EulerRichardson MethodAs
can be seen from the proceeding discussion, the algorithm for obtaining
a numerical solution of a differential equation is not unique, and
there are many algorithms that reduce to the same differential equation
in the limit
. It might occur that it would be better to compute
the velocity at the middle of the interval, rather that at the beginning
or at the end of the interval. The EulerRichardson algorithm is
fusion of this idea together with the simple Euler method. This
algorithm is particularly useful for velocitydependent forces,
but does as well as other simple algorithms for forces that do not
depend on the velocity. The algorithm consists of using the Euler
method to find the intermediate position and the velocity at the time
. Then we compute the force,
, and the acceleration at . The new position and velocity at time is found using and . We summarize the EulerRichardson algorithm as follows:
so
that the particle velocity and position are obtained from
Even
though we need to do twice as many computations per time step, the
EulerRichardson algorithm is much faster because we can make the
time step larger and still obtain better accuracy that with either
the Euler or EulerCromer algorithm. Formal justification of the
EulerRichardson algorithm can be obtained, as usual, from the appropriate
Taylor series expansion.


