One of the most popular methods for the
numerical integration (cf.
Integration, numerical)
of diffusion problems, introduced by
J. Crank
and
P. Nicolson
[a1]
in
1947.
They
considered an implicit
finite difference
scheme to approximate the solution
of a non-linear differential system of the type which arises in
problems of heat flow.
In order to illustrate the main properties of the
Crank–Nicolson method,
consider the following initial-boundary value problem
for the
heat equation
To approximate the solution
of this model problem on
,
one introduces the grid
where
and
are positive integers, and
and
are
the step sizes in space and time, respectively. One looks for approximations
to
,
and to this end one
replaces derivatives by finite-difference formulas. Setting
,
then the Crank–Nicolson
method for the considered problem takes the form
,
,
with boundary conditions
and numerical initial condition
where
is an approximation to
.
Note that each
term in the difference equation can be interpreted as an
approximation to the
corresponding one in the differential equation at
.
Whenever the theoretical solution
is sufficiently smooth,
it can be proved, by
Taylor series
expansions that there exists a
positive constant
,
which depends only on
and
,
such that the
truncation error
satisfies
,
.
Thus, the Crank–Nicolson scheme is
consistent
of second order both in time and space.
The stability study
([a2],
[a3])
in the discrete
-norm
can be made by Fourier analysis or by the energy method.
One introduces the discrete operator
,
,
where it is assumed that
;
and sets
.
There exists a positive constant
,
which is independent of
and
,
such that
.
The stability estimate holds without any restriction on the step
sizes
and
;
thus, the
Crank–Nicolson method is said to be
unconditionally stable.
Convergence is derived by consistency and stability and, as
and
,
one finds
.
Stability can also be established in the discrete
-norm
by means of an energy argument
[a3].
Denoting
,
and
,
where
,
then the following stability estimate holds:
.
Therefore, if the theoretical solution
is sufficiently smooth, then
.
Note that the above inequality implies a convergence estimate in the
maximum norm.
An important question is to establish a
maximum principle
for the
approximations obtained with the Crank–Nicolson method,
similar to the one
satisfied by the solutions of the heat equation.
Related topics are monotonicity properties and, in particular,
the non-negativity (or non-positivity) of the numerical approximations.
Maximum-principle and monotonicity arguments can be used to derive
convergence in the
-norm.
It can be proved
([a2],
[a3])
that a choice of the step sizes such that
and the condition
imply
,
.
Note that if
and
,
then the following stability estimate in the maximum norm holds:
where
and
.
This estimate is still valid with
if
(see
[a4]),
and also it holds
without any restriction between the step sizes but then a value
has to be accepted in the bound
(
[a3],
[a5]).
From a computational point of view, the
Crank–Nicolson method involves a tridiagonal linear system to
be solved
at each time step. This can be carried out efficiently by
Gaussian elimination
techniques
[a2].
Because of that and its
accuracy and stability properties, the Crank–Nicolson method
is
a competitive algorithm for the numerical solution of
one-dimensional problems
for the heat equation.
The Crank–Nicolson method can be used for multi-dimensional
problems as well. For
example, in the integration of an homogeneous
Dirichlet problem
in a
rectangle for the
heat equation, the scheme is still unconditionally stable and
second-order accurate.
Also, the system to be solved at each time step has
a large and sparse matrix, but it does not have a tridiagonal form, so
it is usually solved by iterative methods. The amount of work
required to solve such a system is sufficiently large, so
other numerical schemes are also taken into account here,
such as
alternating-direction implicit methods
[a6]
or
fractional-steps methods
[a7].
On the other hand, it
should be noted that, for multi-dimensional problems in general
domains,
the finite-element method is better suited for the spatial
discretization
than the finite-difference method is.
The Crank–Nicolson method can be considered for the numerical
solution of a
wide variety of time-dependent partial differential equations.
Consider the
abstract Cauchy problem
where
represents a partial differential operator
(cf. also
Differential equation, partial;
Differential operator)
which
differentiates the unknown function
with respect to the space variable
in its space domain in
,
and
may be a vector function.
In the numerical integration of such problem, one can distinguish two
stages:
space discretization and time integration.
For the spatial discretization one can use
finite differences, finite elements, spectral techniques,
etc., and then a system of ordinary
differential equations is obtained, which can be written as
where
is the space discretization parameter (the spatial
grid size of a finite-difference or finite-element scheme, the
inverse
of the highest frequency of a spectral scheme, etc.) and
is a suitable approximation to the theoretical initial condition
.
The phrase
"Crank–Nicolson method"
is used to express that the time integration is carried out in a
particular
way. However, there is no agreement in the literature as to what
time integrator is called the Crank–Nicolson method, and the
phrase
sometimes means the
trapezoidal rule
[a8]
or the
implicit midpoint method
[a6].
Let
be the time step and introduce the discrete time levels
,
.
If one uses the trapezoidal rule, then the full
discrete method takes the form
while when the implicit midpoint method is considered, one obtains
where
,
and
is the approximation to
.
Both methods coincide for linear autonomous problems, but they are
different in
general. For instance, the midpoint rule is symplectic
[a9],
while the trapezoidal rule does not satisfy this property.
The Crank–Nicolson method applied to several problems
can be found in
[a8],
[a10],
[a11],
[a12],
[a13]
and
[a14].