Hello!
May 23 will start
the coding period and I'm trying to figure out how I will implement my
functions before the midterm 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 RungeKutta 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 RungeKutta 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.

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].