FINITE DIFFERENCE FORMULAE FOR THE
SQUARE LATTICE
By W. G. BICKLEY (Imperial College, London)
[Received 20 October 1947]
SUMMARY
The paper gives approximate formulae for derivatives (including combinations
like V2 and V4), and integrals, of a function of two independent variables, in terms
of its values at nodes of a square lattice, primarily for use in the numerical solution
of partial differential equations. Consideration is given to the form, as well as to
the magnitude, of the leading terms in the error, and what is believed to be for
most purposes optimum combinations are thus selected for the simpler compact
sets of nodes.
1. Introduction
THE use of numerical methods in solving problems in applied mathematics
is becoming increasingly common for partial, as well as for ordinary,
differential equations. The first step in such a process is usually the
replacement of differential equations by their finite difference approximations. For ordinary differential equations, approximations of various
orders of accuracy to derivatives and integrals are well known and listed, f
but for partial differential equations it is as yet rare to find any but the
crudest approximations employed. J§
For many purposes formulae of the Lagrangian type are convenient.
In dealing with partial differential equations with two independent
variables, equidistant, and equal, intervals in the independent variables
will usually be employed, so that the required function is computed at the
nodes of a square lattice (net). The object of this paper is to explore the
possibilities of refining approximations to the quantities—derivatives and
integrals—which occur in the formulation of problems in applied mathematics.
Considerable use is made of symbolic operators, which are powerful in
developing such formulae; a posteriori proofs of the formulae are readily
constructed—if they are deemed necessary.
f W. G. Bickley, 'Formulae for numerical integration', Math. Qaz. 23 (1939), 352-9;
id., 'Formulae for numerical differentiation', ibid. 25 (1941), 19-26; L. J. Comrie, Interpolation and Allied Tables, H.M. Stationery Office (reprinted 1942).
f Some attempt is made by A. Thorn, 'Arithmetical solution of equations of the type
W = const.1, A.R.C.B. and M. (1933), 1604.
§ Formulae of various types are given by L. Collatz, Eigenwertproblem und ihre numerische
Behandlung, Akad. Verlag. (Leipzig, 1945).
36
W. G. BICKLEY
We refrain from giving examples of the use of the formulae since this
is quite straightforward. Nor do we advocate that the refinements should
be used in the early stages, or on the coarser nets. Their proper place is
in the final stages, where they may add considerably to the accuracy of
a solution with but little extra labour and without further reducing the
mesh length.
2. Symmetric sums
We cover the (x, y) -plane with a square lattice of mesh length a, and
label a typical point 0 (see Fig. 1). Neighbouring points are numbered
IO
II
6
2
5
3
O
1
7
4
8
9
12
FIG.
1.
1, 2,..., 12, as in the jiire, and these numbers will be used as subscripts
to indicate that the value of any quantity (usually f(x, y)) is to be taken
at the point in question.
We commence by recalling the symbolic form of Taylor's series in one
variable,
•>
f(x+h) = ev>f(x),
(1)
where
D == d/dx.
To deal with two independent variables {x, y) with a mesh length a it
is convenient to introduce the symbolic operators
f = a d/dx,
t) = a djdy.
(2)
FINITE DIFFERENCE FORMULAE FOR THE SQUARE LATTICE 37
In terms of these we may write
/ i = ef/o.
/2 = eVo )
/ s = e-*/o.
f* = e*>fo
h = et+%,
(3)
J
and so on.
We note that
?+yf = o2V2
and find it convenient to write
(4)
Now it will usually happen that the quantities occurring in the mathematical equations will be independent of the choice of axes—invariant
for rotation of these axes—and clearly values of such quantities at 0 can
be represented or approximated only by sums of values at points symmetrically disposed about 0. Of such 'symmetric sums' three only (apart
from /„ itself) are simple (in the sense that they include only four terms),
and involve points not too remote from 0, namely,
+e-v)fo = 2(cosh^+coshr?)/0
(6)
f-ri+e(-ri)f0
= 4 cosh £ cosh 77/0
+....
(8)
The next symmetrical sum involves eight terms:
f(±2a,±a)
2
and f(±a, ±2a).
3. Approximations to V /
By equation (6) we have immediately the well-known and most frequently applied approximation to V2/, namely,
V2/0 = {S1-if0}la2+0(a^),
(9)
usually attributed to Liebmann.
38
W. G. BICKLEY
Equations (7) and (8) lead to what is essentially the same result for
mesh lengths aV2 and 2a respectively.
The problem of improving the accuracy conveniently is not simple.
Clearly the use of ^ and # 2 cannot eliminate both V4 and S 4 , and although
we can eliminate both by using 8X and S3, we use points distant 2a from 0,
which complicates procedure at a boundary, and also we cannot then
obtain any help from S2.
The use we can best make of 8t and 82 is to eliminate the term in ^ 4 ,
since this is not invariant for rotation of axes. The resulting 'error' term
in a4 then involves only V4/0, and is thus (a) invariant, (6) calculable (at
least approximately) from the values of V2/, and (c) small (theoretically
zero) for Laplace's equation.
The resulting formula is
4SH-$-= 20/ 0 +6a2V 2 / 0 +KV 4 /o+> 6 (V 4 +2^ 4 )V 2 / 0 +..., (10)
and from thisf
V2/o = {4^+^-20/ 0 }/6a 2 -ia 2 V 4 / 0 +O(a 4 ):
(11)
4
One may use (9) to compute V /0 from the (approximate) values of
V2/, with an error of order a2, so that if the term — ^a2V4/0 is allowed for
in this way, the error in (11) is of order a4. This procedure is equivalent
to the use of a fourth difference correction (as advocated by Fox),| and
has, indeed, the advantage that it is a iwo-dimensional correction, whereas
all Fox's fourth (and higher) differences are differences of one-dimensional
sequences.
We can, alternatively, eliminate completely the a4 terms between (6)
and (8), obtaining
1 6 ^ - 3 , = 60/ 0 +12a 2 V 2 / 0 -Aa 8 (V 4 -3^ 4 )V 2 / 0 ...
(12)
2
or
V /o = {lft^-$-60/ 0 }/12a«+O(a«).
(13)
If/ 0 is an approximate solution of Laplace's equation, then the last
term written in (12) .<' )ws that the 'error' of (13) is of order higher than
a4. But, as already indicated, the superior accuracy of (13) is purchased
by wider spread of the points used.
Using Sv S2, and #3 we may eliminate V2/o and ^ 4 / 0 , and obtain the
well-known approximation,
V4/0 = {20f0-8S1+2S2+S3}/a^+O(ai),
(14)
and clearly nothing better is available unless we include additional points.
t This formula is given, effectively, by P. M. and A. M. Woodward, 'Four-figure tables
of the Airy function in the complex plane', Phil. Mag. (7) 37 (1946), 259.
J L. Fox, in his paper 'Some improvements in the use of relaxation methods for the
solution of ordinary and partial differential equations', Proc. Roy. Soc. A, 190 (1947), 31-59,
corrects the crude finite difference approximation by the use of higher differences.
FINITE DIFFERENCE FORMULAE FOR THE SQUARE LATTICE 39
4. Other derivatives
First-order derivatives occur infrequently in the governing differential
equations, but are common in boundary conditions.
The crudest approximation to (df/dx)0 is Euler's, namely, (A—/<>)/«, but
this is an approximation 'centred' at the mid-point of the mesh 01. For
the value centred at 0 we use
fi-f* = 2sinh|/ 0
so that
(8f/dx)0 = (/ 1 -/ 3 )/2a+O(a 2 ).
(16)
To improve this we must clearly use approximations to (8fjdx)2 and
)^ in equal proportions. Now
= 4 sinh £ cosh Tjf0
It is clear that potentially the most useful combination will be one in
which (since they cannot be eliminated) the third-order terms constitute
a multiple of V^df/dx). We readily find the combination
= Ua(df/dx)0+2a?V2(df/dx)0+O(aS)....
(18)
By using the approximation similar to (9) in the second term on the
right-hand side of (18), the relative error in (8f/8x)0 may be made of
order a4.
When the point 0 is a boundary point, the use of the above formulae
necessitates values at 'fictitious' points, outside the boundary, but the
use of such points is a commonplace method of the 'relaxation' and other
techniques.
Boundary derivatives at points other than nodes, and normal or tangential derivatives for curved boundaries, do not lend themselves to any
simple treatment if any improvement upon the crudest approximation is
desired.
The formula for {df/8y)0, corresponding to (18), may immediately be
written down.
5. Surface integrals
In many problems the integral of a function taken over some region,
determined at the nodes of a lattice, is required.
40
W. G. BICKLEY
The simplest approximation (corresponding to the trapezoidal rule in
one dimension) associates the value of the integrand at the node with the
square of side a centred at the node (Fig. 2), and applies a factor £ to
nodes on a boundary and J to nodes at corners.
6
2
7/ V/
/ / //
3
7
4
FIG.
8
2.
An obvious improvement is to use Simpson's rule in both directions.
According to this
J jfdxdy = jiWo+^+S,),
(19)
—a —a
but this is not the best that can be done with the nine nodes 0, 1,..., 8.
We have
a
a
1 1
-a
—1 - 1
J J f{x,y)dxdy = a2 J J e'twfodrds
-a
-4a
smhjjjinhrj
_
/o
l
+...}/ 0 .
(20)
By the use of f0, Sv and 82 we can obtain the correct amounts of any
three of f0, V2/o, V4/0, and i^4/0—but not of all four. If one must remain,
clearly it should best be V4/0. We find, with no difficulty,
J
(21)
FINITE DIFFERENCE FORMULAE FOR THE SQUARE LATTICE 41
The numerical coefficients have become larger-—beyond the scope of
mental arithmetic—but the formula has the merit that the value of the
leading term of the error is readily calculable, and is small for solutions
of the Laplace equation.
We may also compare (19) with (20), and the result is
J
(22)
in which the term in a6 is not readily computed.
Two cruder formulae may be mentioned, giving approximations to the
same double integral, namely
').
(23)
•).
(24)
Finally, it may be worth recording the approximation to the integral
over the square of side a centred at the point 0 (Fig. 2), in terms of the
values at the points 0, 1,..., 8.
la
J
-ha
ia
r ,,
. , ,
-ha
•
i
i
•
J.
i
•.
. , sinn *t sinn *ri .
I f(x,y)dxdy = 4a2
2|
Uf0
?7?
}f0,
(25)
and comparison with (6) and (7) leads to the result
J
Jf(x, y) dxdy = ^ ( 1 2 4 4 / , , + 38$+1 IS,) -s»fea8VVo+0(a»).
-la -Ja
.
(26)
It is at first sight attractive to apply this successively to all nodes of
a region. The result is that the factor for all internal nodes is unity—
which clearly cannot be expected to give the best results. The explanation lies in the fact that the regions to be associated with nodes on the
boundary are not complete squares but half-squares or (at corners) quartersquares. The contributions from these are not given by symmetric sums,
so that the corrections must be applied to a border of nodes on and within
the boundary. The situation is parallel to the Gregory integration formula
in one dimension. Corresponding correction terms could be worked out,
but their use would be a complication which (21) or (22) completely
avoids.
42 FINITE DIFFERENCE FORMULAE FOR THE SQUARE LATTICE
()()()
06}
(7)
(Wi)
-Jo - i o
12a
FIG. 3. Computational molecules.
6. Computational molecules
It seems serviceable to represent pictorially the formula given above,
in the form of diagrams indicating the factor to be applied to the function
value at the corresponding node. In each case the result of adding the
given multiples of the function at the corresponding points is given, but
for the 'error' terms reference must be made to the appropriate equation.