Phi functions and general exponential schemes

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).

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.

• 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].