Hello!
May 23 will start
the coding period and I'm trying to figure out how I will implement my
functions before the mid-term evaluation.
With the advice of
my mentors Marco and Jacopo, I decided to start with the
implementation of the phi functions, necessary to calculate the
matrix functions in the two schemes for a general
exponential Runge-Kutta and Rosenbrock integrator.
These schemes will
not be really fast and efficient, but I will use them as a reference
when I go to implement the official methods. It will be useful to
verify the correctness of my codes.
As regards the
implementation of the phi functions I will refer to
[BSW 07] “EXPINT — A
MATLAB Package for Exponential Integrators”, Havard Berland, Bard
Skaflestad and Will M. Wright, 2007,
DOI: 10.1145/1206040.1206044, webpage (software without a license).
DOI: 10.1145/1206040.1206044, webpage (software without a license).
While the general
schemes that then I'm going to implement are as follows:
-
Exponential Runge-Kutta integrators
Consider a problem
of the form
,
the numerical
exponential Runge Kutta scheme for its solution is
where
and the coefficients
and
are
constructed from exponential functions or approximations of such
functions.
and
are
constructed from exponential functions or approximations of such
functions.-
Exponential Rosenbrock integrators
Consider a problem of the form
,
the numerical
exponential Rosenbrock
scheme for its solution is
where
for details about
formulas see [HO 10].
