Truncation error or local truncation error is error made by numerical algorithms that arises from taking finite number of steps in computation. It is present even with infinite-precision arithmetic, because it is caused by truncation of the infinite Taylor series to form the algorithm.
Use of arbitrarily small steps in numerical computation is prevented by round-off error, which are the consequence of using finite precision floating point numbers on computers.
Let 
represent the difference equation approximating the PDE at the 
-th mesh point, with exact solution 
. For example,
represent the difference equation approximating the following PDE 
at the -th mesh point. If is replaced by 
at the mesh points of the 
E, where is the exact solution of the PDES, the value of 
is called the local truncation error 
at the -th mesh point. clearly measures the amount by which the exact solution values of the PDE at the mesh points of the E do not satisfy the E at the point 
.
Continuing the above example gives us
Using Taylor expansions, it is easy to express in terms of power of 
and 
and partial derivatives of at . This leads us to the computation of the local truncation error.
It is sometimes possible ot approximate a parabolic or hyperbolic equation by a finite-difference scheme that is stable (i.e. limits the amplification of all the compoonents of the initial conditions), but which has a solution that converges to the solution of a different differential equation as the mesh lengths tend to zero. Such a difference scheme is said to be inconsistent or imcompatible with the PDE.
The real importance of the concept of consistency lies in a theorem by Lax which states that if a linear finite-difference equation is consistent with a properly posed linear initial-value problem then stability guarantees convergnce of to as the mesh lengths tned to zero. Consistency can be defined in either of two equivalent but slightly different ways.
The more general definition is as follows. Let 
represent the PDE in the independent variables 
and 
, with exact solution . Let 
represent the approximating finite-difference equation with exact solution . Let 
be a continuous function of and with a sufficient number of continuous derivatives to enable 
to be evaluated at the point 
.
Then the truncation error 
at the point is defined by
If 
as 
and 
, the difference equation is said to be consistent or compatible with the PDE. Most authors put 
because .
For example, let us consider the following parabolic equation
with
and
where 
and 
.
Backward Euler + second order central finite difference discretization.
We split equally the domain 
into 
parts and the domain 
into 
parts, i.e. 
-th mesh point is of the following form 
. We are now in a position to discretize the problem as follows
where 
. At 
, that is 
, one has
and at 
, that is 
, one obtains
The local truncation error and consistency
The above discretization can be rewritten as following
where with the following boundary conditions
and
Now we have
By Taylor’s expansion
and
and
Therefore,
which yields
Thus, the principle part of the local truncation error is
Hence 
Consistency.
In view of the local truncation error, one can easily see that the local truncation error is a polynomial of two variables and which implies that the method is consistent.
