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.

 Relaxation equation: Euler / Leapfrog: Start Applet

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