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4.2.6 Numerov's Method
Mostly used for a certain kind of initial value problems (IVPs)
that arise when boundary value problems (BVPs) are rewritten as IVPs.
Many BVP has the general form
with given boundary values and .
Example: one-dimensional Poisson equation
with given at and . Thus and
.
Let us assume we have, instead of and ,
the full set of initial values and
.
Divide the interval
into sub-intervals of length and at each intermediate point
expand into a power series. Adding the Taylor formulae
for and we find
Inserting
we have
The fourth derivative is approximated by
In this way we find Numerov's formula
EXERCISE:
Write a code that permits to solve a given second-order equation of motion
by various algorithms. Apply the program to problems of point mechanics and
explore the stabilities and accuracies of the diverse techniques.
Next: 4.3 Boundary Value Problems
Up: 4.2 Initial Value Problems
Previous: 4.2.5 Symplectic Algorithms
Franz J. Vesely Oct 2005
See also: "Computational Physics - An Introduction," Kluwer-Plenum 2001