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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