,t) f-t<x>
_1 iWXA(21T) ~ e v(w, t)dw,
_00
v(w,O) f(w). (8.5)
i sin whh
iwxe
we get, by substituting (8.5) into (8.4),
v(w,t + k) = (1 + iA sin wh)v(w,t), A k/h,
i.e. ,
Therefore,
v(w, t)t/kA
(1 + iA sin wh) f(w).
v(x, t)
-t<x>
-~f t/k iwxA
(21T) (1 + fA sin wh) e f(w)dw.
_00
(8.6)
28
Thus
u()(,t) - v(x,t)
and by Parseva1's relation
wh) t/k_ iWt) iwx"s~· e e .f~)~,
+00
Ilu(x,t) - v(x,t)112 =/ 1(1 + iA sin wh)t/k_eiwtI2If(w)12dw.
_00
Let w be a fixed frequency. Then
t/k iwtand therefore, (1 + iA sin wh) converges for every fixed w to e
·Most physical processes can be well described by a finite number of frequencies, or
better, by initial values f(x) for which f(w) =0 for Iwl > R where R is some fixed num
ber. Then
j+R
Ilu(x,t) - v(x,t) 11 2= 1(1 + iA
-R
and we have convergence as k + 0, h + O. Thus there is apparently no trouble in using
(8.3). However, we cannot compute without error, and the rounding errors will intro
duce higher frequencies. Consider, for example, (8.4) with k = 20h and f =O. Then
v(x,t + k)
v(x,O)
v(x,t) + 10(V(x + h,t) - v(x - h,t»)
o
and v(x,t) =O. Assume now that we make, for x
the initial values v(x,O) =0 by
0, a rounding error E, i.e., we replace
o for x 'f 0
v(x,O)
E for x = 0
The disastrous influence of this rounding error can be seen from
o
o
o
o
10E
o
E -lOE
E 0 o
o
o
o
o
The reason for this behavior is that this error has introduced high frequencies Ivhose1T .
amplitudes grow very fast. For example, for wh = 2 we have
v(w,t) = (1 + i20)t/kf (w).
Thus, the assumption that the very high frequencies are not present is not realistic.
We have to allow general initial values, and only those methods which guarantee conver
gence for all initial values are useful for computational purposes.
29
A necessary and sufficient condition is that we have convergence for the low fre
quencies and that the high frequencies are essentiall~ not amplified. This can"for
example, be achieved by replacing (8.4) by
v(x,t + k) = t (v(x + h,t) + vex - h,t») + kDov(x,t),
which we consider for
Here Ao is some constant. Then
k/h";;l.
v(w, t)
It is obvious that
t/k"(cos wh + iA sin wh) few).
and
t/k ( ) t/k iwt w2h2tlim(cos wh + iA sin wh) = 1 + ikw + O(w2h2) - e + ---k--k+Q
for every fixed frequency w. Thus we get, for every R,
+R
Ilu(x,t) - v(x,t) 112 =)( 1(cos wh + iA sin wh)t/k_eiwtI2If(w)12dw
-R
+ 11 (cos wh + fA sin wh) t/k_eiwt12 1f (w) /2 dw
Iwl;;;'R
+R
,,;; const (R~htf 1i (w)12dw +
-R
and the convergence is obvious.
We will now make our preceding remarks precise by introducing the concepts of sta
bility and consistency. The most general homogeneous difference approximation is of the
formP
Q_lv(x,t + k) = ~Qjv(x,t - jk).
j=O
(8.7)
Here the Q11
Q11 (x, t, h,k), 11 = - 1, 0, . ., p are difference operators,
where the Av(x,t,h,k) are matrices which depend sufficiently smoothly on x, t, h, k, and
the E. are translation operators,J
• • , x j ' . . , .• , x. + h, ... , x ) .J s
We shall always that -1 exists for all t and is bounded operator. Thus, ifassume Q-l a we
know v(x,t) for to, to - k~ , to ,- pk, then we can compute v(x.t) for all later
30
times t -- to + Vk, v =1, 2, . • . • Let us introduce the vector
;(x,t) = (v(x,t), v(x,t - k), ••..• , v(x,t - pk»)'and the norm
p
11;11 2 = L:11v(x, t - jk) 112
•
j=O
We can find a solution operator S(t,t o) such that
v(x,t) = S(t,to);(x,t o)'
Here S(t,t o) has the following properties:
We now state
Definition 8.1. The difference approximation (8.7) is stable for a sequence k + 0,
h + 0 if there are constants as' Ks such that for all to' t with ~to and all v(x,t o)
the estimate(8.8)
holds.
Now consider the Cauchy problem for a system of differential equations
We approximate this system by
au/at
u(x,O)
p(x,t,a/ax)u + G,
g.
with initial values
p
Q_1w(x,t + k) = L:Qjw(x,t - jk) + kF(x,t)
j=O
(8.9a)
We now make
w(x,pk) fp(X), ••. , \v(x,O) f 0 (x) • (8.9b)
Definition 8.2. The difference scheme (8.9a), (8.9b) is accurate of order (ql' q2)
for the particular solution u(x,t), if there are constants c.;;'O and a function C(t),J
bounded on every finite interval [O,T], such that for all sufficiently small k,h
p
IIQ_1u{x,t + k) - L:Qju(x,t - jk) - kFII~kC(t)(hql + kq2
)
j=O
Ilfjex) - u(x,jk)II~Cj(hql + kq2).
(8.10)
If (8.10) is valid for all sufficiently smooth solutions, then we say that the approxima
tion is accurate of order (ql' q2) without reference to a special solution.
It is quite simple to decide the accuracy of a given difference approximation, at
least when the solution is sufficiently smooth. Then one needs only to develop u(x,t)
31
into a Taylor series. Therefore, the function C(t) represents, in general, some bound
for sufficiently high derivatives of u. It should be pointed out, as we have already
seen for ordinary differential equations, that for a correct evaluation of a given
method, it is also necessary to compute C(t).
To derive an error estimate we need
Lemma 8.1. Consider (8.9) for fixed k and h and assume that the inequality (8.8) is
valid. Then
(8.11)
where
l' °0.
Proof: It is easily seen that we can write the solution of (8.9) in the form
w(x,nk) S(nk,O)w(x,O) +n-l
K~S(nk,Vk)Q-1G(x,Vk),'LJ -1
V=O
(8.12)
where G(x,Vk) = (F(x,Vk,O, , 0»'. «8.11) is nothing but the difference analogue
of Duhamel's formula). Then the estimate (8.10) follows without difficulty from (8.8).
From lemma 8.1 we at once obtain the
Theorem 8.1. Assume that for fixed k and h the estimates (8.8) and (8.10) are
valid. Then we get for all t = V~k
II-;:;(x,t) - ;(x,t) II";;;KseCl.st(hq1 + kq2)(tcf~j=O j)
+ K sup IIQ-1 11· sup IT' (hq1 + k
q2) ~ ft,Cl.s)~
so,,;;;~t -1 o,,;;;T<t \ ~ 'J
Therefore, if (8.8) holdS for a sequence h 7 0, k 7 ° and Q=~ is uniformly bounded for
this sequence, then w(x,t) converges to u(x,t).
Proof: From (8.10) it follows that
p
Q_1u(x,t + k) = ~Qju(x,t - jk) + kF + kH(x,t),
j=O
where for H, the truncation error, the estimate
IIHII";;;C(t) (hq1
+ kq2
)
holds. Subtracting (8.9a) from (8.13) gives us a difference scheme for v
the estimate follows from lemma a.l.
(8.13)
u - wand
32
We have seen that, as for ordinary di·fferential equations, it is the truncation
error and the stability properties represented by as and Ks
which determine the speed of
convergence.
Let us now again consider
ou/ot
u(x,O)
P(x, t.,o lox)u
f
and assume that it is well posed., Le., there are constants a and Ksuch that
Assume that a is as small as possible. If we want to integrate the differential equa
ti.on for a longtime it is ·desirable that a <!ia.. Therefore,we makes
Definition 8.2. The difference approximation is strictly stable if (8.8) holds
with a <a.s
We :will be primarily in.terested in st.rictly stable methods.
33
9. DIFFERENCE APPROXIMATIONS FOR HYPERBOLIC SYSTEMS
Consider the Cauchy problem for a hyperbolic system with constant coefficients:
au/at = p(a/ax)u =~.au/ax.,L:JJ J
(9.1)
u(x,O) = f(x).
We know from chapter 7 that its solutions satisfy the estimate
Ilu(x, t) 11,qzllu(x,O) 11, ° < t < 00.
Thus the solutions of (9.1) are uniformly bounded. Approximate (9.1) by the difference
scheme
y(x,t)
p
Q_1v(x,t + k) = ~Qjv(x,t - jk).
j=O
Kt/kei\VX~(w) ~ °is a solution of (9.2) if ~, K satisfy the characteristic equation
where
(9.2)
(9.3)
(9.4)
w.h.J
From (9.4) we immediately get the
Theorem 9.1 (Von Neuman condition). A necessary condition that the difference
approximation (9.2) has no exponentially growing solutions is that the solutions Kj
of
satisfy the inequality
lA p+lDet Q_1K ° (9.5)
This condition is not sufficient.
and approximate it by
IK .1,,;;;1.J
Consider, for example, the differential equation
Let k hand
V(w,t) is the solution of
v(x,t) i\VXA ( ) 7.e v w,t r 0.
V(w,t + k)
34
i.e. ,
v(w,t) (I
- (I
_ 4 . 2. wh .(0 1))t/kA( 0).S1n 2 0 0 v w,
4t . 2. wh (0 1)) A ( 0)- k S1n 20 o. v w, •
It is obvious that these solutions are not bounded for all time.
General necessary and sufficient conditions for a difference approximation to have
no growing solutions are very complicated. However, there is a class of difference
approximations for which these conditions are relatively simple. These are dissipative
approximations, i.e., for the solutions Kj
of (9.5) there is an estimate
IK.I,,;;; 1 - olwhl2.r
foro-;;lwhl";;;1T·J
o > 0 is some constant and 2r, r a natural number, is called the order of dissipativity.
Let us consider some examples.
Example 1. Consider the hyperbolic system
ilu/ilt = Ailu/ilx,
where A is a constant, symmetric matrix, and approximate it by
v(x,t + k) = (1(1 + khO"D+D_) -i- kADo)V(X,t) = Qov(x,t),
where 0" > 0 is a constant.
A is symmetric so there is a unitary matrix U such that
(9.6)
(9.7)
~) *• ,U U
)1n
I, )1j real.
Therefore, the solutions of the characteristic equation (9.4),
KI<j>
are given by
If we choose A
(1(1 - 40"A sin2.~/2) + iAA sin ~)<j>, ~
k/h and 0" in such a way that
wh, A k/h,
I)1.1 A < 20"/ 1)1.lq, jJ J
then there exists a constant 0 > 0 such that
1, 2, ... n,
1K. 1";;;1 - 0 1~ 12., for r~ 1";;;1T·J
The approximation is dissipative of order 2.
Example 2. (Lax-Wendroff method). The accuracy of (9.7) is generally only (1,1).
We proceed to derive a method of higher accuracy. For any sufficiently smooth solution
35
of the differential equation we have
u(x,t + k) = u(x,t) + kou(x,t)/Ot + (kZ/2)OZu(x,t)/ot'Z + O(k 3)
u(x,t) + kAou(x,t)/ox + (kZ/2)AZOZU(X,t)/oxZ + O(k 3)
u(x,t) + kADou(x,t) + (kZ/2)AZD+D_U(X,t) + 0(k 3 + kh z + kZh).
v(x,t + k) = (I + kAD o + (kZ/2)AZD+D_)V(X,t)
is therefore accurate of order (2,2). In this case
Kj
and if A is nonsingular then
The approximation is accurate of order (2,2) and dissipative of order 4.
Example 3. (Leap-frog). We approximate (9.6) by
v(x,t + k) = v(x,t - k) + 2kAD v(x,t),o
which is accurate of order (2,2). The characteristic equation (9.4) is
I(KZ - 1) - 2KAiA sin ~ = O.
Therefore,
(KZ - 1) - 2 iKA~. sin ~ 0,J
and
and
IK.I = 1 for Amaxlll.l~l.J J J
(9.8)
Thus the approximation is not dissipative. We can, however, easily make it dissipative
by changing (9.8) to
then
v(x,t + k) (I - E ~: D~ D~)V(X,t - k) + 2kAD ov(x,t), (9.9)
IK. I = 1 - E sin4~/2 for IAI~l - E, E <1.J
The approximation is dissipative of order 4 and still accurate of order (2,2).
It should be pointed out that there are quite a number of ways to make the approxi
mation dissipative. For example, we could have replaced (9.8) by
(9.9a)
Then the approximation is dissipative for all E > O.
36
Example 4. We approximate (9.6) by
v(x, t + k) = v(x,t - k) + 2kA(i Do (h) - t Do (2h)) v(x, t), (9.10)
which is accurate of order (4,2). Then for sufficiently small A, we again have IK.I = 1J
and the approximation is not dissipative. Consider instead
which is also accurate of order (4.2) but dissipative of order 6.
The Lax-Wendroff scheme is dissipative. However, the amount of dissipation can only
be coritrolled by changing h, the grid interval. This is an impractical procedure. It is
generally much better to add a dissipative term to a scheme which is not dissipative. In
this case the amount of dissipation can be controlled.
The importance of dissipativity lies in
Theorem 9.1. Let (9.2) be an approximation which is accurate of order 2m - 2 or
2m - 1 and dissipative of order 2m. It is strictly stable.
Similar results hold for equations with variable coefficients.
In most applications one can prove the stability of the difference approximation
using another method, namely by the energy method. We now prove a rather general sta
bility result for approximations of the form
(I - kQJV(t + k) = 2kQ ov(t) + (I + kQl)v(t - k). (9.12)
Theorem 9.2. The approximation (9.12) is stable if there is a constant n such that
for all v,w
(v,Qow) = - (Qov,w), kllQo 11 < 1 - n.
Real (v,Q1v)"';o.
Proof: Write (9.12) in the form
v(t + k) - v(t) = 2kQ ov(t) + kQ 1(v(t + k) + v(t - k)).
After multiplication withv(t + k) + v(t) we get·from (9.14)
(9.13) implies
L(t + k) = Ilv(t +k)11 2 + Ilv(t)11 2 - 2k Real (v(t + k),Qov(t))
".;llv(t)11 2 + Ilv(t - k)112 - 2k Real (v(t),Qo v(t - k)) = L(t),
and
Therefore, the approximation is stable.
(9.13)
(9.14)
37
Many conunon1y used difference methods can be written in the form (9.12). To illus
trate this technique we again prove the stability of the approximation (9.9a). We begin
with the
Lenuna 9.1.
E.U(X ,J 1
Therefore,
Proof: Integration by parts gives us
The second statement follows from the relations
2hD Oj
For (9.9a) we have
By lemma 9.1
E. - 1.J
ADO'
Furthermore,
max kllAD vii,;;; max kI1~llllv(x + h) - v(x - h) 1IIlvll=l 0 Ilvll=l
Therefore the approximation is stable for
38
10. ON THE CHOICE OF A DIFFERENCE SCHEME
In this chapter we ,~ant to discuss different methods of integrating the scalar
equation
which has the solution
/ /2TIiwxau at = - cau ax, u(x,O) = e ,
u(x,t) = e 2TIiw(x - ct).
(10.1)
We ignore any errors due to discretization in time, Le., we consider the differential
difference equation
- cD o(h)v(x, t)aat vex, t)
which has local truncation error O(hZ).
i2TIWXIf v(x,O) = e then (10.2) has the solution
where
(10.2)
(10.3)
The phase error, el' is
Cl (w) = csin 2TIWh
2TIWh(10.4)
A fourth order approximation is
e (w)1
2TIwt(C - Cl (w») " (10.5)
If, as before, v(x,O)
where
~t v(x,t) = - c(~ Do(h) - ~ Do(2h»)V(X,t).
i2TIWxe then (10.6) has the solution
__ e i2TIW (X - cz(w)t)vex, t)
(10.6)
(10.7)
The phase error, ez' is
c (w)zsin 2TIwh - sin
12TIWh(10.8)
e z (w) = 2nwt (c - C z (w») "
We. now look for conditions such that the solutions (10.3) and (10.7) satisfy
(10.9)
(10.10)
(10.11)
for 0 ~ e ~ land 0 ~ t ~ 1- j denotes the number of periods we want to compute ini 2 wc'
time. It is easily seen from (10.4), (10.5), (10.8), and (10.9) that e l and e z are
increasing functions of t. Therefore, (10.10) and (10.11) are satisfied for
o ~ t ~l- if we choose N = (wh)-l such thatwc
( ") = 2 j (1 _ sin (2TI/N»)=e l W,] TI 2TI/N. e (10.12)
39
and
N denotes the number of points per wave length.
e. (10.13)
We develop the left hand sides of (10.12) and (10.13) in power series in (2~/N) and
retain only the terms of lowest order. Then we have
(10.14)
and
(10.15)
Consider NI and N2
as functions of j. Let e be the maximum phase error allowed.
Utilizing (10.14) and (10.15) we have
(10.16)
and
A similar computation for the sixth order scheme
(3 3 1 )v = - c - D (h) - - D (2h) + -- D (3h) v(t)
t 2 0 5 0 . 10 0
yields
If e 0.1 then
NI (j) 20j 1/2
N2
(j) 7j 1/4
Na(j) 5j 1'6
and if e 0.01 then
NI (j) 64j 1/2
N2(j) 13j 1/4
Na(j) 8j 1/6
(10.17)
(10.18)
(10.19)
Observe that the operation count of the sixth order method is approximately 3/2
times that of the fourth order method. The fourth order method "has approximately twice
the operation count of the second order method. The table above clearly illustrates the
superiority of the fourth and sixth order schemes over the second order scheme. The
superiority is much more pronounced for smaller errors. However, considering the addi
tional effort the sixth order method requires over the fourth order method the table
above illustrates that little or nothing is gained by using the sixth order scheme,
as long as we allow an error of 1% and the integrations are not over extremely long
40
time intervals, which is natural for many meteorological calculations. The superiority
of the higher order methods is even greater when the computations are extended over long
time intervals since N grows like j 1/2, N like j 1/~, and N like· j 1/6 • Thus, for long1 ·23
integrations the sixth order method. is more economical but the saving is small.
We now consider even higher order approximations to the differential operator a/ax,
Let us now approximate the problem (10.1) by
~•• ~. = - CD[2m
l(h)v, v(x,O)i21TWX
e
where m
D[2m] (h) = LAvDo(Vk), Av
v=l
- 2.(- 1)v(m!)2(m + v) ! (m - v)!
When m = 1, 2, 3 we have the second, fourth, and sixth order schemes discussed earlier.
As before we let N2m = (Wh)-1 denote the number of points per wavelength and j.= cwt the
number of periods to be computed. In this case it can be shown, Kreiss and Oliger
(1972), that N2m(j) -7 2 as 2m -7 00. Thus we must always have at least 2 points per wave
length.
Observe that the amount of work the above 2m th order method requires is approxi
mately m times the work of the second order scheme. In light of (10.18) it it doubtful
that difference methods of order greater than six have any practical advantage for
meteorological calculations.
There is an.other method for increasing the order of accuracy, namely Richardson
extrapolation. The basis for Richardson extrapolation is that the solution of (10.2)
can be expanded in a series
where the w,(x,t) are the solutions of certain inhomogeneous equations:J
aW,/at = - caw,/ax. + rJ,(x), wj(x,O) = O.J J J
Let us determine the w,. Substituting (10.26) into (10.2) yieldsJ
c(Dou + h 2DoW1 + h~w2 + ... )
2 ~
DoU = ou/ox + ~! o3 u/ ox 3 + ~! o5u / ox 5 +
(10.27)
and the corresponding expansions hold for the wj(x,t). Introducing these expressions
into (10.27) gives us, after collecting terms in powers of h,
OW2 OW2 C 5 5 C 3 3-- = - c -- - -- a u/ox - -3·I i:lw /ox ,ot ax 5! • 1
41
u = e 2TIiw(x-ct) and therefore wI
is the solution of
aWl c (2TIiw) 3- C ax - 3~
2TIiw(x - ct)e
wI
(x,O) 0,
i.e. ,
c(2TIiw)3 t 2TIiw(x - ct)3! . e
Correspondingly we get
C 5 t 2TIiw(x - ct) + (2TIiw)6 • c 2 • t 2 2TIiw(x - ct)w2(x,t) = - 5T (2TIiw) • • e 3~ • 3: • 2 e
Let us compute v(x,t) = v(x,t,h) for a specific h o and then also for 2h o' We get
and·therefore,
Thus, if we neglect higher order terms, we have after j periods in time
where N is defined as before. Corresponding to equation (10.17) we have
1
15j 1/2. for e = 0.1N = 2TI • (2n) 1/2 • (1/12e) 1/" • j 1/2 =
26.8 jln for e = 0.01
Thus the improvement over the original leap-frog method (see equation (10.16)) is not so
impressive for the 10% error limit but is substantial for the 1% error limit. In any
case, the fourth order method (10.18) is better.
One can of course also compute v(x,t,h) for h
the h" term in (10.26) and obtain
3ho' Then we can also eliminate
1
12.9j 1/2 for aN =
19.0jl/2 for a
10% error
1% error
Thus not much·is gained. The fourth order method (10.18) is again better.
42
11. TRIGONOMETRIC INTERPOLATION
Let N be a natural number, h = e2N + 1)-1 and x =·vh, V = 0, ±l, ±2, • . . ConsiderV
a one-periodic function vex}, vex) = vex + 1), whose values Vv = v(xv} we know at the grid-
points xV' We want to approximate vex) by a trigonometric interpolation polynomial
such that
wex)
N
~..' e.)' 21fiwxt...J a W,e
w=-N
(11.1)
2N (11.2)
We want to show that this interpolation. problem has a unique solution. tve begin by de
fining the discrete scalar product and norm
We prove
2N
(u(x}, v(x}\ = L u(xvJ v(xv)h, lIull~v=O
(u,u)h
Lemma 11.1.
(..21finX21firnx), = j•.'°e , e . h
1
if 0 < Im-nl .;; 2N
if m = n
Proof: If m = n the lemma is obvious. Otherwise
(. 21finx·,e- ,
This lemma gives us
e21firnx
)
2N
=L e 21fi (m-n }vh h
v=O
h (1 l1fi (m-n) )
1 _ e 21fi (m-n)ho
Theorem 11.1. The interpolation problem (ll.l)" (11.2} has a unique solution. The
coefficients a(w} are given by
t.'· 21fi\1x,e ,2N
=~ v(xv)e-21fi\1xV h
v=O
Proof: If we form the scalar product of (11.2.) with e21fi
\1x we obtain
(11.3 )
(" 21fi\1x (. )~.".'" =e , w x,} h
N
LW=-N
t 21fi\1x 21fiWx). _a(W} \ e , eh - a(\1)
utilizing lemma ll.l. If we expand the left hand side of this equation and use the con
ditions (11.2) we obtain (11.3.). If we substitute (11.3) into (ILl) and use lemma 11.1
we verify that (1l.2) holds. (11.2) is a system of 2N + 1 linear equations for the 2N + 1
unknowns a (w) . Uniqueness follows from the fact that (11. 3.) gives us a sol ut ion for ar
bitrary v(xv)' V = 0, •• " 2N which implies that the matrix is invertible.
The usefulness of trigonome.tric interpolation stems from the fact that the smoothness
properties of the function are preserved and the convergence is rapid for sufficiently
smooth functions. The L2-scalar product and norm are defined by
43
1
(U,V) =£uvdx, Ilul/2 = (U,U)
We have
Theorem 11.2. (Smoothness properties)
N
/Iw(x) I 12 = Ilv(x)ll~ = L la(w)12
W=-N
(11.4)
(11.5)
Note: It is crucial that v is one-periodic for (11.5) to hold. If vex) is only defined
for 0 ~ x < 1 then (11.5) is only correct if we extend it as a one-periodic function. If
the extension is discontinuous then 1Inivl Ih = O(h-j ).
Proof: (11.4) follows immediately from lemma 11.1 and Parseval's relation. Therefore,
N
L(jJ--N
2 .(21TW) J Ia(w) 1
2
and N
II~W=-N
(
21Ti ha(w) e h
This completes the proof.
w-N
= (_21T)2j j j .• 11 d Wj dxJ 11
2
w=-N
Let p(a,M) be the class of all functions vex) which can be developed in a Fourier
series00
vex) ~ v(w)e21T:iwx
W=-OO
(11.6)
withIv(w)1 ~ __M__
121TWla + 1
(11.7)
The following lemma is well known.
aLemma 11.2. If vex) is a one-periodic function and V(X)EC , the class of all functions
with a continuous derivatives on _00 < x < 00, then vEP{a,M) with M = 11 dav/dxall.
44
If v(x) is a one-periodic roof function
~then v(x)EP(2,M) for some M. Let
v(x)
+00L v(w)e21TiWx
W=-OO
(11.8)
be a one-periodic function and interpolate by
w(x)
NL a(w)e21TiWx
W=-N
(11.9)
such that (11.2) holds. We want to express the a(w) in termS of the v(w).
Lemma 11.3. For the coefficients of (11.8) and (11.9) we have the relation
a(Jl)
+00
L v(Jl + j(2N + 1», IJlI < Nj=_oo
(11. 10)
Proof: Any integer W can be written in the form
W= fw] + j(2N + 1)
where [W] is an integer with -N < [Ul] ..;; N and j is another integer. From (11. 3) and (11. 8)
we have
(11.11)
+00... I: v(w) (e21riWX, e21riJlX) h
lIJ...-ClO
Now
21riWXV 21ri[W]xve • e
and therefore,
(e21TiWX, e21TiJlX) h ... ( e21r
i[lIJ]X, e21TiJlX) h ... { : :: ::~ : :
(11.10) follows from (11.11).
We can now investigate the rate of convergence of. the interpolation po1yn~miai to the
true function.
Theorem 11.3. Let v (x)EP(a,M) with a > 1. Then
(11.12 )
45
Proof: We write (11.6) in the form
where
N
" "( ) 21fiWXL...J vwe ,
w=-N
We also write w(x) in the form
L v(w)e21fiWX
Iwl>N
where
N
wN(x) L a (N) (w)e21fiWX , a (N) (00)
w=-N(.
21fiwx () )e , vN
xh
N
" (R) ( ) 21fiwxL-Ja we ,
w=-N
(R) ( ) = (21fiWX () )aWe , vR
xh
wN(X) is simply the interpolation of vN(x) and therefore, by theorem 11.1,
wR(X) represents the interpolation of vR(x) and therefore, by lemma 11.3
-t-oo
a(R) (00) = L v(w + j(2N + 1», IW'I.-:; N
j=-"',j*O
From (11.7) we obtain the estimates
'"_1_ .-:; __-=2=.:M=---__ • N-0'.+1
000'. (a - 1)(2n)a
and
Thus
Finally (11.12) follows from
Theorem 11.3 establishes uniform convergence if a > 1. This theorem thus applies to
the roof function and similar piecewise smooth functions. Similar convergence theorems
hold for more space dimensions.
46
12. THE FOURIER METHOD
Let N be a natural number, h = 1/ (2N + 1) and Xv = vh, v = 0,. 1, • • ., 2N. Consider
a one-periodic function vex), i.e., vex) = vex + 1), whose value we know at the gridpoints
xv' Vv = v (xv) . A very accurate method of approximating dV(Xv)!dx is to interpolate the
function values v(xv) by a trigonometric polynomial
V(x) ". A ( ) 21Tiwx~ vwe ,x=xvIwl~
v(w)
2N
=Lv=O
( )-27Tiwxv
v Xv e
(12.1)
and to differentiate this polynomial obtaining
L 27Tiwv(w)e27TiWxV
Iwl~
(12.2)
This can be achieved by two fast fourier transforms (FFT) and N complex multiplications.
We introduce the vectors v = (v0' ••• , V2N ) and W= ( dvo!dX, ••• , dV2N!dX)'. Then we
can write the above process in operator form
w Sv
where S is a (2N + 1) x (2N + 1) matrix. Let a scalar product and norm be defined by
2N
(v,u)N = L VjUj , Ilvll~j=O
(v,v)
It is obvious±N.0, ±l, • • •w
of vj
• We have
11 S11 N = 27TN.
e27TiW (2Nh))'Let = (1 e 27TiW (h) ••ew ' ,Proof:
where v. denotes the complex conjugateJ
Lemma 12.1. S is skew-hermitian and
that
Se- = 27Tiw6w w (12.3)
1. e., 27TiW are the eigenvalues and 6W the corresponding eigenfunctions of S. Also, by
lemma 11.1,
2N
(ej
,6k
) = L: e27Ti (k-j)Vk =
v=O 11 i.f k = j
Oifk=l=j
and, therefore, the eigenfunctions form an orthogonal basis. Observing that the eigenvalues
are purely imaginary and their absolute values bounded by 2TfN the lemma follows.
We now replace the differential equation
au/dt = -du/dx (12.4)
u(x,O) = f(x), u(O,t) = u(l,t)
f(x) =L f(w)e2TfiUJX
(v
47
by the system of ordinary differential equations
dv/dt = -cSv
v(O) = g
g is defined by
It follows from (12.3) that the solution of (12.5) is given by
~ f(W)e2~iW(xv-ct)
Iwl~
Thus the first 2N + 1 frequencies, Iwl ~ N, are represented exactly.
(12.5)
Therefore, using this" method we need only two points per wavelength to represent the
wave exactly, compared to seven points for the fourth order scheme allowing an error of
10% and 13 points allowing 1% error.
Now approximate (12.5) by the leap-frog scheme
vet + k) = vet - k) - 2ckSv(t)
It follows from lemma 12.1 that the approximation (12.6) is stable for 12~Nckl < 1.
(12.6)
Since each FFT on 2N + 1 points requires approximately N log2
(2N) complex multiplica
tions and 2N log 2 (2N) complex additions the number of operations per time step for (12.6)
is approximately
8N log 2 (2N) real multiplications and "8N log 2 (2N) real additions (1~.7)
The fourth order scheme requires 4N real multiplications and 6N real additions. We
need approximately 4-7 times as many points for the fourth order scheme. Thus we must com
pare (12.7) with
l6N - 28N real multiplications and 24N - 42N real additions
Therefore, the Fourier method is in this case at least as economical as a fourth order
scheme as long as we compute no more than 16 wave numbers. The advantage of the Fourier
method i~ more evident for longer time integrations. Furthermore, the storage requirements
are reduced by a factor of 4-7 for every space dimension. The dissipation and data filter
ing problems are much more easily handled by the Fourier method.
Some additional difficulties arise when this method is applied to equations with
variable coefficients. Consider, for example, the equation
dU/dt = C(X,t)dU/dX = Tu (12.8)
Let the L2
scalar product and norm be defined by
1
(u,v) =f ;-vdx, IIul1 2
o
Then (12.8) implies
(u,u) (12.9)
(u,Tu) + (Tu,u) «T + T*)u, u) (- dC-)- v -v, dX (12.10)
48
where T*u = - a: cu is the adjoint of T. Therefore (T + T*)u = - (~ ) u is a bounded opera
tor. This is precisely the reason the problem is well posed.
We approximate (12.8) by
where
C
c (xo.t) 0
o c(x1.t) 0
• 0
• 0
(12.11)
Then
o
In general CS - SC is not bounded independent of N. Thus we cannot use (12.10). This
difficulty is easily avoided. We write (12.8) in the form
and approximate it by
au 1 (au a )at = 2" c ax + ax (cu) .1 dc
- -- u2 dx (12.12)
dv = 1:. (CS + SC)v - 1:. v ~ Cdt 2 2 dx
CS + SC is skew-hermitian and therefore
(12.13)
(..- dc -)- v - v
• dx
which is the same equality as (12.10).
It is at the present time not clear what the accuracy of the Fourier method is for
equations with variable coefficients. in particular when discontinuities are present. Some
preliminary calculations have shown that if the solution is discontinuous then the number
of necessary frequencies has to be increased substantially. A rough preliminary estimate
is that we need twice as many modes as for equations with constant coefficients to achieve
the same accuracy.
We now want to derive error estimates for the Fourier method. As we have seen, the
situation is very good for equations with constant coefficients. The error is never larger
than the error we commit by approximating the initial function by a truncated Fourier series2~iwx .of 2N + 1 terms. The reason being that the functions e are the eigenfunctlons of the
problem. For equations with variable coefficients the situation is not as favorable. In
this case the error also depends on how well we can approximate the first derivatives by
truncated Fourier series.
Consider the differential equation (12.12) with initial values
N
f (x) L f (w)e2~iwxw=-N
49
For everyt its solution can be developed into a Fourier series
u(x.t) =~ G(w.t)e2~iWK
w
~ ~ O(w.t)e2~iWK + ~ G(w.t)e2~iWK
Iwl<N Iwl>N
In the same. way
~ A 2~iWXd(x.t) c(x)~(x.t) =~ d(w)e = dN(x.t) + dR(x.t)
w
Now we can write (12.12) in the form
where
1 (dUR d d dC dUR)G = '2 c dX + dX (cuR)N + dX (CU)R - dX uR - 2~
We now consider (12.14) on the 2N + 1 gridpoints x = Vh. V = O. 1. 2.V
can be written as·
d-~ 1 (- ( _)) 1 dC - -dt = '2 CS~ + S C~ - '2 dx ~ + G
where C is defined in (12.11) and
(12.14)
2N. Then it
(12.15)
~(O.t)
~=
G(O.t)
(T0 0
c'(h) o • 0- dCG = dx =
G(2Nh.t)
Let v be the solution of (12.13) with initial values
v(O) = ~(O)
then w= ~ - v is the solution of
dw 1 - (-» 1 de - -dt = '2 (CSw + S CW - '2 dx w + G.
Proceeding as in the proof of lemma 8.1 we have
*Theorem 12.1. Let dC - ( dC) .,;; 20.. Then
dx dx
w(O) o
(12.16)
50
If v(x,t) denotes the trigonometric polynomial which interpolates the gridvalues
vlVh,t), then by Parseval's relation
11~(x,t) - v(x,t) 11 = 11~(t) - vet) II N
Therefore, (lZ.16) gives us an estimate of how well v(x,t) approximates the first ZN + 1
modes of u(x,t). For G we have
ZIIGII N ~ m~xlcl'lIa(u - ~)/axIIN + Ila(cu) - (cu)N)/axIlN
+ zlla(u - ~VatlIN + ~x, ~~ "II(u - ~)IIN + Ila(cuR)NjaxIIN
But
Ila(cuR)~axll = I la (cuR)N/axl 1~ I Ia(cuRyax 11
~ m~xlcl'lla(u - ~Vaxll + ~xl ~~ l'llu - ~II
so IIGIIN
can be estimated in terms of the approximations of u, au/ax, a(cu)/ax, au/at by
~, a~/ax, a(cu)Njax and a~/at, respectively.
There is another way to construct Fourier methods, the Galerkin procedure. Let
~_N(x), •• " ~o(x), • • " ~N(x) be a system of linearly independent functions. We want
to approximate the solution of
+N
au/at = p(x,t,a/ax)u, u(x,O) f(x) ~ fj~jj=-N
by an expression
+N
v(x,t) ~ ej (t H j (x)
j=-N
where ej (0) < fj . The Ga1erkin procedure requires that
(av/at - p(x,t,a/ax)v,~v(x») = 0, v = 0, ±l, •••
i.e. ,
(12.17)
(12.18)
(12.19)
+N
L ej(t)(p(x,t,a/ax)~j'~V)' V = 0, ±l, ••• ±N
j=-N
(12.19) is a system of ordinary differential equations, which determines the e.(t). WeJ
can write it in the form
de ()-A dt = B t e,
where
51
. . .
. . ., B =A=
,In general A is a dense matrix and can be ill-conditioned. If the ~j(x), j 0, ±1, •••
±N, are orthogona1," i.e. ,
for j = k
for j * k
then A = I and (12.19) has the form
(12.20)
The Ga1erkin procedure has the advantage that the resulting method is stable if the opera
tor p(x,t,a/ax) is half-bounded, i.e.,
(Pv,v) + (v,Pv) .;;;; 2allvl1 2 (12.21)
for all linear combinations (12.18).
Theorem 12.2. The solutions of (12.19) satisfy
Ilv(x,t) 11 .;;;; eat 1jv(x,O) 11 (12.22)
if P satisfies (12.21).
Proof: Let (12.18) be a solution of (12.19). Multiply each of the equations (12.19)
by ev(t), respectively, and add the resulting equations. Then
(av/at - p(x,t,a/ax)v,v) + (v,av/at - p(x,t,a/ax)v) 0
and (12.21) implies
(12.23)
a~ 11 vii 2 .;;;; 2a 11 v11 2
(12.22) follows immediately.
Let ~(x,t) be the b~st approximation of u(x,t) in the form (12.18) and define
uR(x,t) = u(x,t) - ~(x,t)
Now we can write (12.23) in the form
a~/at - p(x,t,a/ax)~ = ~' ~ = -auRjat + PUR
~(x,O) = fN(x)
Thus
Let v(x,t) be the solution of (12.19) with initial values
52
Then w = ~(x,t) ~ v(x,t) satisfies
and therefore,
(aw/at - Pw,w) =- (~,w )
If P is half-bounded we then get
and finally obtain the estimate
Ilw(t)11 .;;; sup 11 R('r) 1I· eat - 1~""t a
We can now state
Theorem 12.3. If the operator P is half-bounded then we have the estimate
· at 1 (aIlv(x,t) - ~(x,t)ll.;;;e a- • sup 11 --at+p(X,T,a/aX)\(u- u
N)1I
~""t I(12.Z4)
( t) 21fi(V-j)xdc x, e x
(12.24) is, essentially, of the same form" as (12.16). Let P be a differential opera
tnr of order m. Then v(x,t), the solution of the Galerkin procedure, approximates the
solution u(x,t) = ~(x,t) + uR(x,t) of the differential equation well if u - UN' a~ (u - UN)and the first m space derivatives of (u - ~) are small.
As an example let us consider the equation (12.8) and take
iPj
(x) = eZ1fijx
Then the Galerkin equations (12.19) are of the form (12.20) with
bjV = _Z1fij /
1
o
If c(x,t) = Co = constant then the integrals can be comput~d explicitly and, in fact, the
resulting system is precisely the system (12.5) which we derived earlier. Consequently,
the earlier comparison of that method with difference methods applies here.
We now consider the more general case where c is a function of x and t. When a
Galerkin method is used to solve the system (12.20) the b. have to be computed at every]V
time step. This can be done using numerical quadrature techniques. The resulting method
can be considered as a difference method and nothing is gained. Alternatively, we can
develop c(x,t) in a Fourier series
c(x,t) = L ~(t)e21fi11X
11
and compute the integrals by using FFT-techniques. This technique has been extensively
investigated by Orszag. He shows, Orszag (1971), that the bjV can be computed by six
53
complex FFTl s on 2N points. Therefore, the first Fourier method we discussed is 50% more
efficient since it only requires four FFTl s • This advantage is more pronounced for systems.
Furthermore, it is not necessary to write the differential operator in anti-self-adjoint
form, provided one adds a suitable dissipation mechanism. Then the first method should
be even faster.
54
13. IMPLICIT DIFFERENCE METHODS
Let us consider the differential equation
u - au + ae2~iPxx
(13.1)
with initial values
u(x,O)2~icrx
e (13.2)
where a~ 0 and a are real and p and cr are natural.numbers. The solution of the above
problem is given by
u(x,t)Qe2~ipx )~ (1 _e- at + 2~ipt + e2~icr(x + t) - ata - 2~ip
(13.3)
We consider two difference approximations:
(13.4)21Ticrvh
e
and
(13.5)2'Jiicrvh.= e+ ke21TiPVh, vv(O)
2~icr(x + t)Then u(x,t) = e and the solution oscillates as fast in theLet a = a = o.t-direction as it does in the x-direction. Thus, the spatial and temporal resolutions
should be of the same size, i.e., k/h -1. Therefore, the stability requirement, k/h ~ 1,
of (13.4) is no restriction. In this case there is no point in using the method (13.5)
which, ·by theorem 9.2, is unconditionally stable.
Now let a > O. Then u(x,t) converges as t + 00 to the steady state solution
(Se2~iPx)/(a _ 2~ip). Thus, for large t, u(x,t) oscillates much slower in the
t-direction than it does ~n the x-direction. In this case the method (13.5) may be
advantageous, provided that the system of equations can be solved easily for vv(t + k).
There are two types of methods to consider for solving this system, iterative methods
and direct methods. For linear systems the iterative methods are not competitive with
the direct methods. So we now only consider direct methods.
Let k N- I , N a natural number. Then Vv + N(t) = vv(t) arid we consider the equa-
tions (13.5) for V = 0,1, N - 1. We let
v F
55
(13.5) can be written
1 + ak k0 k
2 • 4h + 4h
k0+ 4h
Av v = F
0• k
- 4h
k 0• k
1 ~ ak- 4h + 4h 2
Since-the matrix A is of band structure it is easily solved using Gaussian elimination.
As we have seen before it is usually better to use a fourth-order method in the x
direction. In this case the corresponding matrix A is again a banded matrix with a band
of width 5.
Let us now consider the general problem of solving linear systems with banded
matrices. Let
o
A aal.
o 'ann-a
an-an
ann
(13.6)
be an n x n matrix of band width 2a - 1. Then the solution of the equation Ax = b
requires less than n(a2 + a-I) multiplications and n«a - 1)2 + 2(a - 1)) additions._
In the periodic case
A+
o o
o
'0
an,a-l
aIn
aa-l,n
o
(13.7)
we can use the above formula with a replaced by 2a.
Let us now consider the Cauchy problem
3u/3t = (1 0) 3u + 10 (0 0) 3u (u(l)) (13.8)o 0 3x o 1 3x ' u u (2) ,
with initial values
Its solution is given by
21Tiwt (2)-e ,u (x,t)
l/s1Tiwxe .
l/s1Tiwx 21fiwte • e •
56
Thus u(r) and u(2) have the same behavior in time. Now approximate (13.8) by the leap
frog scheme.
(13.9)
Suppose we are interested in approximating the solution of (13.8) with an error of about
1%. Then we must have h = 1/(64w). (See chapter 10. Obviously one should use a fourth
order scheme.) The stability condition for (13.9) requires that k = 1/(64Ow). This is a
hopeless situation. There are two ways of improving the situation. We first consider
(13.10)
By theorem 9.2 the stabili,ty condition for (13 .10) is k .;; h = 11 (64w) • By treating the
term9(0,. O,),ou'0 lox
implicitly we do not destroy the accuracy even if we take k as large as k = h - 0,
° < 0 « 1 if the initial values u(2)(x,0) do not oscillate faster than 1/10 the speed of
)l)(x,O). If u(z)(x,O) = e21Tiwx then the phase error of the approximation is about 50%.
If the initial values are of the form
then the term Ee21Tiwx introduces rapidly oscillating noise in the system which is never
dissipated. This can be disastrous for nonlinear equations.
The other alteEnative is to use (13.9) but replace
by
where C is a smoothing operator which decreases the amplitudes of the high frequencies.
As an example of this technique we consider
Then the' approximation (13.9) is stable if k/h';; 1 and
k 10 sin 2TIWh .,;:h 1 + 16 sin4TIWh ~ 1.
If IT = 11 (64w) then the accuracy is not affected if,
1,60 sin4 TIWh 160 (1T)4 = 10-2 ,10 ~ 10,4 • ·,64
i.e., 0 ~ 106:. If 0 ;;;. 6'" then (13.11) holds for all k/h .;; 1.
Now consider the equation
ou"t =, au + bu ,u x y
with the initial values
(13 .11)
(13.12)
u(x,y,O)
57
An unconditiona11y·stab1e ·difference approximation is
(13.13)
where D ,D denote the central difference operators in the x and y directions, respecox oytive1y. The difficulty with this approximation is in the inversion of I - ~ (aD ox + bDoy)'
It is more economical to use an approximation of the form
(13.14)
which is again unconditionally stable. To soive the equation
(I - ~ aD ) (I - ~ bD )v(t + k) = F2 ox \ 2 oy
we introduce
(I - ~ bDOy)V(t + k) = w(t + k)
as an auxiliary variable. Then
(I - ~ aD )W(t + k) = F2 Ox
(13.15)
(13.16)
Theorem 13.1.
is, on every line y = constant, a triagona1 system of the form (13.7) with a = 2. In the
same way, on every line x = constant, (13.15) is also of the form .(13.7). Thus we can
solve the equation (13.14) by 2N inversions of simple banded matrices. The approximation
(13.13), on the other hand, requires the inversion of a block tridiagona1 matrix which is
much more time consuming. The truncation error of both methods. is O(k2. + h2.). (13.14)
is an example of the splitting technique. We can prove the following general theorem:
Let Q ,Q be bounded operators with1 2.
Real (v,Qjv) = (v,Qjv) + (Qjv,v) ~ 0, j 1,2.
Then the approximation
(13.17)
is stable.
Proof: Let
y.
Then we can write (13.17) as
and therefore,
Real (z + y, z - y) Real (z + y, Ql(z + y)) ~ O.
Thus
and the approximation is stable.
58
As another example we approximate (13.12) with truncation error 0(h4 + k 2). We
simply let
Q = ~6 a (4D (h) - D (2h») , Q1 V Ox ox 2
= ~ h(4D (h) - D (2h»).6 Oy oy
The situation is more complicated if we consider non-linear equations. For example
Ut = (1 + u)ux ' (13.18)
We examine these approximations:
(1 - t (1 + V(t»)Do)V(t + k) (1 + t (1 +v(t))D o) v(t), (13.19)
(1 k(1 + v(t + k») Do)V(t + k) (1 + t (1+ v(t») Do)V(t), (13.20)-"2
(I - kDo)V(t + k) = (I + kDo)V(t - k) + 2kv(t)Dov(t). (13.21)
The truncation error of (13 .19) is 0 (k + h 2) ·and that of. (13.20) and (13.21) is 0 (k2 + h 2
).
(13.19) and (13.20) are unconditionally linearly stable while (13.21) is only linearly
stable for vk/h ~ 1. It should be noted that (13.21) requires the storage of an addi
tional time-level. Both (13.19) and (13.21) are linear in the implicit part and the
resulting linear systems can be solved by Gaussian elimination. (13.20) requires the
solution of a nonlinear system of equations which can be written as
(1 + A(y») Y F, (13.22)
(13.24)
where y = (Yl' • YN)' and A denotes a N x N matrix depending on y. The most elemen-
tary method to solve (13.22) is the iterative procedure
In general, this method only converges for sufficiently small k/h and the convergence may
be slow. A better procedure to use is Newton's method: Assume that we have computed the
approximation y(n). Then we write the solution y of (13.22) in the form
y = y(n) + 0
and substitute it into (13.22)
(I + A(y(n) + 0)) (y(n) + 0) = F.Now
A(y(n) + 0) = A(y(n») + B(o),
where B(o) is a matrix depending linearly on O. Thus, neglecting terms of order 0(0 2),
we get the linear system
c(y(n»)o = B(O)/n) + (I + A(y(n»))o = F - (I + A(/n»)h(n).
for O. This system for 0 can easily be solved because C is again a banded matrix. The
by Y(n+l) __ y(n)
statement of the algorithm is completed + O.
If also C is slowly varying as a function of t then one can fix C
If the matrix C is slowly varying with n then we can replace
c(v(t»).
C(y(n») by c(y(o») =
= C{v(t»)
for a number of time-steps and store C = L D, where L is a lower triangular matrix and D
is an upper triangular matrix. In this case the solution of (13.24) is quite economical.
A detailed discussion of this procedure is found in Gustafsson (197la).
59
14. NONLINEAR INSTABILITY
Consider the shallow water equations
v t + uv + vu + </>x y y
a
a
</> + (u</» + (v</» = at x y
for t ~ a with periodic boundary conditions." In matrix form these equations are
a (14.1)
Consider approximating these equations by a centered difference approximation of leap
frog type. For example
- 2k G au
a
w(t + k) w(t - k) -
1
v
a(14.2)
Here w(t) = (~(x,y,t), v(x,y,t), ~(x,y,t»)' denotes the solution of the difference equa
tions and Dox ' Doy are the central difference approximations in the x and y directions,
respectively.
Remark: There are quite a number of ways to decrease the amount of work in solving
these equations by using staggered grids. The following discussion also applies to these
methods.
Let us first consider the equations (14.2) with constant coefficients, Le., we
replace the coefficient matrices by
(~o
</>0
a
where u o' v o ' </>0 > a represent a constant flow. It is easy to see that the approximation
is stable provided k/h is small enough.
If we introduce a new variable
]1a a)1 a w
aKathen the difference approximation becomes
]1(t + k) = ]1(t - k) - wkQ ]1(t)o(14.3)
60
where the difference operator
0 :') ~
.~ ) DOYC C<Po1:
Qo = \ u D + .. <p 2Vo0 ox 0
<Po 0 uo 0 0 Vo
(14.4)
has symmetric coefficients. By theorem 9.2 the approximation is stable if
Let
:~)1:
. ,~((:~0
C<p 2
~ ) dn n)0
"kQ o uo sin t; + <p~ Vo0
I/J~ 0 0 0 Vo0 0
denote the Fourier transform of kQo' We know that
The eigenvalues of kQ areo
n) ,Thus
and the approximation is stable if
(14.5)
~ < max(lul + Ivl + %)-1.h x,y
If we compute solutions of (14.3) with this restriction no instability appears. Simi
larly, if we linearize the equations (14.2) and replace u, ;, ~ in the coefficient ma
trices by a smooth flow U(x,y), V(x,y), ~(x,y) then no instability occurs if we replace
(14.5) by
On the other hand, if we integrate the nonlinear equations (14.2) the solutions become
unbounded after a relatively short time. This phenomena is called nonlinear instability
and was discussed for the first time by N. Phillips, Phillips (1959), and later by R. D.
Richtmyer, Richtmyer and Morton (1967). We shall discuss the phenomena from a somewhat
different point of view which is based on Fornberg (1972b) and Kreiss and Oliger (1972).
Consider the differential equation
for t ;;;. 0 and 0 ~ x ~ 1 with periodic boundary conditions. We want to solve (14.6) with
the help of a system of ordinary differential equations. Let N be a natural number and
denote gridpoints by Xv vh, v = - 1,0, ••. , N and a gridfunction by vv(t) = v(xv,t).
We approximate (14.6) by
0,1,2, •. , N - 1 (14.7)
61
with boundary conditions
Let us consider the case where, for t = 0,
vo(O) = va(O) = O.
It follows from (14.7) that vo(t) = va(t) = 0 and therefore that v 1(t), v2(t) are com
pletely independent of the other variables. We get
i.e. ,
Now assume that, for t 0,
(v
1(t))
v 2 (t) .(14.8)
It then follows that
for all t and that the absolute values of the solutions of (14 •. 8)· are monotonically in
creasing. If both Iv1(t)l, Iv2
(t)1 « h then the increase is slow. If on the other hand
Iv 1(t) I, Iv 2 (t) I .» h then the increase is quite rapid.
This behavior occurs not only for nonlinear equations but also for linear equations
with sufficiently rough variable coefficients. Consider, for example, the equation
and approximate it by
If
then we get
8u/8t = a(x)8u/8x
dVV/dt = a(Vh)Dov (t).V
"-a(O) a(3h) = 0, a(h) < 0 < a {2h)
(14.9)
0-4.10)
~t ~~) = ~h (_~(2h)(hb) (~~)and therefore vj(t), j = 1,2 are growing like exp (~hJ!;(h)a(2h)l) if la(h)l,
la(2h)I > 0, and a(h) and a(2h) have opposite sign. (Observe that the solutions of the
differential equation are uniformly bounded.) If a(x) is a smooth function then
Jla(h)a(2h)I O(h 2 ) and, therefore, the growth is restricted and this exponential
increase is of no consequence. If a(x) is a very rough function then the growth can
be disastrous. This behavior is in sharp contrast to the behavior when a(x) does not
change sign. Let a(x) ~ 0 > O. Then we can write (14.10) in the form
-1a (vh)dv /dt = D v (t)
V 0 V(14.11)
62
and therefore
Thus the weighted L - norm is constant. Observe that the function a(x) need not be smooth2
at all for this estimate to hold.
Correspondingly, Bengt Fornberg, Fornberg (1972b), who performed an extensive series
of experiments for the equation (14.6) was never able to produce nonlinear instabilities
if a(x) ~ 0 > O. Nonlinear instabilities only occurred if u(x) oscillated around zero.
The genera~ pattern for the behavior of the leap-frog approximation
is that at some time t = to there appears, for some x, the situation that vv(to) =
vV+3(t O) ~ 0 and vv+l (to) < 0 < Vv+2(t O)' Then 1VV+l (to)l, 1Vv+2(t) I start to grow rap
idly up to the point that v (t ), v,~ (t ) are no longer small. Thereafter, these largeV 0 v-.-3 0
values of IVV+l (to)l, IVv+2(t) I propagate over the whole interval. Then the process
restarts at this higher level.
How does one avoid this instability? One can make the approximation (14.7) dissipa
tive, i.e., replace it by
dv (t)/dt = v (t)Dov (t) + ahD+D v (t).V V V - V
If the constant a is sufficiently large then the solutions vv(t) stay bounded for all
time. The only trouble is that the amount of dissipation necessary (size of a) is so
large that it damps the solution too much. Another way is to write the differential
equation (14.6) in the form
and approximate it by
dVV/dt = t (vvDovv + DoV~)'
Then it follows from (f,Dog) = - (Dof,g) that
(14.12)
(14.13).
Thus, the solutions are uniformly bounded in time. The shallow water equations can be
written in a similar way. Thus the nonlinear instability can be completely avoided as
long as we do not discretize the time derivative. If we use a Crank-Nicholson procedure
for the time derivative then this is also true for the complete difference approximation.
If we, on the other hand, use a leap-frog procedure, i.e.,
(14.14)
then we have given examples in Kreiss and Oliger (1972) which show that instabilities
can occur. These can be controlled by dissipation. In this case the amount of dissipa
tion necessary is much smaller. In fact it need not be applied at every time-step.
63
As we have seen earlier, one should use a fourth order accurate scheme, i.e., one
should instead use the approximation
dV)dt = v,,(t)(t Do(h) - t Do(2h»)V,,(t).
Again'nonlinear instabilities arise. However, they can be controlled by a very small
amount of dissipation. In Kreiss and Oliger (1972) we have shown that the solutions of
where K(X",t) = ID v (t)! are uniformly bounded. (Observe that we use a nonlinear dissi-v 0 " .
pativ~ term.) The test calculations in Kreiss and Oliger (1972) also show that one can
integrate the shallow water equations, using the corresponding fourth order method, with
out adding any dissipation at all, provided one restricts oneself to a reasonable time
intervaL
64
15. INITIAL BOUNDARY-VALUE PROBLEMS FOR HYPERBOLIC EQUATIONS
Examples and Definitions
Consider the equation
au/at = aau/ax for 0 ~. x < 00 and t ;;;. 0 (15.1)
Let initial values be given by
u(x,o) = f(x), 0 ~ x < 00 (15.2)
The solution of (15.1), (15.2) is given by
u(x,t) = f(x + at) (15.3)
a < 0 x a > 0 x
The solution is constant on the lines x + at = constant, which are called characteristic
lines. If a > 0 then u(x,t) is uniquely determined by (15.3) and it is not appropriate to
specify a boundary condition at x = O. If a < 0 then u(x,t) is only determined in the tri
angular region x + at ~ O. In this case a boundary condition
u(o,t) = g(t), t;;;' 0 (15.4)
is required to determine the solution for x + at > O.
The solution u(x,t) is continuous in a neighborhood of x +. at
g are continuous and satisfy the compatibility condition
o if and only if f and
f(O) = g(O)
Now consider a system
au/at (-!I. 0 )0 1 11.2. au/ax,
u(x,o) f(x),
x;;;' 0,
x;;;' 0
t ;;;. 0 (15.5)
(15.6)
Here
> 0
• • Ar
o
o
o
Ar
o
are positive matrices and
Iu
IIu
ur
65
We can also write (15.5) in the form
(15.7)
(15.8)
The solution of (15.8) is determined by the initial values fII(x) as is the solution of
equation (15.1) for a > O. Correspondingly, uI has to be specified on the boundary x = O.
We require
(15.9)
and assume its compatibility with the initial values. Here SI is a rx(n-r) matrix and theIII . I II
term S u (O,t) represents the dependence of u on u Because of the different direc-I dII II Itions of the characteristics of u an u , u is considered an outgoing variable and u
an ingoing variable on the line x = O. SI then represents a generalized reflection ofII
u
Let us now consider (15.5) for 0 ~ x ~ 1, t ~ O. In this case we must also specify
boundary conditions at x 1. Proceeding in the same way as before we require
11 11 I 11u (l,t) = S u (l,t) + g (t)
·IIwhere S is an (n-r)xr matrix representing a generalized reflection at x = 1.
(15.10)
If a = 0 in (15.1) then u (x,t) = f(x) and we do not need to specify any boundary con
ditions. Similarly, if Al or A2 are not positive definite then the components uJ.(x,t) corIIresponding to a A = 0 can be considered as outgoing variables and will be included in u
jIfor x = 0 and in u for x = 1, i.e., we always assume that Al > 0 for x = 0 and A
2> 0 for
x = 1.
More generally we consider the system
with initial values
dw A dW forat = dX o ~ x ~ 1, t ~ 0 (15.11)
w(x, 0) = f (x) (15.12)
where A can be transformed to real diagonal form by a nonsingu1ar transformation T, i.e.,
Let u = T-lw. Then the system (15.11) is transformed to the form (15.5) and the boundary
conditions must be of the form (15.9) and (15.10). Therefore, any boundary conditions
specified for the original system (13.11), must transform to boundary conditions of the
form (15.9), (15.10). In particular, the number of boundary conditions at x = 0 must
equal the number of negative eigenva1ues of A and ~he number of boundary conditions at
x = 1 must correspond to the number of positive eigenva1ues.
66
Example. Consider the equation
The eigenvalues K. of A areJ
° ,.;; x ,.;; 1, t ;;;. ° (15.13)
Let ° < a < 1. Then A has two positive and one negative eigenvalue. Thus we have to spec-
ify one boundary condition at x = ° and two boundary conditions at x 1. Let us consider
(15.14)
We have to show that these conditions are, after transformation, of the form (15.9) and
(15.10), respectively. An easy calculation yields
T
The conditions (15.14) become
which are obviously of the desired form.
We consider the variable coefficient problem for a system
dU/dt = A(X,t)dU/dX + Bu, 0";; x ,.;; 1, t;;;' 0, u(x,O) = f(x) (15.15)
where A can be smoothly transformed to real diagonal form. To obtain proper boundary con
ditions for the problem (15.15) at the point x = 0, t = to we consider the system
dW/dt = A(O,tO)dW/dX (15.16)
with constant coefficients on the half-plane x ;;;. 0, t ;;;. 0. Any boundary conditions which
are proper for (15.16) are also proper for the original system at the point x = 0, t = to'
The corresponding process is carried out on the line x = 1. We shall not include a proof
of the correctness of this procedure. After transforming A to diagonal form and neglecting
the non-differentiated terms the proof is concluded via the method of characteristics.
Now consider the equation
dU/dt adu/dx + bdU/dy (15.17)
for t ;;;. 0, x ;;;. ° and _00 < Y < 00, with initial values
u(x,y,O) = f(x,y)
The solution of (15.17), (15.18) is
u(x,y,t) = f(x + at, y + bt)
i.e., the solution is again constant along the characteristic lines
x + at = constant, y + bt = constant.
(15.18)
(15.19 )
(15.20)
If a ;;;. ° then u(x,y,t) is completely determined by f. If a < ° then we have to specify
boundary values
67
u(O,y,t) = g(y,t)
Thus the additional space dimension, y, does not change things, it is only the sign of
11 a" ~vhich determines whether boundary values are necessary or not.
Next we consider the equation (15.17) in a region t ~ 0, x, yEQ eR x R with boundary
an. Let (xo'Yo) be a point contained in an. The necessity of specifying boundary values
at (xo'Yo) is again determined by the direction of the characteristics (15.20).
y
x
This can be formalized by introducing a new coordinate system with origin (xo'Yo) and axes
directed as the tangent L and the internal normal n (see figure).
x = (x - x o) cos a - (y - Yo) sin a, y (15.21)
where a is the angle between the y and L axes. In the new coordinate system the equation
(15.17) has the form
where
au/at aau/ax + bau/ay (15.22)
a = a cos a - b sin a, b = a sin a + b cos a
The sign of a determines whether or not we need to specify boundary data at (xo'Yo)'
Next consider the hyperbolic system
(15.23)
for t ~ 0, x ~ 0, _00 < Y < 00, with initial values
u(x,y,O) = f(x,y) (15.24)
Here u is a column vector with n components and A,B are nx n matrices. In the same way as
above the term Bau/ay has no influence on the boundary conditions. Therefore, they are of
the same form as in the one-dimensional case, namely those given by (15.9). Correspondingly,
if we consider (15.23) on the domain t ~ 0, (x,y)£n, the form of the boundary conditions at
a point (xo'Yo)£an is determined by the matrix
A = cos a A - sin a B (15.25)
UnfortunatelY,not all boundary conditions of the above form for systems of equations
yield well posed problems. Consider the half-plane problem
a: w = ( ~O -~ ~ ) a~ w + ( ~° y . b 13
(15.26)
w(x,y,O) f(x,y), x ~ 0, _00 < Y < 00 (15.27)
(15.28)
68
((1) (2) (a»).. 'Here w = u . ,1.1 ,u is a vector function and a, a, S, Y > 0 and b 1a , b 2a * 0 are
real constant$.
t
f(x,y)
x
If f(x,y) = f(x) is a function of x alone, then u(j)(x,y,t) ::; u(j)(x,t) is a function of
only x and t. Then (15.26) reduces to
a (aOO)a- w = .. 0 -.S 0 - wat 0 0 y . ax
Therefore, the boundary conditions must necessarily be of the form (15.28). We want to
show that the problem is not well posed for al.l values of "a" and start with
Lemma 15.1. For any constants a, S, Y > 0 and b13
, b 2a * 0 we can find boundary con
ditions of the form (15.28) with b = 0 such that the problem (15.26)-(15.• 28) has solutions
stw(x,y,t) = e f(x,y)
with
(15.29)
Also
Real s > 0 sup If (x, y) 1< 00 :
x,y
w (x,y,t) = esptf(px,Py), p > 0p
are solutions. Thus we can construct solutions which grow arbitrarily fast.
Proof:
obtain
A iwyAssume that f(x,y) = f(x)e and substitute (15.29) into (15.26). Then we
(15.30)
IrIn addition assume that
where Real K < O. (15.30) gets the form
0 0 Xf(l) ~ 0 )('(1)er b u . f
S+SK o ~(:) = iw 0 01'(2)
b 2 a. fo .. iiU )0 s-YK if() b 13 b
23
69
i.e. ,
(s - CXK)i(l) _ -wb £(3)13 .
(S + f3K)~(2·) '" -wb . £(3)23 .
(S - YK)f(3) '" W(b l3"f(1)
The first two equations are fulfilled if
wb 13
s-CXK
A(2) bf W 23
£(3) = - S+f3K
Then the third condition is
2(b~3 b~3)S - yK = -w s-CXK - S+f3K (15.31)
The boundary conditions (15.28) are fulfilled if
A(2) A(l). A(3)f = af + ibf
i.e. ,
(15.32)
Therefore,
w(x,y,t) (
A(l»)est+iWY+KX ~(2) ,
A(3)f
Real s > 0, Real K < 0 (15.33)
is a solution if K,S are solutions of (15.22), (15.23).
Let us now consider special cases.
(1) Let a = O. Then
and (15.22) becomes
This equation has no solution of the desired form because
Real (s - YK) > 0
(2) Let b O. Then by (15.23)
andb~ 3
Real w2-- < 0S-CXK
ab 13(s - CXK) = ---- (s + eK)
b 23
and (15.22) can be written as
(15.34)
(s - yK) (15.35)
70
From (15.25) we obtain
~a+f3 )
s ab 13
a+--f3b 23
(15.36)
It is obvious that Re'al (s - aK) > 0, Real (s - YK) > O. Therefore, it is necessary that
Real (s + f3K) < 0 (15.37)
.By, (15.36) we have that (15.37) is equivalent to
ab 13
a + -b- f3 < 023
i. e. ,
a-"(3 (15.38)
If (15.38) is fulfilled we get from (15.34) for any s.> 0 a K < 0 and we can then satisfy
(15.35) by an appropriate choice of w. Thus we have constructed a solution of the form
(15.29). Observe that if (15.38) is not fulfilled then there are 'no solutions of this form.
This is, for example, the case if
ab 13
-b- > 023
or lal is sufficiently small. This example shows that one must be very careful when posing
boundary conditions. Unfortunately, a thorough investigation is needed in every particular
case. Even if there are no solutions like those just discussed there can be solutions
which lose all their smoothness in a finite length of time. A thorough 'discussion of
relevant examples is carried out in Elvius and Sundstrom (1972).
71
16. INITIAL BOUNDARY-VALUE PROBLEMS FOR PARABOLIC EQUATIONS
The simplest parabolic equation is the heat equation
dU/dt (16.1)
If we give periodic initial values
u(x,O) = L: f(w)eiwx
then its solution is given by
f(x) (16.2)
(16.3)
which illustrates the smoothing properties of (16.1), Le., even if we start with "rough"
initial values the solution is smooth for any fixed t = to > 0. The larger "a" is the
smoother the solution. This behavior is typical for all parabolic equations.
Now consider (16.1) for t ;;;. 0, °.;;; x .;;; L We want to specify boundary conditions for
x = 0, 1. If the problem is well posed, then we must be able to solve it by Laplace
transform. Let
u(x,s)
Then u(x,s) is the solution of
I -st .e u(x,t)dt (16.4)
su(x,s) = adzu/dxz + u(x,O) (16.5)
This is an ordinary differential equation, which has a unique solution if we specify the
boundary conditions
Correspondingly, boundary conditions for (16.1) are given by
° (16.6)
(16.7)
This type of boundary condition, namely a linear combination of u and the first derivatives
of u always leads to well posed problems. Consider, for example, the system
(16.8)
in a region t;;;' 0, (X,y)En. Here u is a column vector with n components and Aj
, Bj
, Care
nx n matrices where AI' Az are positive definite. The initial boundary-value problem is
well posed if we specify
u = ° for (X,y)Edn
This is called a Dirichlet boundary condition.
(16.9)
72
17. DIFFERENCE APPROXIMATIONS FOR INITIAL BOUNDARY'-VALUE PROBLEMS. STABILITY DEFINITION
Consider the differential equation
au(x,t)/at = au(x,t)/ax
in the quarter plane x ~ 0, t ~ 0 with initial values
(17.1)
(17 • 2)
From chapter 15 we know that the solution u(x,t) = f(x + t) is constant along the charac
teristics x + t = constant. Therefore, we do not need to prescribe any boundary condi
tions for x 0, t ~ O.
We want to solve the above problem using the 1eap~frog scheme
with initial values
(17.3b)
and assume that A = k/h < 1.
It is immediately apparent that the solution (17.3) is not uniquely determined. We
must give an additional equation for v o' Let us first consider the relation
O. (17.4)
This relation is obviously not consistent. In general this will destroy the convergence.
For example: let f (x) :: L Then u(x,t) - 1 and
where yv(t) is the solution of
1,2, .... ,
(17.5)
and (17.4) becomes
10(t) = ~ L
(17.5), (17.6) is an approximation to the problem
aw/at - - aw/ax
w(x,O) =0, w(O,t) =-1,
i.e. ,
o for t < x
w(x,t) =
-1 for t .;;; x.
(17.6)
73
Therefore,
1 for t < x vh
v1 - (- 1) for t ~ x vh
v
1
1 x
This behavior is typical for all nondissipative centered schemes. Therefore, one
needs to be very careful when overspecifying boundary conditions. The situation looks
nicer if the approximation is dissipative because the oscillations will be damped. How
ever, neat the boundary the errors are as bad. If one considers a system of equations
this error can be propagated into the interior of the region by the ingoing character
istics of coupled variables.
We now replace (17.4) by
(17.7)
which is an extrapolation. We can also eliminate V o in
to get a one-sided difference forumula. Thus (17.3), (17.7) is consistent with (17.1),
(17.2). This approximation is only usful if it is stable. If we choose
1 for V = 1
vV(k) - 0 for all V,
o for V > 1
as initial values and use the scheme (17.3a) an easy calculation shows that I Iv(t) IIhgrows like
!><constant • (t/k) 2
where
It can be shown that this growth-rate is worst possible and therefore, one might consider
(17.7) to be a useful boundary condition. However, one is seldom interested in half
infinite x-intervals. Let us therefore consider an example of B. Gustafsson; consider
(17.1), (17.2) for t ~ 0, 0 ~ x ~ 1. Then we have to specify boundary conditions at
x = 1. We use
u(l,t) O. (17.8)
74
Correspondingly, we consider (17.3) for V 1,2, ••. , N - 1, Nh
conditions
1 with boundary
v = V1 0 '
We look for solutions of the form
which grow like
(17.9)
Ilv(t) Il h ~ constant Nt / z •
Substituting (17.10) into (17.3a) and (17.9) yields
(17.10)
•• , N - 1, (17.11)
(17.12)
(17.11) is an ordinary differential equation with the general solution
where K1
, Kz are the roots of the characteristic equation
(zz_ l)K = A(Kz - l)z, A = k/h.
Substituting (17;13) into the boundary conditions (17.12) yields
There is a nontrivia1 solution if
KN KN1 Z
Det O.
K1-1 Kz-1
(17.13)
(17.14)
(17.15)
By (17.14) Kz- l/K and therefore (17.15) is equivalent to
1
Let N be even. The last equation has a solution
The corresponding z is
We have a solution
_ ( 1 log N)z - - . 1 + "2 A N1
- 1 + "2 k log N.
which grows like (17.10).
75
This behavior can be explained as follows: At the boundary x = 0 a wave is created1
which grows like N~ This wave is reflected at the boundary x = 1 and is increased by~another factor N when it hits the boundary x = 0 again, and so on. Instead of (17.9).we
could use higher-order extrapolation on the boundary x = 0, i.e.,
Dlvo(t) =0. (17.16)
However, the situation only gets worse. In this case there are solutions which grow like(j-~)tN .
Now consider a first-order system of partial differential equations
3u(x,t)/3t = Aou(x,t)/ox
in the quarter plane 0 ~ x < 00, t ~ O. Here u(x,t) = (u(l)(x,t),
is a vector function and A is a symmetric matrix of the form
A1
0
A with A1
< O,Az > O.
o Az
... ,
(17.17)
u(n)(x,t»),
The solution of (17.17) is uniquely determined if we.give initial values
u(x,O) = 0, 0 ~ x < 00
and boundary conditions
(17.18)
(17.19)
Here uI (u(l), ... , u(t») I, uII = (u(t+l),
tion of A, and S is a rectangular matrix.
... , correspond to the parti-
Remark. Here we adopt the attitude that solving the Cauchy problem is trivial.
Therefore,it is no restriction to assume that the differential equations (17.17) and the
initial conditions are homogeneous. If this is not the case then we fi~st solve the ap
propriate Cauchy problem and subtract its solution.
0,k,2k, .•• , by a differenceWe approximate (17.17) for V = 1,2, .•• and t
scheme1,2, . , (17.20)
Here
Q
is a difference operator with matrix coefficients. For the solution of (17.20) to be
uniquely determined it is necessary to specify initial values
and boundary conditions
VV(O) = 0, V = 1,2, ..• , (17.21)
q
~Cj~Vj(t + k) + F~(t), ~j=O
- r + 1, ... , o. (17.22)
76
Remark. For simplicity we formulate the theory only for explicit one-step methods.
However all results also hold for general implicit and explicit multi-step schemes,
Gustafsson et al. (1971).
We always make the obvious
Assumption 17.1. The difference approximation (17.20) is stable for the Cauchy
problem.
For convenience we make
Assumption 17.2.
are nonsingular.
The coefficient matrices A and A of the difference operator Q-r p
Let us introduce the following notations:
00 00 00
Ilv(t) II~ = Llv\l(t) 12h, IIF(t) II~ = LIF(O'k) 1
2k, Ilv(t) II~k = Lllv(O'k) II~k.
\1=1 O'~O 0'=0
We can now define stability for the initial boundary-value problem:
Definition 17.1. The difference approximation (17.20) to (17.22) is stable if there
are constants ao
and Ko such that for all F~(t) with.1 IF~(t)1 Ik < 00 and all a > a o an
estimate
o(a - a o) Ile-atv(t) II~k .;;; K~ L Ile-atF~(t) II~
~=-r+l
holds.
There are quite a number of other sta.bility definitions which we could choose. How
ever, this definition will not allow behavior like that exemplified by (17.10) and the
algebraic conditions can be generalized to problems with two boundaries and equations
with variable coefficients. The following theorem holds:
Theorem 17.1. Consider the equation (17.20) in the interval 0 < x < 1 and add for
x = 1 boundary conditions of type (17.22). The approximation is stable if the correspond
ing left and right quarter-space problems are stable. Assume that the coefficients
A. = A.(x) of the difference operator Q are two times continously differentiableJ J .
functions of x. Replace A.(x) by A.(O) or A.(l). If the corresponding left and rightJ J J
quarter-plane problems with constant coefficients are stable then the original problem
is also stable.
Another advantage of the above stability definition is that it leads to rather
simple algebraic conditions which we now derive:
The following resolvent equation is related to (17.20) and (17.22):
(zI - Q)<P\I = 0, \I = 1,2, ... , 11<pll h < 00, (17.23)
q
<P~ = LCj~<Pj(t + k) + G~, ~j=O
- r + 1, ••• , O. (17.24)
77
(17.23) is an ordinary difference equation with constant coefficients. Therefore, its
general solution <l>v with 11 <1>11 h < 00 can be written in the form
Here the Kj
are the solutions of the characteristic equation
p
det IZI - Q(k) I ... det IZI - ~:)rj 1
j=-r
o (17.26)
(17.27)
with IK.I < 1 and P.(V) are polynomials in V with vector coefficients. The degree ofJ J
Pj(V) is one less than the multiplicity of the corresponding Kj
• The following lemma is
essential:
Lemma 17.1. For Izl > 1 the characteristic equation (17.26) has no solution K. withJIKjl = 1, and the number of K
jwith IKjl < 0, counted according to their multiplicity, is
equal to the number of boundary conditions (17.24).
Definition 17.2. The Ryabenkii-Godunov condition is said to be fulfilled if the
homogeneous resolvent equations (17.23) and (17.24), i.e., G~ = 0, have no nontrivia1
solution <I> for Izl > 1.
We can determine the coefficients of the polynomials Pj(V) by introducing (17.25)
into (17.26). The following theorem then gives us algebraic stability conditions.
Theorem 17.2. The Ryabenkii-Godunov condition is fulfilled if and only if (17.23)
and (17.24) have no eigenva1ue zwith Izl > 1, i.e., (17.23) and (17.24) have a unique
solution for every Izl > 1. The approximation is stable if "and only if there is, in
addition, "a constant K > 0 such that for all z with Izl > 1 and all G~ the estimate
o(lzl - 1) II<I>II~ < K L:
holds.
Remark. We have thus strengthened the Ryabenkii-Godunov condition by also requiring
the estimate (17.27) which is an extra condition for Izl = 1 only. (If the Ryabenkii
Godunov condition is fulfilled then (17.27) holds for all z with Izl ~ 1 + 0 where 0 > 0
is any constant.)
Often the following sufficient stability conditions are useful.
Theorem 17.3. The approximation is stable if instead of (17.27) an estimate
o,,;:;; K L IG~12
~=-r+1
(17.28)
holds, i.e., we can estimate the solution at the boundary points i~ terms of G .~
We again consider the leap-frog scheme (17.3) with the boundary conditions (17.7)
and shall show that the estimate (17.27) does not hold. The equations (17.11), (17.16)
73
are now given by·
(Z2 - l)Q>V = XZ(Q>V + 1- Q>V - 1)'Dll- = G+'t'o o'
(17.29)
(17.30)
Let z (1 + T), T > O. Then the characteristic equation (17.14) has a solution
Therefore Q>v = ~lK~ and ~l is determined by
~lDtQ>o = ~l (x-jTj + O(T
j + 1)) = Go'
Thus
which shows that the estimate (17.27) cannot hold.
79
18. DIFFERENCE APPROXIMATIONS FOR THE INITIAL BOUNDARY-VALUE PROBLEM
·Some Stable Methods
In this chapter we present some stable difference methods for the initial boundary
value problem for hyperbolic equations. These approximations are all within the general
theory outlined in chapter 17. Here we present several useful methods for second and fourth
order centered schemes. The verifications of the stability of these methods and discus
sions of many other methods are to be found in Gustafsson, et al. (1972), E1vius and
Sundstrom (1972), and 01iger (1972).
Consider the system (17.17), (17.19) with g(t) _ O. We first approximate (17.17) by
the dissipative scheme
where C > 0 is a matrix that can be transformed to diagonal form together with A. We use
the condition
I IIV o = Sv0
IIand consider the following possibilities for specifying Vo
(18.2)
j a natural number (18.3a)
II II k II [ II 1 ( II II ~JVo (t + k) = Vo (t - k) + 2 h A VI (t) - 2 Vo (t + k) + Vo (t - k'l
We can now state
(18.3b)
(18.3c)
Theorem 18.1. The approximation (18.1) is stable in the sense of definition (17.1)
with the boundary condition (18.2) in combination with anyone of (18.3a), (18.3b), (18.3c).
We next approximate (17.17) by the non-dissipative leap-frog scheme.
v (t + k)V
We then have the following
v (t - k) + 2kAD v (t)V 0 V
(18.4)
Theorem 18.2. The app~oximation (18.4) is stable in the sense of definition (17.1)
with the boundary condition (18.2) in connection with either (18.3b) or (18.3c), but not
with (18.3a).
Now consider approximating (17.17) by the non-dissipative O(h~ + k 2) scheme
(18.5)
In this case we must have, in addition to the condition .(18.2), additional equations for
both V = 0 and V = 1. We consider
80
V~I(t + k)
(18.6a)11 Ir rr J+ l8vV-H (t) - gVv+2 et) + 2vV+3 et)
for V = 0 and
vvCt - k) + 3~ A [-2VV_1 (t) - t (VvCt + k) + vvCt - k»)
+ 6vv+l (t) - vv+2 (t)]
at V =" 1.
We can now state
(18.6b)
Theorem 18.3. The approximation (18.5) is stable in the sense of definition (17.1)
with the boundary condition (18.2) and the e~uations (18.6a) and (18.6b).
Note that we must use an additional equation at V = 1 for the computation of both v~
and v~I. This is a fundamentally different situation than one has with the second order
scheme (18.4). vI can be considered the incoming variable and vII the outgoing variable.
The second order scheme only requires an additional equation for vII, the outgoing variable.
This difference forces one to be extremely careful with the fourth order method. The equa
tion (18.6b) is the only equation known to the authors which generalizes to more space
dimensions easily and which is stable when used with (18.5) and (18.2) for both the ingoing
and outgoing components. Most stable methods require the use of different equations for
the ingoing and outgoing components. This can be quite troublesome with vector equations.
If we compare theorems (18.1) and (18.2) we see that the approximation (18.1) is stable
with equation (18.3a) but that the approximation (18.4) is not. This is an example of a
rather general situation: one must be more careful specifying the additional equations re
quired near the boundaries with non-dissipative schemes than with dissipative schemes.
Inspection of the extra boundary equations given here shows that we are utilizing equa
tions of lower orders of accuracy at the boundaries than our approximations at interior
points. Recent convergence theorems, Gustafsson (197lb), show that one can often sacrifice
one order of accuracy in both space and time and still retain the higher orders in the con
vergence estimates.
Now consider the equation
u = Au + Bu, 0';;; x < 00, -00 < Y < 00, t ~ 0t x Y
(18.7)
and assume A to be of the same form as in (17.17). The theory is still sketchy in this
case. Recent experiments, Oliger (1972), have established the stability of the obvious
analogues of (18.1) and (18.4) with (18.2) and (18.3c) for certain choices of A and B in
(18.7). However, it has been found that the obvious generalizations of the one-dimensional
methods are not always useful. In this case the complications arising from the second space
dimension can often be alleviated by adding a dissipative operator in the tangential direc
tion on the boundary.
81
General, irregular boundaries have only been treated by ad hoc methods. However,
these problems can perhaps be best vielved by at least thinking of locally mapping the
irregular boundary to a tangent plane and then introducing a local net with points on
planes normal and tangent to the plane tangent to the boundary. These local problems ~an
then be handled like (18.7). There remains the problem of interpolating between these nets.
19. THE SHALLOW-WATER EQUATIONS
In this chapter we consider the linearized shallow water equations
where
w =Aw +Bw +Cwt x y
w=(:), A=(~:~)' B=(::~)' c= (-~~~)$ c 0 u 0 c v \0 0 0
(19.1)
for U2 + V2 < c 2 in the region 0 .,;;; x";;; 1, 0·";;; Y .,;;; 1 and t ;;. O. We assume that u, v and $
are one-periodic in x, i.e.,
~u). (U).. v· =.. v..•·~&·~1 (19.2)
for 0 .,;;; y .,;;; L Assume V < O. For y
conditions
1, 0 .,;;; x .,;;; 1 we consider the following boundary
and
v = 0
$ = 0
(19.3a)
(19.3b)
with (19.3c)
For y
and
O,O";;;x";;; 1, we consider the corresponding boundary conditions
v = u = 0
$ = u = 0
(19.4a)
(19.4b)
(19.4c)
Remark: The homog~neous boundary conditions specified above are not realistic for ac
tual problems. However, it· is sufficient to consider this case for the subsequent analysis.
We have chosen the number of boundary conditions at y = 0, 1 in accordance with the
discussion in chapter 15. The boundary conditions (19. 3c) and (19. 4c) are chosen to guar
antee the energy estimate
11.(tlll'" Ilw(Olll', IHI' ·lllul' + lvi' + 1.I'dx""
Since dllw(t)112/dt = d(w,w)/dt the estimate (19.5) follows from.
d~ (~,w) = 2(W' ~~) = 2(w,A ~:) + 2(W,B ~;)
(19.5)
IY=l
w*Bwdx,
ly=O
-1(V0 o)(U) y=;l
(u,v,<!»0 v c v dxo c V <!> y=;O=1
83
)""1
vlul 2 +.v2 /vI 2 + 2cV<!>v +.v2 1<!>1 2dx
)""0
as a consequence of the assumption of (19.3c) and (19.4c). An energy estimate of this form
does not exist for the other b.oundary conditions. The only estimates which exist are of
the form
Therefore, the solutions are less smooth than the initial values. This results from the
fact that the amplitude of certain waves is substantially increased when they are reflected
from the boundaries.
In our opinion the best difference approximations are centered in space and time, of
leap-frog type. Therefore, we approximate (19.1) by
wet + k) = wet - k) + 2k(ADox + BDOy)W(t) + kc[w(t + k) + wet - k)] (19.7)
at all interior grid points. This scheme is, according to chapter 14, stable for the
Cauchy problem if k';;; h/CV'2c + lul + IVI). For x = 0, 1 and 0·< Y< 1 the equation (19.7)
is used with the aid of the periodicity condition
-I, +1
Unfortunately, this is not enough to determine the solution uniquely. We must add extra
conditions as discussed in chapter 18. For example, using the approximation (18.3c) for
the boundary conditions (19.3a), (19.4a) we have
at y = 1 and
(~) (t + k) = (*) (t - k) + 2k I(~~) DOx (*) (t)
+ ~ (~ ~) [t [( *)(t + k) + (~) (t - k)]
- E;l (¥) (t)]I~(t + k) = ~(t - k) + 2k lUDox~(t)
+ ~ C[EyV(t) - t [vet + k) + vet - k)] ]
+ ~ V[Ey~(t) ~ [~(t + k) + ~(t - k)])
(19.9)
(19.10)
at y = O. Recall that E w(x,y,t) = w(x,y + h,t). This approximation has recently beeny
demonstrated stable, E1vius and Sundstrom (1972).
84
As we discussed in chapter 10 one should use a scheme which is fourth order accurate
in space and second order accurate in time. This scheme is obtained from (19.7) by re
placing the operator Do by the operator
The necessary boundary equations can be constructed analogous to the example above utiliz
ing approximations of the type (18.6a) and (18.6b). Recent results indicate that a dis
sipative term in the x-direction is often necessary.
In many meteorological problems U2 + V2 ~ C2 and we have near geostrophical balance,
i.e. ,
ccjJ + fv"'" 0x
ccjJ - fu "'" 0Y
We can write (19.1) in the form
Uw+Vw+Fwx Y
where
Fw Aw +Bw +Cx Y
and
(19.11)
(
0 0 C)A = 000,
cOO
(19.12)
If it can be assumed that the term Fw is unimportant for high frequencies then we can use
a half-implicit scheme. For example,
(I - k(ADox + BD oy + C)) wet + k)
(I + k(Anox + BDoy + C)) wet - k)
+ 2k(un + VD )w(t)ox oy
The stability condition for this scheme for the Cauchy problem is k ~ h/(Iul + IVI). Thus,
one can use much longer timesteps. However, all waves traveling faster than (k/h)-l are
slowed down artificially and their contribution to the solution is completely erroneous.
It should be much better to eliminate these completely, or at least their dynamical effects,
by a filtering process. It is easy to derive such a filtering process as discussed earlier
for the Cauchy problem in chapter 13. The effect of boundaries on these filtering processes
has not been adequately treated.
New problems appear if we replace the periodicity condition(19.2) with boundary con
ditions of the form (19.3a), (19.3b), (19.4a), (19.4b) because it is not at all clear if
the differential equation problem is we11-posed. l
IBoundary conditions of type (19.3c) and (19.4c) always yield well-posed problems sinceone can derive proper energy estimates.
85"
Consider. for"examp1e, the differential equation
(19.13)
for 0 ~ x < 00, 0 ~ y < 00. t ~ 0 with boundary conditions
$ = av for x = 0, y ~ 0(19.14)
v = S$ for x ~ O. y = 0
(19.15)I
X
.; ":'y'= 0
then the differential equation becomesI I
X + y<7. 0y,' ;;. 0
-y ~ J!. ~ y'
Introduce new variables xr = X - y, y'= y - X,
with boundary conditions
. cl> = ctv for t;>.O , x' =' -y"
v = M for t ~ O. x' =" y'
The solutions of this equation can be considered as waves which are reflected from the
boundaries x' + y' = O. J!. -" y' = o. If Ias I > 1 then the amplitude will increase a fixed
amount each time the wave travels back and forth. Since the time to travel across the
region goes to zero as the origin is approached the solution grows arbitrarily fast in
the neighborhood of the origin.
This example sho~s that one must be careful with corners. Indeed. there is some ex
perimental evidence that the equation (19.1)" with a boundary condition of type (19.4b) in
a corner does not represent a well-posed problem.
It should be pointed out that the problem (19.13). (19.14) becomes well-posed if we
add a dissipative term
(~~)(;) + (~ ~}(;)xx yy
No new boundary conditions are needed.
A similar modification of (19.1) can be made to ensure the we11-posedness of (19.1)
with the boundary conditio~ (19.4b).
86
20. GRIDS
Often, the solutions of partial di~ferential.equationsare much smoother in some parts
of the region in which a solution is sought than they are in others. Furthermore, there
is often good a priori knowledge of this behavior, e.g., boundary layer phenomena. Sup
pose we want to solve these problems using difference methods. It is well known that the
complexity of the solution requires a certain net spacing to obtain given accuracy. It is
attractive for economical reasons to consider using different mesh intervals in different
parts of the region in this situation.
In this chapter we consider using refinement techniques for time dependent problems
whose behavior is essentially hyperbolic.
We now examine a refinement procedure for hyperbolic equations which is similar to a
procedure used by E. Isaacson, Isaacson (1961) to handle discontinuous coefficients of
parabolic equations. Consider the Cauchy problem for the equation
(20.1)
with initial values
w(O,x) = f(x) (20.2)
We approximate (20.1) using the leap-frog scheme with a refinement at x = O. Let
h l > h2
> 0 denote the different mesh intervals and define gridpoints by
y = (" - 1. ) h" 2 2
for " = 0, 1, 2, • • •I
x2
Let k > 0 denote the time step and
Iyo .
Io
t
and
u,,(t) = u(x",t), v,,(t) = v(y",t)
0, k, 2k, • •• the gridfunctions. Approximate (20.1) by
u,,(t + k) = u,,(t - k) - aAl(uv+l(t) - u"_l (t») (20.3)
(20.4)
for " 1,2, • • " where
The solution is uniquely determined if we give initial values for t
continuity conditions
0, k and impose the
(20.5)
at the interface x O.
87
We assume that (20.3) and (20.4) are stable for the related Cauchy problems, 1. e. ,
o .;;; at.. l' at.. 2. .;;; l.
We can consider (20.3), (20 .• 4). (20.5) as an initial-boundary value problem in the
quarter-space x;;;' 0, t ;;;. O. Associated with (20.3), (20.4), (20.5) is the following resol
vent problem:
·0 + ., = to + t, +~, IUo - u1= -p (v0- V1) + g2.
(20.6)
(20.7)
(20.8)
where p = h 1/h2. > 1. The method is stable by theorem .(17.3) if (20.6)-(20.8) have, for
Izl > 1, a unique solution with
00
where
~ (luvl 2. + IVvl2.) < 00
v=0
and there is a constant K such that
Let Izl > 1, the general solutions of (20.6) and (20.7) satisfying (20.9) are
A V 0, 1, 2,u = P1 K1' V =V
A V 0, 1, 2, eu = P2. K2.' V=V
KJ = (_l)J (2A~' ",- 1-V(,A~' ",- 1)' + 1)IK.I < 1, j 1,2, are the solutions of the characteristic· equations
J2.
K2. + (_l)j+1 -.l:..- z - 1 K - 1 = 0, j = 1, 2t..ja z
Substitute (20.11) into (20.8) to obtain
It is obvious that the estimate (20.10) holds if
detC(z) = (p + 1)(1 - K1K2.) + (p - 1)(K 1 - K2.) * 0
for Izl ~ 1. Suppose detC = 0, then
(20.9)
(20.10)
(20.11)
(20.12)
Since IK.I < 1 for Izl > 1 this last relation cannot hold for Izl > 1. Let zJ
0\ = (t.. 1a )-1 then
i8e and
88
K1 '" -iasin6 + ./ 1 - (asine) 2.
K2. '" ip-1asin6 - ./1 - ( ap- 1sine)2.
If lasin61 > 1 then IK11 < 1 and (20.12) cannot hold. If lasin61 .-; 1 then
-Im(detC(z» '" (p + 1)~sin6 (./1 - (ap-lsin6r + p-l ./1 - (asin6)2. )
+ ~sin6(p - 1)(1 + p-l) * 0 for sin6 * 0
If sin6 '" 0 we have K1 '" I, Kz '" -1 so detC(l) '" 4p * O. Thus the approximation is stable.
Consider computing an approximate solution of the problem
with initial data
and boundary conditions
u(x,O) sin'lfx
We use the scheme
u(o,t) = u(l,t) 0
where
and
DOVV(t) = (vV+1 (t) - V\r1 (t»)/ 2h
D+Vv(t) = (vV+1 (t) - V)t»)/h
In figure 20.1 we show the result of this computation for £ '" 10- 3 with h '" 10-2. and
k '" 10- 3 at time t '" 0.52. We have also computed an approximation using the previously
defined refinement procedure. We have a 5:1 refinement in the right half of the interval.
The continuity equations (20.5) are centered about the point 0.495. The mesh interval in
the coarser part of the net is hc
'" 10-2. and that in the finer portion hf '" 0.2 X 10-2..
For this computation we have used k '" 10-4 in both nets. The result at t '" 0.52 is shown
in figure 20.2.
89
1.0
.9
.9
.1
.6
> .5oll~
.~
.3
.2
.1
00 .1 .2 .3 .4 .5 .6 .7 .8
X
Figure 20.1
.9 UI
i.o,-----------------::;;;I
.9
.6
.'2
o-l--------~------_____1
The improved accuracy using the refinement is apparent. In practice the refinement
should be introduced much further to the right (nearer 1) to reduce the number of net
points, and a larger time step used in ,the coarser net. This can be done by using the
continuity conditions (20.5) at the times nk , where k is the time interval in the coarserc c
net, and then interpolating in the fine net for the first two net points at the intermediate
times mkf
•
It has been stated that the use of nonuniform grid intervals invariably causes reflec
tions. The last example shows that this need not be a problem. However, there is another
phenomenon which can cause trouble and is, in a sense made precise later, intrinsic in the
technique and unavoidable. Any wave which is poorly represented in a coarser grid will
change phase speed when passing through an interface into a finer grid. If this wave later
passes from the fine grid back into the coarse grid, a serious interaction can result with
that part of the wave which has remained in the coarse net. We begin this discussion by
first considering the related problem of using difference methods for a first order hyper
bolic equation in a quarter space with homogeneous initial data and inhomogeneous boundary
data. This discussion establishes a quantitative estimate of the change in signal speed
of waves passing through the interface of two grids.
Consider the differential equation
-ux
for x ~ 0, t ~ 0 with initial values
(20.13)
u(x,O)
and boundary condition
u(O,t)
The solution of this problem is given by
o
iate
(20.14)
(20.15)
__ {oeia(t-X)u(x,t)
x";;;t
x > t
(20.16)
90
Thus the signal speed is 1.
Approximate this problem by
late t=0,k,2k,···
(20.17)
(20.18)
(20.19)
This scheme is stable for A = k/h < 1 where k >0 denotes a time increment, h > 0 the mesh
interval in the x-coordinate and
We want to determine the signal speed f.orthe solution to these difference equations. For
this reason we introduce a new variab1e,w,,(t), defined by
(20.20)
where
{I sinakl ,,;;: 1_ . a A .• ""
a = sinak.• 'fk'arCdnA)I-A--/ > 1
We substitute (20.20) into (20.17)-(20.19) and ·obtain
2iAsinph
w (0) = w(k) = 0
" "w (t) = ei(a-Ci)t
o
We can rewrite (20.21) as
eiak(w/t + k) - w,,(t))k- l = e-iCik(w/t - k) - w,,(t))k- l
(e-f3h
(wV+-1 (t) - w/t))h- l + ef3h
(w/t) - w"_l (t))h-l)
_ k-I((eiCik _ e-iCik ) _ A(e-13h _ e(3h))w,,(t)
If Isin (ak) I AI ~ 1 then Ci = a and we can choose 13 = io, 0 real, such that
2isinak =eiak _ e-iak = A(e13h _ e-l3h)
We can then consider (20.22) as an approximation to the differential equation
Y = cosoh Y _ byt - cosak x - - x
y(x,O) = 0, y(O,t) = 1
and the solution of (20.17)-(20.19) is approximately
{.•
..ei(at-OX)y x ~ bt
v(x,t) =Ox> bt
(20.21)
(20.22 )
(20.23)
(20.24)
(20.25)
91
Thus the signal speed of the solution of the difference approximation is approximately
b ~ coaohcosqk
The relation (20.23) can be written
Sirtak . o.h--1..- = S1,n
Let us assume for simplicity that we compute with very small time steps.
(20.27) become
sinoh "'" all
b "" cosoh "'" J 1 - (ah) 2 = V1 - e;r
(20.26)
(20.27)
Then (20.26) and
where N = 2~/ah denotes the number of mesh points per wavelength in the x-direction. This
shows that N has to be quite large for b to be near 1. For example:n 32 16 8 7 6b 0.98 0.92 0.64 0.48 0
Now assume that ISi~akl > 1. Then we have
a.= k- 1arcsinA, A-1sina. = 1
and
(20.28)
for Sh . ~1,2
(eia.k _ e-ia.k) _ A(e-Sh _ eSh) = 0
Then (20.23) approximates the differential equation
and the forced wave does not propagate into the interior of the region.
If a wave is already well represented in the coarse net the previous analysis indi
cates that this wave should propagate through the interface of the coarse and fine nets
without difficulty. Computations confirm this. On the other hand, this analysis also in
dicates that there will be difficulty with the propagation of any wave which is not well
represented in the coarse net through the interface of the nets. Several computations
confirm this.
Consider the problem
ux
' x';;; 1, t;;;' 0
with initial values
u(x,O) 0
and boundary values
u(l,t) = sin2~wt
We approximate (20.29) by the difference equation
(20.29)
(20.30)
We have a mesh refinement to the right of 0.495 as in the previous computational example
and hc
= 10-2, hf
= 0.2 X10-2, k = 10-4 as before. We use OOi
= 2%, 3,%., 5%. for
92
i = 1. 2. 3 respectively. In the fine net we have Nf(OOi
} .. 5~:: .. 40. 30. 20 mesh points
per wavelength and therefore expect a gpod approxmation there. However. in the coarse net
we have only N too... ): .. 100 = 8,. 6, 4 mesh. points per wavelength and the previous analysisc\' 1. w!.
indicates that the forced wave should only propagate with the speeds d(Wi
): .. 0.64. 0, 0
respectively. Figures 20.3. 20.4 and 20:.5 are the results of the computations: at t = 1. 84.
In figure 20.3 we have 00 = 001. =; 2,% and E: = 10-6 , in figure 20.4 00 = 00
2= 3% and
E: = 0.5 X 1O~5 and in figure. 20.5 00 = 003
= 5~ and E: = 10- 5 •
1.0
1.5:T------------------,1.
1.0
.5
> ./\ AAoil 0::> V VV
-.5
-1.0-
.5
o
-.5
-1.0
/\ 11 I\IiIiAAAAAAA~hAftA \~V V v· v vv VVv vu uvu
.8 1.0.6.4..1.5,+-..,,---,----,,---.--,----,-.....-.,--,----1
-1.0 -.8 -.6 -.4 -.2 0 ~2X
Figure 20.4
1.0
Figure 20.3
-1.5+--'----''--'----'--'-,-,---r--r-I.----rl.:----l-la -.8 -.6 -.4. -.2 0 .2 .4 .6 .8
X
1.5
1.0
.5
>oil 0::>
-:5
-1.0
-1.5+----,,..--.--...-----.-..,...-.---,---r---.--j-1.0 -.8 -.6 -,4 -.2 0 .2 .4. .6 .8 loll
X
Figure: 20.5
93
We now consider mesh re~inement for the two-dimensional problem
(20.31)
with initial values
u(x,y,O) = sin (21T{6x + 3y»
and boundary conditions
u(x,O,t) = u(x,l.t)
u(O,y,t) = u(l,y,t)
t = 0, k, 2k, •
(20.32)
Define -a grid function v -et) = v (vLix , \.It.y) for positive increments t.x > 0, t.y > 0 andV,\.l
We approximate (20.31) by the leap-frog scheme
Vv \.l(t + k) = Vv (t - k) -'2k ~(Do + Do )vv \.l(t)'_ ,\.l / I ,x . ,y ,
where the additional subscripts of the D operator are used to indicate the coordinate direc
tion in which the previously defined operator acts. We use a 5:1 mesh refinement in the
center of the region. We refine about the line segments
11
: x = 01., OI.o;;;yo;;; 13
1 • x = 13, OI.o;;;yo;;; 132'
13
: 01. 0;;; X 0;;; 13, y 01.
1~: 01. 0;;; X 0;;; 13, y 13
where 01. = X + }/g 0 and 13 = 73 - X0 • We use t.x = t.y =}( 5 in the coarse net andc c
t.xf = t.Yf 1/(5 x 45) in the fine net. The refinement procedure is carried out in the
same manner as described earlier. Linear interpolation is used to provide the additional
fine net values required. In the corners of the refined area where the equations (20.5)
in the ~ and y-directions both define values for the points (t.x; + t.y;)Y2 from the corners
we use their average value. This integration was carried out with k = 10- 2 and the result
at t = 0.75 is shown in figure 20.6.
94
Figure 20.6
The situation here is much worse as is evidenced by the computation. The dependence
of the approximate solution on the mesh interval produces a phenomenon very much like the
propagation of waves in materials of varying density. There is interference of the waves
which have passed through the refined region and those which have not. It is obvious how
one can construct examples with variable coefficients which double the amplitude at selected
points. It is also obvious, since all difference methods have phase errors which are func
tions of the grid interval, that this phenomenon is present with any difference method used
with such a refinement.
We now present another example ?f the effects of phase errors in grids with non
constant mesh intervals. This is often encountered in computations in polar coordinates
where it is common to reduce the number of points as the origin is approached to avoid
being forced to use very small time steps for stability.
We study the effect of using grids with variable net spacing h(6) = ~A(6) (A is the
longitudinal coordinate and 6 the latitudinal coordinate) on the harmonic wave solutions
1/J(A,6,t) = - Ba2cos6 + CPn(cos6) + ASin[m(A - ~ t)]P~(cos6)
(Neamtan (1946~ of the vorticity equation
(~ _ I ~ ~ + I 1t ~)dt a 2sin6 dt.. d6 a 2sin6 d6 dA
x (~ 1t I ~) d1/J _d62 + cot6 d6 + sin26 dA2 + 2w aI - 0
(20.33)
(20.34)
95
Pn denotes ti~ Legendre polynomials and P~ the associated Legendre polynomials. This solu
tion, Neamtan (1946), satisfies the equation
~ .. - 2.~ (20.35)Clt m ClA
We approximate (20.35) by the differential-difference equations
.~ = - ; Do.,A (h(S»1/!(t)
of accuracy O(h 2 ) and
~ v (4 . . 1 ~at = - ID 3 Do,A(h(S» - 3 Do,A(2h(S»1/I(t~,
of accuracy O(h~). h(S) is assumed to be a smooth function of Sand
D (a)1/I(t) = 1/1(1. + a,S,t) - 1/1(1. ~ a,S,t)0,1. . 2a
(20.36)
(20.37)
The solution 1/11 of (20.36) and 1/12 of (20.37) with initial data 1/I(A,S,0) given by (20.33)
and boundary conditions
1/Ii(0,S,t)
are given by (see chapter 10)
1/Ii (21T ,S,t), i 1, 2
1/1. (A,S,t)~
1, 2, are given by
- Ba2cosS + CP (cosS)n (20.38)
and
c (h(S» = .Y. (Sinmh)1 m mh
If h is an increasing function of S it has the obvious effect of tipping the waves, i.e.,
the upper portion of the waves is travelling slower than the lower portion so the waves
tip to the left. We can further ana1yze this effect. It was noted by Starr (1948) that
the sloping of waves like that resulting from increasing h(S) produces a transport of
angular momentum from higher latitudes to lower latitudes. We can determine this effect
by computing the meridional f1uxes of angular momentum
m(a) =f.IT uvd'
where- ~u - - ClA' v = Cl1/!
ClS
If we let m.(S) denote the flux corresponding to 1/!., i = 1,2, we find~ 1
[A J2 Clc.
mi (S) = - a 2 sinS P:cosS mVt1T ()~
The function m(S) for the solution 1/1 of (20.35) is identically zero but we find that
96
oC l oC 2 ohm
l(8) and m
2(8) only vanish if ---as- and --as vanish and this is only true if as - O. Other-
omwise we find that ~~ > ° implies o~ < 0, i = 1, 2, and vice versa.
This error is of considerable importance since this meridional transport is one of
the most interesting quantities in the study of solutions of the vorticity equation (20.34).
A difference method which includes an erroneous transport process can produce extremely
misleading results. We can conclude that h(8) should be constant in those regions where
there are appreciable meridional flows like those exemplified by the solutions (20.33).
An analysis by Williamson and Browning (1972) of grids on the sphere yields the same con
clusion. They utili~e a smoothing operator in the longitudinal direction to relax the
stability condition of a grid with h(8) constant. This seems to be an. excellent technique.
97
21. DISCONTINUITIES
In chapter 17 we considered the di£ferential equation
aulat = aulax
in the quarter plane x ~ 0, t ~ 0 with initial values
u(x,O) = f(x)
and approximated by the leap-frog scheme
(21.1)
(21.2)
v = I, 2 •• It,(21.3)
According to chapter 16 there is no need to specify an~ boundary values at x = 0 for the
continuous problem (21.1),· (21.2). On the other hand, an extra boundary condition is
needed for the difference approximation. As we have shown in chapter 17, specifying
(21.4)
leads in general to a rapidly oscillating wave which travels in the direction opposite to
the characteristic and which is completely erroneous.· In Kreiss and Lundqvist (1968) it has
been shown that this is typical for nondissipative approximations. Let us now add to
(21.3) a di~~ipat~on term and consider instead
v (t + k)·V
(21.5)
with the extra boundary condition (21.4). Then, as is shown in Kreiss and Lundqvist (1968)
the erroneous wave will be damped. As an illustration we shall give the results of some
numerical experiments (figures 21.1-21.4) where we have solved the equations (21.5) with
initial values fv(O) = 0 and boundary data V o(t) == 1. k = 0.2 and h = 0.01 were used.
V 11 1 1
V -I I 1
-
0 A'f\A'f\
VV0 V U
-I -
-I -
1 I 1 11 I I
-hO h 1 X -hOh I 2 X
Figure 21.1 Figure 21.2
t=1.0, d=O t=2., d=O
V I1 I
98
I - -V I1 I
-fI
0f\_
V y
0 ,r-..
V-I - -
-I I- -
I I I11 I
-hOh I X -hOh I X
F'igur-e 21.3 Figure 21.4
ValId for botht=l and t=2 Valid for both t=l and t=2d=O.25 d=O.5
From the experiments it is obvious that the erroneous wave is most effectively damped
by a large amount of dissipation,. The only trouble with the large dissipation is that it
also affects the accuracy of the solution sought. Therefore, one should use a higher order
dissipation and (or) restrict the dissipation to the neighborhood of the boundary.
These procedures will also work for sys,tems provided that there is no coupling between
the dependent variables corresponding to the ingoing and the outgoing characteristics. To
be more precise, consider the system
A > 02
(21.6)
and d(xv
) ~ 0 is different from zero at only a few points near the
the boundary conditions for (21. 8) corresponding to (21. 7) given by
where Vv =
boundary.
and assume that the boundary conditions for the differential equations (21.6) are
Iu (O,t) = g(t)
which specify the ingoing variab1es • We approxlmate (21. 6) by
vv(t + k) = (I + d(Xv)h2D+D_)Vv(t - k) + 2kAn oVv(t)
( I II)'vv,vv '
If we use
(21. 7)
(21. 8)
then the whole approximation is accurate of order h 2110g h I away from the boundary.
Remark: It would be still better to use the dissipation operator only on the vari
ables vII, i.e., replace d(xv)h2D+D_Vv by
d!xv)(:h':+DX~~Then the accuracy is O(h2) away from the boundary •. Unfortunately, the situation is gener
ally not this nice. It is often the case that we can write the boundary conditions in the
99
form (21.7) but that the differential equations have lower order terms, i.e., instead of
(21.6) we have
(21. 7)
The corresponding differenceThen the ingoing and outgoing variables are coupled through B.
approximation has the form
vv(t + k) = (I + d(Xv)h 2D+D_)Vv( t - k) + 2kAD ovv (t) + 2kBvv (t)
IIIn a few points near the boundary the outgoing variables Vv (t) are completely wrong. This
error will be transmitted to the ingoing variables v1(t) through B and it can then be sho\YnV
that this error, which travels in the ingoing variables into the interior will be O(h).
Thus the accuracy is not very high.
No accuracy at all is left if the ingoing and outgoing variables are coupled through
the boundary conditions, i.e.,
Therefore, the same is truecompletely wrong.
the interior.
IISu (o,t)u1(O,t) =
if the difference approximation v~I(t) is
for vI(t) and this error will spread intoo
As we have shown in chapter 18 there are better ways to specify extra boundary con
ditions. Thus we can always avoid this problem altogether.
In actual computations the solutions one wants to compute often have discontinuities
which travel with the flow. These discontinuities can be considered as interior boundary
lines. Therefore, the above results apply:
1. If we do not follow the discontinuity lines and compute differences crossing these
lines then the error will be 0(1) in its neighborhood.
2. If the approximation is nondissipative then rapidly oscillating waves of the same
kind as discussed earlier will be generated. These waves travel in the opposite direction
to the characteristics.
3. If the approximation is dissipative then the error will be O(h) in the whole region
which can be reached by the characteristics which originate from the discontinuity line.
The last fact is rather disturbing. It says that we can only attain a high order of
accuracy in two ways.
1) Discontinuity fitting, i.e., we follow the propagation of the discontinuity line
and do not compute differences crossing that line. This procedure works rather satisfac
torily in one space dimension. In more space dimensions this is a very cumbersome method
and rather time consuming. See Richtmyer and Morton (1967) section 13.9 for a discussion
of this technique.
2) Instead of following the discontinuity line exactly one only follows it approxi
matively and covers it with a finer net. The main problem here is the interpolation between
the different grids. The results in chapter 20 indicate that this method should be very
useful. Experiments with shock calculations in one space dimension, e.g., Moretti and
Salas (1971), also support this method.
10Q
Finally, it should be pointed Qut that one should use a method which is as accurate
as possible and has a dissipatiye tePID such that the dif~e~ence between the order of dis
sipation and the order" of accuracy is as small as possible. The reason ;for this is that
the discontinuities can be approximated locally by stepfunctions. Stepfunctions can be
developed into Fourier series which can be modexately well approximated by their first
N terms, prow-i:ded N is large enough. As we have seen in chapter 10, the number of points
necessary to describe these frequencies decreases with increased order of accuracy. In
the difference appro~imation there are always a large number of frequencies present which
are completely in error. The amplitude of these frequencies is most effectively damped
by a dissipative term as described above.
101
REFERENCES
Chapters 1-5:
Dah1quist (1956), Gear (1971), Henrici (1962).
Chapter 6:
Be11man (1960), Forsythe and Mo1er (1967), Isaacson and Ke11er (1966), Liusternikand Sobo1ev (1961), Sneddon (1957), Varga (1962).
Chapter 7:
Aronson and Serrin (1967), Courant and Hi1bert (1962), Friedman (1964), Kreiss (1963),Ladyzhenskaya (1963), Petrovsky (1954), Thomee (1969), Sundstrom (1969b, 1972).
Chapters 8-9:
Godunov and Ryabenkii (1964), Kreiss (1962), Kreiss (1964), Kreiss and Wid1und (1967),Richtmyer and Morton (1967), Smith (1965), Thomee (1969).
Chapter 10:
Burstein and Mirin (1970), Crow1ey (1967), Fornberg (1972a), ~k1and (1958), Robertsand Weiss (1966), Rusanov (1968), Sundstrom (1969a).
Chapter 11:
Isaacson and Ke11er (1966), Lanzos (1956).
Chapter 12:
Baer and King (1967), E1iasen, et al. (1970), E11saesser (1966), Fornberg (1972a),Kreiss and 01iger (1972), Machenhauer and Rasmussen (1972), Orszag (1969, 1971a,1971b), Robert (1966, 1968a, 1968b), Swartz and Wendroff (1969).
Chapter 13:
E1vius and Sundstrom (1972), Gustafsson (1971a), Johansson and Kreiss (1963),Kurihara (1965), Marchuk, et al. (1968), McPherson (1970), Richtmyer and Morton(1967), Robert (1968b).
Chapter 14:
Fornberg (1972b), Kreiss and 01iger (1972), Phi11ips (1959a), Richtmyer and Morton(1964, 1967), Strang (1964a).
Chapter 15:
Friedrichs (1954), Kreiss (1970), Lax and Nirenberg (1966), Osher (1972a).
Chapter 16:
Friedman (1964).
Chapter 17:
Godunov and Ryabenkii (1964), Gustafsson (1971b), Gustafsson, et al. (1972), Kreiss(1968, 1971), Kreiss and Lundqvist (1968), 01iger (1972), Osher (1969a, 1969b,1969c, 1969d, 1972b), P1atzman (1954), Strang (1964b, 1966), Varah (1970, 1972),Wid1und (1966, 1970a, 1970b). .
Chapter 18:
E1vius and Sundstrom (1972), Gustafsson (1971b), Gustafsson, et al. (1972), 01iger(1972).
Chapter 19:
E1vius and Sundstrom (1972), Kreiss and 01iger (1972), 01iger (1972), Osher (1972a),Sundstrom (1972).
102
Chapter 20:
Browning et al. (1973), Ciment (1971), Osher (1970), Phi11ips (1959b), Wi11iamsonand Browning (1972).
Chapter 21:
Ape1krans (1968), Kreiss and Lundqvist (1968), Kreiss and Wid1und (1967), Hedstrom(1968), Osher (1969d).
103
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