** Next:** 4.1.4 Implicit Methods
**Up:** 4.1 Initial Value Problems
** Previous:** 4.1.2 Stability and Accuracy

##

4.1.3 Explicit Methods

- Euler-Cauchy (from DNGF; see above)

- Leapfrog algorithm (from DST):

This is an example of a *multistep technique*, as timesteps
and contribute .
Stability analysis for such algorithms is as follows:

Let the explicit multistep scheme be written as

Inserting a slightly deviating solution
and computing the difference, we have

We combine the errors at subsequent time steps to a vector

and define the quadratic matrix

Then

Stability is guaranteed if

__EXAMPLE 1:__
Leapfrog / Relaxation equation:

Therefore

which means that
, and , and
the matrix
is

with eigenvalues

For real
we find that always, meaning
that the leapfrog scheme is unstable for the relaxation (or growth)
equation.

__EXAMPLE 2:__
Leapfrog / Harmonic oscillator:

and

For the amplification matrix we find
(with
)

with eigenvalues

But the modulus of this is always .
The leapfrog
algorithm is *marginally stable* for the harmonic oscillator.

** Next:** 4.1.4 Implicit Methods
**Up:** 4.1 Initial Value Problems
** Previous:** 4.1.2 Stability and Accuracy
* Franz J. Vesely Oct 2005*

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