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##

4.1.2 Stability and Accuracy of Difference Schemes

Let
be the exact solution of a DE, and
an error: at time , the algorithm produces
.

?
For EC,
.
Generally,

Expand around the correct solution:

or

The matrix
is called *amplification matrix*. All its
eigenvalues must be within the unit circle:

__EXAMPLE:__
*EC + Relaxation equation*

For this condition is met whenever
.
Check the previous figure!

__EXAMPLE:__
*EC + Harmonic oscillator*

The amplification matrix is

with eigenvalues
, so
that

EC applied to the harmonic oscillator is never stable.

** Next:** 4.1.3 Explicit Methods
**Up:** 4.1 Initial Value Problems
** Previous:** 4.1.1 Euler-Cauchy Algorithm
* Franz J. Vesely Oct 2005*

See also: "Computational Physics - An Introduction," Kluwer-Plenum 2001