COMMUNICATIONS ON PURE
AND APPLIED
MATHEMATICS,
\'OL.
IX, 267293
(1956)
Survey
of
the
Stability
Linear Finite Difference
Equations*
P. I>. LAX and R. D. RICHTMYEK
PART I
AN EQUIVALENCE THEOREM
1.
Introduction
Beginning with the discovery by Courant, Friedrichs and Lewy
[l]
of the conditional stability of certain finite difference approximations to
partial differential equations, the subject of stability has been variously
discussed in the literature (see bibliography at end). The present paper is
concerned with the numerical solution of initial value problems by finite
difference methods, generally for a finite time interval, by a sequence of
calculations with increasingly finer mesh, Thus if
t
is the time variable
and
dt
its increment, we are concerned with limits as
At
+
0
for fixed
t,
not with limits as
t+co
for fixed
(although often the stability con
siderations are similar). The basic question is whether the solution con
verges to the true solution of the initial value problem as the mesh is
refined. The term
stabiZity,
as usually understood, refers to a property
of the finite difference equations, or rather of the above mentioned sequence
of finite difference equations with increasingly finer mesh. We shall give a
definition of stability in terms of the uniform boundedness
a certain set
of operators and then show that under suitable circumstances, for linear
initial value problems, stability is necessary and sufficient for convergence
in
a certain uniform sense for arbitrary initial data. The circumstances
are first that a certain consistency condition must be satisfied which
essentially insures that the difference equations approximate the differen
tial equations under study, rather than for exampre some other differentid
equations, and secondly that the initial value problem be properly posed,
in a sense to be defined later.
We shall not be concerned with rounding errors, and in fact assume
that all arithmetic steps are carried out with infinite precision. But it will
*The
work
for this paper
was
done under
Contract
AT([email protected])1480
of the Atomic
Energy Commission.
267
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P.
D. LAX, AND R.
D.
RICHTMYER
be evident to the reader that there is an intimate connection between
stability and practicality of the equations from the point of view of the
growth and amplification of rounding errors. Indeed, O’Brien, Hyman and
Kaplan
[8]
defined stability in terms of the growth of rounding errors.
However, we have
a slight preference for the definition given below, because
it emphasizes that stability still has to be considered, even if rounding
errors are negligible, unless, of course, the initial data are chosen with
diabolical care so
as to be exactly free of those components that would
be unduly amplified if they were present.
The basic notions will be spelled out ill considerable detail below in
an attempt to motivate the definitions given and to justify the approach
via the theory of linear operators in Banach space.
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 Spring '08
 Iskandarani,M
 Partial differential equation, difference equations, finite difference, von neumann

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