Splitting methods for
perturbed problems
We have collected a number of splitting methods
for perturbed systems. There are more than 30 methods and they have been
classified according to their structure since they are built for different
purposes.
Real and positive coefficients (only second
order methods)
Real coefficients and high order (some negative
coefficients)
Real coefficients, but the perturbation is not
exactly solvable
Complex coefficients
Real-Complex coefficients (a set of
coefficients is real and positive and the other one are complex)
Methods with processor
Explicitly time-dependent problems (for the
dominant part)
All fortran programs with a pdf file to explain
them are available here. You can also
find useful the talk I gave at SciCADE13 where these methods were presented (slides).
The coefficients of the splitting methods with
real coefficients and high order were obtained in APNUM2013
and tested in different problems in the Solar System in CELMEC2013. Some frotran codes which implement the
2-dimensional perturbed Kepler used in APNUM2013 are available in:
The
perturbed Kepler problems.
Splitting methods for the
Schödinger equation
We have prepared a pseudo-method which
corresponds to a collection of a number of splitting methods for solving the
Schrödinger equation. Given a symmetric real matrix, H, and a unitary vector,
u0, it computes the product u(t) =
exp(-i t H) u0. The algorithm requires as inputs (similar to what is required
by Chebyshev methods):
-The value of t
-The initial conditions, u0.
-Bounds to the spectrum of H, i.e. Emin and
Emax such that Emin<= sigma(H) <=Emax
-A given tolerance.
-A subroutine which computes the vector-matrix
products H u0.
The method provides an approximation to u(t) with an error under this tolerance. The
pseudo-method has a collection of splitting methods, each one with different
number of stages, error bounds and optimized for different purposes and it
chooses the method which provides the desired result using the smaller number
of vector-matrix products. In general, this is considerably faster than using
the Chebyshev method. Here it is provided both the programs for the splitting
methods as well as for the Chebyshev method (and for the
Frortran programs are available here.