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2
Professor D. M. Causon & Professor C. G. Mingham
Introductory Finite Difference
Methods for PDEs

3
Introductory Finite Difference Methods for PDEs
© 2010 Professor D. M. Causon, Professor C. G. Mingham & Ventus Publishing ApS
ISBN 978-87-7681-642-1

4
Contents
Contents
Preface 9
1. Introduction 10
1.1 Partial Differential Equations 10
1.2 Solution to a Partial Differential Equation 10
1.3 PDE Models 11
1.4 Classification of PDEs 11
1.5 Discrete Notation 15
1.6 Checking Results 15
1.7 Exercise 1 16
2. Fundamentals 17
2.1 Taylor’s Theorem 17
2.2 Taylor’s Theorem Applied to the Finite Difference Method (FDM) 17
2.3 Simple Finite Difference Approximation to a Derivative 18
2.4 Example: Simple Finite Difference Approximations to a Derivative 18
2.5 Constructing a Finite Difference Toolkit 20
2.6 Simple Example of a Finite Difference Scheme 24
2.7 Pen and Paper Calculation (very important) 28
2.8 Exercise 2a 32
2.9 Exercise 2b 33

5
Contents
3. Elliptic Equations 34
3.1 Introduction 34
3.2 Finite Difference Method for Laplace’s Equation 34
3.3 Setting up the Equations 37
3.4 Grid Convergence 38
3.5 Direct Solution Method 38
3.6 Exercise 3a 41
3.7 Iterative Solution Methods 42
3.8 Jacobi Iteration 43
3.9 Gauss-Seidel Iteration 45
3.10 Exercise 3b 47
3.11 Successive Over Relaxation (SoR) Method 47
3.12 Line SoR 49
3.13 Exercise 3c 51
4. Hyperbolic Equations 52
4.1 Introduction 52
4.2 1D Linear Advection Equation 53
4.3 Results for the Simple Linear Advection Scheme 55
4.4 Scheme Design 60
4.5 Multi-Level Scheme Design 67
4.6 Exercise 4a 69
4.7 Implicit Schemes 70
4.8 Exercise 4b 76

6
Contents
5. Parabolic Equations: the Advection-Diffusion Equation 77
5.1 Introduction 77
5.2 Pure Diffusion 78
5.3 Advection-Diffusion Equation 81
5.4 Exercise 5b 83
6. Extension to Multi-dimensions and Operator Splitting 84
6.1 Introduction 84
6.2 2D Scheme Design (unsplit) 84
6.3 Operator Splitting (Approximate Factorisation) 92
7. Systems of Equations 105
7.1 Introduction 105
7.2 The Shallow Water Equations 105
7.3 Solving the Shallow Water Equations 106
7.4 Example Scheme to Solve the SWE 109
7.5 Exercise 7 111
Appendix A: Definition and Properties of Order 112
A.1 Definition of O(h) 112
A.2 The Meaning of O(h) 113
A.3 Properties of O(h) 113
A.4 Explanation of the Properties of O(h) 114
A.5 Exercise A 114

7
Contents
Appendix B: Boundary Conditions 115
B.1 Introduction 115
B.2 Boundary Conditions 116
B.3 Specifying Ghost and Boundary Values 118
B.4 Common Boundary Conditions 120
B.5 Exercise B 121
Appendix C: Consistency, Convergence and Stability 123
C.1 Introduction 123
C.2 Convergence 124
C.3 Consistency and Scheme Order 124
C.4 Stability 126
C.5 Exercise C 133
Appendix D: Convergence Analysis for Iterative Methods 135
D.1 Introduction 135
D.2 Jacobi Iteration 136
D.3 Gauss-Seidel Iteration 137
D.4 SoR Iterative Scheme 139
D.5 Theory for Dominant Eigenvalues 139
D.6 Rates of Convergence of Iterative Schemes 142
D.7 Exercise D 143

8
Professor D.M. Causon and Professor C.G. Mingham
Department of Computing and Mathematics, Manchester Metropolitan University, UK
To our parents and to Mags

9
Preface
Preface
The following chapters contain core material supported by pen and paper exercises together with
computer-based exercises where appropriate. In addition there are web links to:
 worked solutions,
 computer codes,
 audio-visual presentations,
 case studies,
 further reading.
Codes are written using Scilab (a Matlab clone, downloadable for free from http://www.scilab.org/) and
also Matlab.
The emphasis of this book is on the practical: students are encouraged to experiment with different input
parameters and investigate outputs in the computer-based exercises. Theory is reduced to a necessary
minimum and provided in appendices. Web links are found on the following web page:
http://www2.docm.mmu.ac.uk/STAFF/C.Mingham/
This book is intended for final year undergraduates who have knowledge of Calculus and introductory
level computer programming.

10
Introduction
1. Introduction
This book provides an introduction to the finite difference method (FDM) for solving partial differential
equations (PDEs). In addition to specific FDM details, general concepts such as stability, boundary
conditions, verification, validation and grid independence are presented which are important for anyone
wishing to solve PDEs by using other numerical methods and/or commercial software packages. Material
is presented in order of increasing complexity and supplementary theory is included in appendices.
1.1 Partial Differential Equations
The following equation is an example of a PDE:
)
y
,
x
,
t
(
f
U
)
y
,
x
,
t
(
c
U
)
y
,
x
,
t
(
b
U
)
y
,
x
,
t
(
a yy
x
t 

 (1.1)
where,
 t, x, y are the independent variables (often time and space)
 a, b, c and f are known functions of the independent variables,
 U is the dependent variable and is an unknown function of the independent variables.
 partial derivatives are denoted by subscripts: 2
2
yy
x
t
y
U
U
,
x
U
U
,
t
U
U








 etc.
The order of a PDE is the order of its highest derivative. A PDE is linear if U and all its partial derivatives
occur to the first power only and there are no products involving more than one of these terms. (1.1) is
second order and linear. The dimension of a PDE is the number of independent spatial variables it
contains. (1.1) is 2D if x and y are spatial variables.
1.2 Solution to a Partial Differential Equation
Solving a PDE means finding the unknown function U. An analytical (i.e. exact) solution of a PDE is a
function that satisfies the PDE and also satisfies any boundary and/or initial conditions given with the PDE
(more about these later). Most PDEs of interest do not have analytical solutions so a numerical procedure must
be used to find an approximate solution. The approximation is made at discrete values of the independent
variables and the approximation scheme is implemented via a computer program. The FDM replaces all partial
derivatives and other terms in the PDE by approximations. After some manipulation, a finite difference scheme
(FDS) is created from which the approximate solution is obtained. The FDM depends fundamentally on
Taylor’s beautiful theorem (circa 1712!) which is stated in the next chapter.

11
Introduction
1.3 PDE Models
PDEs describe many of the fundamental natural laws (e.g. conservation of mass) so describe a wide range
of physical phenomena. Examples include Laplace’s equation for steady state heat conduction, the
advection-diffusion equation for pollutant transport, Maxwell’s equations for electromagnetic waves, the
Navier–Stokes equations for fluid flow and many, many more. The authors’ main interest is in solving
PDEs for fluid flow problems and details, including pictures and animations, can be found at:
http://www.docm.mmu.ac.uk/cmmfa/
1.4 Classification of PDEs
Second order linear PDEs can be formally classified into 3 generic types: elliptic, parabolic and
hyperbolic. The simplest examples are:
a) Elliptic: e.g. )
y
,
x
(
f
U
U yy
xx 
 .
This is Poisson’s equation or Laplace’s equation (when f(x,y) =0) which may be used to model the steady
state temperature distribution in a plate or incompressible potential flow. Notice there is no time derivative.
b) Parabolic: e.g. xx
t kU
U  .
This is the 1D diffusion equation and can be used to model the time-dependent temperature distribution
along a heated 1D bar.
c) Hyperbolic: e.g. xx
2
tt U
c
U  .
This is the wave equation and may be used to model a vibrating guitar string or 1D supersonic flow.
d) x
t cU
U 
 .
This first order PDE is called the advection equation. Solutions of d) also satisfy c).
e) xx
x
t kU
cU
U 
 .
This is the advection-diffusion equation and may be used to model transport of a pollutant in a river. The
coefficients k, c in the above PDEs quantify material properties that relate to the problem being solved e.g.
k could be the coefficient of thermal conductivity in the case of a heated bar, or 1D diffusion coefficient in
the case of pollutant transport; c is a wave speed, usually, in fluid flow, the speed of sound.

12
Introduction
1.4.1 Initial and Boundary Conditions
PDEs require proper initial conditions (ICs) and boundary conditions (BCs) in order to define what is
known as a well-posed problem. If too many conditions are specified then there will be no solution; if too
few conditions are specified the solution will not be unique. If the ICs/BCs are specified in the wrong
place or at the wrong time then the solution will not depend smoothly on the ICs/BCs and small errors in
the ICs/BCs will bring about large changes in the solution. This is referred to as an ill-posed problem. The
PDEs encountered in practice are often non-linear and multi-dimensional and cannot be reduced to the
simple so-called canonical forms of a) - e). However, we need to understand the properties of the solution
to these simple model PDEs before attempting to solve more complicated PDEs.
A second order elliptic PDE such as a) requires a boundary condition on U at each point on the boundary.
Thus, these are called Boundary Value (BV) problems. The BC may be a value of U on the boundary or
the value of its derivative (see Appendix B). Linear parabolic equations such a b) require ICs at the initial
start time (usually t=0) and one BC at each end-point of the spatial domain (e.g. at the ends of the heated
bar). Technically linear hyperbolic equations such as d) require ICs and as many BCs as there are inward-
pointing characteristics (this is an advanced topic which we will not cover) which depend on the sign of
wave speed c, thus:
If c>0, we need ICs: U(0,x) = f(x) and BCs: U(t,0) = g(t);
If c<0, we need ICs: U(0,x) = f(x) but no BCs.
These are called Initial Boundary Value Problems (IBV) problems.
1.4.2 Domain of Dependence
The differences between the types of PDEs can be illustrated by sketching their respective domains of
dependence. So for example, in the hyperbolic case d), point P (x0, t0) in Figure 1.1 can only be influenced
by points lying within the region bounded by the two characteristics x+ct = const and x-ct = const and
t < t0. This region is called the domain of dependence. In turn, point P can influence points at later times
lying within its zone of influence. In the parabolic case, shown in Figure 1.2 information travels
downstream (or forward in time) only and so the domain of dependence of point P (x0, t0) in this case is
the region t < t0 and the zone of influence is all points for which t > t0.

13
Introduction
Figure 1.1 Domain of dependence: hyperbolic case.
Figure 1.2 Domain of dependence: parabolic case.
x
BC
P (x0, t0)
Domain of
dependence
Zone of
influence
IC
x+ct = const
t
BC
x-ct = const
x
BC
P (x0, t0)
Domain of
dependence
Zone of
influence
IC
t
BC

14
Introduction
In the elliptic case, corresponding to subsonic flow (Figure 1.3), information travels in all directions at
infinite speed so the solution at point P (x0, t0) influences all points within the domain and vice versa.
Figure 1.3 Domain of dependence: elliptic case.
Notice in this case that the whole region bounded by the BCs is both a domain of dependence and zone of
influence.
x
BC
P (x0, t0)
Domain of
dependence
Zone of
influence
BC
y
BC
BC

15
Introduction
The type of PDE fundamentally influences the choice of solution strategy. Time dependent hyperbolic
problems and parabolic problems illustrated by Figures 1.1 and 1.2 are solved numerically by time-
marching methods which involves, as its name suggests, obtaining the numerical solution at a later time
from that at an earlier time starting from given ICs.
Elliptic problems, as illustrated in Figure 1.3 are solved numerically by so-called relaxation methods.
1.5 Discrete Notation
We will use upper case U to denote the analytic (exact) solution of the PDE and lower case u to denote the
numerical (approximate) solution. Subscripts will denote discrete points in space and superscripts discrete
levels in time. e.g. n
j
,
i
u denotes the numerical solution at grid point (i, j) in a 2D region at time level n.
1.6 Checking Results
Before applying a numerical scheme to real life situations modelled by PDEs there are two important steps
that should always be undertaken.
1.6.1 Verification
The computer program implementing the scheme must be verified. This is a check to see if the program is
doing what it is supposed to do. Comparing results from pen and paper calculations at a small number of
points to equivalent computer output is a way to (partially) verify a program. Give or take a small amount
of rounding error the numbers should be the same. Another way to verify the program is to find an exact
solution to the PDE for a simpler problem (if one exists) and compare numerical and exact results.
Complete program verification involves testing that all branches, program elements and statements are
executed and produce the expected outcomes. For large programs there exist software verification
programs to facilitate the verification process. For a commercial solver it may not be possible to
completely verify the program if the source code is unavailable.
1.6.2 Validation
Validation is really a check on whether the PDE is a good model for the real problem being studied.
Validation means comparing numerical results with results from similar physical problems. Physical
results may come from measurements from real life or from small-scale laboratory experiments. Either
way, due to measurement errors, scaling problems and the inevitable failure of the PDEs to capture all the
underlying physics, agreement between numerical and physical results will not be perfect and the user will
have to decide what is ‘close enough’.

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soft pearly silver, while mingled with them were others that
seemed laced with sable velvet or purple bands.
The secret of the hand-dipping was plain too, for, as
Thickens softly placed his fingers to the surface, first one
and then another would swim up and seem to kiss the ends,
taking therefrom some snack of rice, to dart away directly
with a flourish of the tail which set the water all a ripple,
and made it flash in the evening light.
Thickens was talking to his pets, calling them by many an
endearing name as they swam up, kissed his finger tips,
and darted away, till, becoming conscious of the presence of
some one in the garden, he started to his feet, but stooped
quickly again to pick up the basin, dip a little water, rinse
out the vessel, and throw its contents far and wide.
“I did not hear you come, Mr Bayle,” he said hastily.
“I ought to have spoken,” replied the curate gravely. “How
tame your fishes are!”
“Yes, sir, yes. They’ve got to know people from being petted
so. Dip your fingers in the water and they’ll come.”
The visitor bent down and followed the example he had
seen, with the result that fish after fish swam up, touched a
white finger tip with its soft wet mouth, and then darted off.
“Strange pets, Mr Thickens, are they not?”
“Yes, sir, yes. But I like them,” said Thickens with a droll
sidewise look at his visitor. “You see the water’s always
gently warmed from the mill there, and that makes them
thrive. They put one in mind of gold and silver, sir, and the
bank. And they’re nice companions: they don’t talk.”

He seemed then to have remembered something. A curious
rigidity came over him, and though his visitor was disposed
to linger by the pool where, in the evening light, the
brightly-coloured fish glowed like dropped flakes of the
sunset, Thickens drew back for him to pass, and then
almost backed him into the house.
“Sit down, please, Mr Bayle,” he said, rather huskily; and he
placed a chair for his visitor. “You got my note, then?”
“Yes, and I came on. You want my—”
“Help and advice, sir; that’s it. I’m in a cleft stick, sir—fast.”
“I am sorry,” said Bayle earnestly, for Thickens paused. “Is
it anything serious?”
Thickens nodded, sat down astride a Windsor chair, holding
tightly by the curved back, and rested his upper teeth on
the top, tapping the wood gently.
Bayle waited a few moments for him to go on; but he only
began rubbing at the top of the chair back, and stared at
his visitor.
“You say it is serious, Mr Thickens.”
“Terribly, sir.”
“Is it—is it a monetary question?”
Thickens raised his head, nodded, and lowered it again till
his teeth touched the chair back. “Some one in difficulties?”
Thickens nodded.
“Not you, Mr Thickens? You are too careful a man.”

“No: not me, sir.”
“Some friend?”
Thickens shook his head, and there was silence for a few
moments, only broken by the dull sound of the clerk’s teeth
upon the chair.
“Do you want me to advance some money to a person in
distress?”
Thickens raised his head quickly, and looked sharply in his
visitor’s eye; but only to lower his head again.
“No. No,” he said.
“Then will you explain yourself?” said the curate gravely.
“Yes. Give me time. It’s hard work. You don’t know.”
Bayle looked at him curiously, and waited for some minutes
before Thickens spoke again.
“Yes,” he said suddenly and as if his words were the result
of deep thought; “yes, I’ll tell you. I did think I wouldn’t
speak after all; but it’s right, and I will. I can trust you, Mr
Bayle?”
“I hope so, Mr Thickens.”
“Yes, I can trust you. I used to think you were too young
and boyish, but you’re older much, and I didn’t understand
you then as I do now.”
“I was very young when I first came, Mr Thickens,” said
Bayle smiling. “It was almost presumption for me to
undertake such a duty. Well, what is your trouble?”

“Give me time, man; give me time,” said Thickens fiercely.
“You don’t know what it is to be in my place. I am a
confidential clerk, and it is like being torn up by the roots to
have to speak as I want to speak.”
“If it is a matter of confidence ought you to speak to me, Mr
Thickens?” said Bayle gravely. “Do I understand you to say
it is a bank matter?”
“That’s it, sir.”
“Then why not go to Mr Dixon?”
Thickens shook his head.
“Mr Trampleasure? or Sir Gordon Bourne?”
“They’ll know soon enough,” said Thickens grimly. A curious
feeling of horror came over Bayle, as he heard these words,
the cold, damp dew gathered on his brow, his hands felt
moist, and his heart began to beat heavily.
He could not have told why this was, only that a vague
sense of some terrible horror oppressed him. He felt that he
was about to receive some blow, and that he was weak,
unnerved, and unprepared for the shock, just when he
required all his faculties to be at their strongest and best.
And yet the clerk had said so little—nothing that could be
considered as leading up to the horror the hearer foresaw.
All the same though, Bayle’s imagination seized upon the
few scant words—those few dry bones of utterance, clothed
them with flesh, and made of them giants of terror before
whose presence he shook and felt cowed.
“Tell me,” he said at last, and his voice sounded strange to
him, “tell me all.”

There was another pause, and then Thickens, who looked
singularly troubled and grey, sat up.
“Yes,” he said, “I’ll tell you all. I can trust you, Mr Bayle. I
don’t come to you because you are a priest, but because
you are a man—a gentleman who will help me, and I want
to do what’s right.”
“I know—I believe you do, Thickens,” said the curate
huskily, and he looked at him almost reproachfully, as if
blaming him for the pain that he was about to give.
He felt all this. He could not have explained why, but as
plainly as if he had been forewarned, he knew that some
terrible blow was about to fall.
Thickens sat staring straight before him now, gnawing hard
at one of his nails, and looking like a man having a hard
struggle with himself.
It was a very plainly-furnished but pleasant little room,
whose wide, low window had a broad sill upon which some
half-dozen flowers bloomed, and just then, as the two men
sat facing each other, the last glow of evening lit up the
curate’s troubled face, and left that of Thickens more and
more in the shade.
“That’s better,” he said with a half laugh. “I wish I had left it
till it was dark. Look here, Mr Bayle, I’ve been in trouble
these five years past.”
“You?”
“Yes, sir. I say it again: I’ve been in trouble these six or
seven years past, and it’s been a trouble that began like a
little cloud as you’d say—no bigger than a man’s hand; and
it grew slowly bigger and bigger, till it’s got to be a great,

thick, black darkness, covering everything before the storm
bursts.”
“Don’t talk riddles, man; speak out.”
“Parables, Mr Bayle, sir, parables. Give me time, sir, give me
time. You don’t know what it is to a man who has trained
himself from a boy to be close and keep secrets, to have to
bring them out of himself and lay them all bare.”
“I’ll be patient; but you are torturing me. Go on.”
“I felt it would, and that’s one of the things that’s kept me
back, sir; but I’m going to speak now.”
“Go on.”
“Well, sir, a bank clerk is trained to be suspicious. Every
new customer who comes to the place is an object of
suspicion to a man like me. He may want to cheat us. Every
cheque that’s drawn is an object of suspicion because it
may be a forgery, or the drawer may not have a balance to
meet it. Then money—the number of bad coins I’ve
detected, sir, would fill a big chest full of sham gold and
silver, so that one grows to doubt and suspect every
sovereign one handles. Then, sir, there’s men in general,
and even your own people. It’s a bad life, sir, a bad life, a
bank clerk’s, for you grow at last so that you even begin to
doubt yourself.”
“Ah! but that is a morbid feeling, Thickens.”
“No, sir, it’s a true one. I’ve had such a fight as you couldn’t
believe, doubting myself and whether I was right: but I
think I am.”

“Well,” said the curate, smiling a faint, dejected smile; “but
you are still keeping me in the dark.”
“It will be light directly,” said Thickens fiercely, “light that is
blinding. I dread almost to speak and let you hear.”
“Go on, man; go on.”
“I will, sir. Well, for years past I’ve been in doubt about our
bank.”
“Dixons’, that every one trusts?”
“Yes, sir, that’s it. Dixons’ has been trusted by everybody.
Dixons’, after a hundred years’ trial, has grown to be looked
upon as the truth in commerce. It has been like a sort of
money mill set going a hundred years ago, and once set
going it has gone on of itself, always grinding coin.”
“But you don’t mean to tell me that the bank is unsafe?
Man, man, it means ruin to hundreds of our friends!”
He spoke in an impassioned way, but at the same time he
felt more himself; the vague horror had grown less.
“Hear me out, sir; hear me out,” said Thickens slowly.
“Years ago, sir, I began to doubt, and then I doubted
myself, and then I doubted again, but even then I couldn’t
believe. Doubts are no use to a man like me, sir; he must
have figures, and figures I couldn’t get to prove it, sir. I
must be able to balance a couple of pages, and then if the
balance is on the wrong side there’s something to go upon.
It has taken years to get these figures, but I’ve got them
now.”
“Thickens, you are torturing me with this slow preamble.”

“For a few minutes, sir,” said the clerk pathetically, “for an
hour. It has tortured me for years. Listen, sir. I began to
doubt—not Dixons’ stability, but something else.”
The vague horror began to increase again, and Christie
Bayle’s hands grew more damp.
“I have saved a little money, and that and my writings were
in the bank. I withdrew everything. Cowardly? Dishonest?
Perhaps it was; but I doubted, sir, and it was my little all.
Then you’ll say, if I had these doubts I ought to have
spoken. If I had been sure perhaps I might; but I tell you,
sir, they were doubts. I couldn’t be false to my friends
though, and where here and there they’ve consulted me
about their little bits of money I’ve found out investments
for them, or advised them to buy house property. A
clergyman for whom I changed a cheque one day, said it
would be convenient for him to have a little banking account
with Dixons’, and I said if I had an account with a good
bank in London I wouldn’t change it. Never change your
banker, I said.”
“Yes, Thickens, you did,” said the curate eagerly, “and I
have followed your advice. But you are keeping me in
suspense. Tell me, is there risk of Dixons’ having to close
their doors?”
“No, no, sir; it’s not so bad as that. Old Mr Dixon is very
rich, and he’d give his last penny to put things straight. Sir
Gordon Bourne is an honourable gentleman—one who would
sacrifice his fortune so that he might hold up his head. But
things are bad, sir, bad; how bad I don’t know.”
“But, good heavens, man! your half-yearly balance-sheets—
your books?”

“All kept right, sir, and wonderfully correct. Everything looks
well in the books.”
“Then how is it?”
“The securities, sir,” said Thickens, with his lip quivering.
“I’ve done a scoundrelly thing.”
“You, Thickens? You? I thought you were as honest a man
as ever trod this earth!”
“Me, sir?” said the clerk grimly. “Oh, no! oh, no! I’m a
gambler, I am.”
The vague horror was dissolving fast into thin mist. “You
astound me!” cried Bayle, as he thought of Sir Gordon’s
doubts of Hallam. “You, in your position of trust! What are
you going to do?”
The grim smile on James Thickens’s lips grew more
saturnine as he said:
“Make a clean breast of it, sir. That’s why I sent for you.”
“But, my good man!—oh, for heaven’s sake! go with me at
once to Sir Gordon and Mr Hallam. I ought not to listen to
this alone.”
“You’re going to hear it all alone,” said James Thickens,
growing still more grim of aspect; “and when I’ve done
you’re going to give me your advice.”
Bayle gazed at him sternly, but with the strange oppression
gone, and the shadow of the vague horror fading into
nothingness.

“I’m confessing to you, sir, just as if I were a Roman
Catholic, and you were a priest.”
“But I decline to receive your confession on such terms,
James Thickens,” cried Bayle sternly. “I warn you that, if
you make me the recipient of your confidence, I must be
free to lay the case before your employers.”
“Yes, of course,” said Thickens with the same grim smile.
“Hear me out, Mr Bayle, sir. You’d never think it of me, who
came regularly to church, and never missed—you’d never
think I had false keys made to our safe; but I did. Two
months ago, in London.”
Bayle involuntarily drew back his chair, and Thickens
laughed—a little hard, dry laugh.
“Don’t be hard on the man, Mr Bayle, who advised you not
to put your money and securities in at Dixons’.”
“Go on, sir,” said the curate sternly.
“Yes: I will go on!” cried Thickens, speaking now excitedly,
in a low, harsh voice. “I can’t carry on that nonsense. Look
here, sir,” he continued, shuffling his chair closer to his
visitor, and getting hold of his sleeve, “you don’t know our
habits at the bank. Everything is locked up in our strong-
room, and Hallam keeps the key of that, and carefully too! I
go in and out there often, but it’s always when he’s in the
room, and when he is not there he always locks it, so that,
though I tried for years to get in there, I never had a
chance.”
“Wretched man!” cried Bayle, trying to shake off his grip,
but Thickens’s fingers closed upon his arm like a claw.

“Yes, I was wretched, and that’s why I had the keys made,
and altered again and again till I could get them to fit. Then
one day I had my chance. Hallam went over to Lincoln, and
I had a good examination of the different securities, shares,
deeds—scrip of all kinds—that I had down on a paper, an
abstract from my books.”
“Well, sir?”
“Well, sir? Half of them are not there. They’re dummies tied
up and docketed.”
“But the real deeds?”
“Pledged for advances in all sorts of quarters. Money raised
upon them at a dozen banks, perhaps, in town.”
“But—I don’t understand you, Thickens; you do not mean
that you—”
“That I, Mr Bayle!” cried the clerk passionately. “Shame
upon you!—do you think I could be such a scoundrel—such
a thief?”
“But these deeds, and this scrip, what are they all?”
“Valuable securities placed in Dixons’ hands for safety.”
“And they are gone?”
“To an enormous amount.”
“But, tell me,” panted Bayle, with the horror vague no
longer, but seeming to have assumed form and substance,
and to be crushing him down, “who has done this thing?”
“Who had the care of them, sir?”

“Thickens,” cried Bayle, starting from his chair, and catching
at the mantelpiece, for the room seemed to swim round,
and he swept an ornament from the shelf, which fell with a
crash, “Thickens, for heaven’s sake, don’t say that.”
“I must say it, sir. What am I to do? I’ve doubted him for
years.”
“But the money—he has lived extravagantly; but, oh! it is
impossible. It can’t be much.”
“Much, sir? It’s fifty thousand pounds if it’s a penny!”
“But, Thickens, it means felony, criminal prosecution, a
trial.”
He spoke hoarsely, and his hands were trembling. “It means
transportation for one-and-twenty years, sir—perhaps for
life.”
Bayle’s face was ashy, and with lips apart he stood gazing
at the grim, quiet clerk.
“Man, man!” he cried at last; “it can’t be true.”
“Do you doubt too, sir? Well, it’s natural. I used to, and I
tried to doubt it; a hundred times over when I was going to
be sure that he was a villain, I used to say to myself as I
went and fed my fish, it’s impossible, a man with a wife and
child like—”
“Hush! for God’s sake, hush!” cried Bayle passionately, and
then with a burst of fury, he caught the clerk by the throat.
“It is a lie; Robert Hallam could not be such a wretch as
that!”

“Mr Bayle, sir,” said Thickens calmly, and in an appealing
tone; “can’t you see now, sir, why I sent to you? Do you
think I don’t know how you loved that lady, and how much
she and her bright little fairy of a child are to you? Why, sir,
if it hadn’t been for them I should have gone straight to Sir
Gordon, and before now that scoundrel would have been in
Lincoln jail.”
“But you are mistaken, Thickens. Man, man, think what you
are saying. Such a charge would break her heart, would
brand that poor innocent child as the daughter of a felon.
Oh, it cannot be!” he cried excitedly. “Heaven would not
suffer such a wrong.”
“I’ve been years proving it, sir; years,” said Thickens slowly;
“and until I was sure, I’ve been as silent as the dead. Fifty
thousand pounds’ worth of securities at least have been
taken from that safe, and dummies filled up the spaces.
Why, sir, a score of times people wanted these deeds, and
he has put them off for a few days till he could go up to
London, raise money on others, and get those wanted from
the banker’s hands.”
“But you knew something of this, then?”
“Yes, I knew it, sir—that is, I suspected it. Until I got the
keys made, I was not sure.”
“Does—does any one else know of this?”
“Yes, sir.”
“Ah!” ejaculated Bayle, with quite a moan.
“Robert Hallam, sir.”

“Ah!” ejaculated Bayle, drawing a breath full of relief. “You
have not told a soul?”
“No, sir. I said to myself there’s that sweet lady and her
little child; and that stopped me. I said to myself, I must go
to the trustiest friend they have, sir, and that was you. Now,
sir, I have told you all. The simple truth. What am I to do?”
Christie Bayle dropped into a chair, his eyes staring, his
blanched face drawn, and his lips apart, as he conjured up
the scene that must take place—the arrest, the wreck of
Mrs Hallam’s life, the suffering that would be her lot. And at
last, half maddened, he started up, and stood with clenched
hands gazing fiercely at the man who had fired this train.
“Well, sir,” said Thickens coldly, “will you get them and the
old people away before the exposure comes?”
“No,” cried Bayle fiercely, “this must not—shall not be. It
must be some mistake. Mr Hallam could not do such a
wrong. Man, man, do you not see that such a charge would
break his wife’s heart?”
“It was in the hope that you would do something for them,
sir, that I told you all this first.”
“But we must see Mr Dixon and Sir Gordon at once.”
“And they will—you know what.”
“Hah! the matter must be hushed up. It would kill her!”
cried Bayle incoherently. “Mr Thickens, you stand there like
this man’s judge; have you not made one mistake?”
Thickens shook his head and tightened his lips to a thin line.
“Do you not see what it would do? Have you no mercy?”

“Mr Bayle, sir,” said Thickens slowly, “this has served you as
it served me. It’s so stunning that it takes you off your
head. Am I, the servant of my good masters, knowing what
I do, to hide this from them till the crash comes first—the
crash that is only a matter of time? Do you advise—do you
wish me to do this?”
Christie Bayle sat with his hands clasping his forehead, for
the pain he suffered seemed greater than he could bear. He
had known for long enough that Hallam was a harsh
husband and a bad father; but it had never even entered
his dreams that he was other than an honest man. And now
he was asked to decide upon this momentous matter, when
his decision must bring ruin, perhaps even death, to the
woman he esteemed, and misery to the sweet, helpless
child he had grown to love.
It was to him as if he were being exposed to some
temptation, for even though his love for Millicent had long
been dead, to live again in another form for her child,
Christie Bayle would have gone through any suffering for
her sake. As he bent down there the struggle was almost
greater than he could bear.
And there for long he sat, crushed and stunned by the
terrible stroke that had fallen upon him, and was about to
fall upon the helpless wife and child. His mind seemed
chaotic. His reasoning powers failed, and as he kept clinging
to little scraps of hope, they seemed to be snatched away.
It was with a heart full of grief mingled with rage that he
started to his feet at last, and faced Thickens, for the clerk
had again spoken in measured tones. “Mr Bayle, what am I
to do?”

The curate gazed at him piteously, as he essayed to speak;
but the words seemed smothered as they struggled in his
breast.
Then, by a supreme effort, he mastered his emotion, and
drew himself up.
“Once more, sir, what am I to do?”
“Your duty,” said Christie Bayle, and with throbbing brain he
turned and left the house.

Volume Two—Chapter Seven.
Christie Bayle Changes his Mind.
“God help me! What shall I do?” groaned Christie Bayle, as
he paced his room hour after hour into the night. A dozen
times over he had been on the point of going to Thickens,
awakening him and forcing him to declare that he would
keep the fearful discovery a secret until something could be
done.
“It is too horrible,” he said. “Poor Millicent! The disgrace! It
would kill her.”
He went to the desk and began to examine his papers and
his bank-book.
Then he relocked his desk and paced the room again. “Julie,
my poor little child, too. The horror and disgrace to rest
upon her little innocent head. Oh, it is too dreadful! Will
morning never come?”
The hours glided slowly by, and that weary exclamation rose
to his lips again and again:
“Will morning never come?”
It seemed as if it never would be day, but long before the
first faint rays had streaked the east he had made his plans.
“It is for her sake; for her child’s sake. At whatever cost, I
must try and save them.”
His first ideas were to go straight to Hallam’s house; but
such a course would have excited notice. He felt that

Millicent would think it strange if he went there early. Time
was of the greatest importance, but he felt that he must not
be too hasty, so seated himself to try and calm the
throbbings of his brain, and to make himself cool and
judicial for the task he had in hand.
Soon after seven he walked quietly downstairs, and took his
hat. It would excite no surprise, he thought, for him to be
going for a morning walk, and, drawing in a long breath of
the sweet refreshing air, he began to stride up the street.
“How bright and beautiful is thy earth, O God!” he
murmured, as the delicious morning sunshine bathed his
face, “and how we mar and destroy its beauties with our
wretched scheming and plans! Ah! I must not feel like this,”
he muttered, as a restful hopefulness born of the early day
seemed to be infusing itself throughout his being.
He had no occasion to check the feeling of content and rest,
for he had not gone a dozen yards before the whole force of
his position flashed upon him. He felt that he was a plotter
against the prosperity of the town—that scores of the
people whose homes he was passing were beginning the
day in happy ignorance that perhaps the savings of a life
were in jeopardy. Ought he not to warn them at once, and
bid them save what they could out of the fire?
For his conscience smote him, asking him, how he, a
clergyman, the preacher of truth and justice and innocence,
could be going to temporise, almost to join in the fraud by
what he was about to do?
“How can I meet my people after this?” he asked himself;
and his face grew careworn and lined. The old reproach
against him had passed away. No one could have called him
young and boyish-looking now.

“Morning, sir,” cried a harsh voice.
Bayle started, and flushed like some guilty creature, for he
had come suddenly upon old Gemp as he supposed, though
the reverse was really the case.
“Going for a walk, sir?” said Gemp, pointing at him, and
scanning his face searchingly.
“Yes, Mr Gemp. Fine morning, is it not?”
Gemp stood shaving himself with one finger, as the curate
passed on, and made a curious rasping noise as the rough
finger passed over the stubble. Then he shook his head and
began to follow slowly and at a long distance.
“I felt as if that man could read my very thoughts,” said
Bayle, as he went along the street, past the bank, and out
into the north road that led towards the mill.
He shuddered as he passed Dixons’, and pictured to himself
what would happen if the doors were closed and an excited
crowd of depositors were hungering for their money.
“It must be stopped at any cost,” he muttered; and once
more the sweet sad face of Millicent seemed to be looking
into his for help.
“I ought to have suspected him before,” he continued; “but
how could I, when even Sir Gordon could see no wrong? Ha!
Yes. Perhaps Thickens is mistaken after all. It may be, as he
said, only suspicion.”
His heart seemed like lead, though, the next moment, as he
neared the clerk’s house. Thickens was too just, too careful
a man to have been wrong.

He stopped, and rapped with his knuckles at the door
directly after, to find it opened by Thickens himself, and, as
the clerk drew back, he passed in, ignorant of the fact that
Gemp was shaving himself with his rough forefinger a
hundred yards away, and saying to himself, “Which is it?
Thickens going to marry skinny Heathery on the sly; or
something wrong? I shan’t be long before I know.”
The brightness of the morning seemed to be shut out as the
clerk closed the door, and followed his visitor into the
sitting-room.
“Well, Mr Bayle,” he said, for the curate was silent. “You’ve
come to say something particular.”
“Yes,” said Bayle firmly. “Thickens, this exposure would be
too horrible. It must not take place.”
“Ah,” said Thickens in his quiet, grave way, “you’re the
Hallams’ friend.”
“I hope I am the friend of every one in this town.”
“And you advise me to keep this quiet and let your friends
be robbed?”
“Silence, man! How dare you speak to me like that?” cried
Bayle furiously, and he took a step in advance. “No, no,” he
said, checking himself, and holding out his hand; “we must
be calm and sensible over this, Thickens. There must be no
temper. Now listen. You remember what I said you must do
last night.”
“Yes; and I’m going directly after breakfast to Sir Gordon.”
“No; I retract my words. You must not go.”

“And the people who have been robbed?”
“Wait a few moments, Thickens,” cried Bayle, flushing, as
he saw that his hand was not taken. “Hear me out. You—
yes, surely, you have some respect for Mrs Hallam—some
love for her sweet child.”
Thickens nodded.
“Think, then, man, of the horrible disgrace—the ruin that
would follow your disclosures.”
“Yes; it is very horrid, sir; but I must do my duty. You
owned to it last night.”
“Yes, man, yes; but surely there are times when we may try
and avert some of the horrors that would fall upon the
heads of the innocent and true.”
“That don’t sound like what a parson ought to say,” said
Thickens dryly.
Bayle flushed angrily again, but he kept down his wrath.
“James Thickens,” he said coldly, “you mistake me.”
“No,” said Thickens, “you spoke out like a man last night.
This morning, sir, you speak like Robert Hallam’s friend.”
“Yes; as his friend—as the friend of his wife; as one who
loves his child. Now listen, Thickens. To what amount do
you suppose Hallam is a defaulter?”
“How can I tell, sir? It is impossible to say. It can’t be
hushed up.”
“It must, it shall be hushed up,” said Bayle sternly. “Now,
look here; I insist upon your keeping what you know quiet

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