Efficient computation of the sinc matrix function for the integration of second-order differential equations

Rational Krylov Methods for the Integration of Second Order Differential Equations

External codes

The code contained in this repository makes use of some Chebfun functions. To add to your MATLAB environment simply run the following command from the MATLAB command window:

unzip('https://github.com/chebfun/chebfun/archive/master.zip')
movefile('chebfun-master', 'chebfun'), addpath(fullfile(cd,'chebfun')), savepath

The other fundamental ingredient is the Rational Krylov toolbox, if you do not have it already you can automatically download and install it, by simply copying and pasting the following two lines to your MATLAB command window:

unzip('http://guettel.com/rktoolbox/rktoolbox.zip'); 
cd('rktoolbox'); addpath(fullfile(cd)); savepath

Among the methodologies available for calculating exponential sums, the expmv code is used. This is added as a Git submodule to the repository. By cloning the repository the code is not added automatically, this can be done by going to the appropriate directory and doing git pull, or from the root directory with

git submodule init
git submodule update

Information about this code can be found at:

• A. H. Al-Mohy and N. J. Higham, "Computing the action of the matrix exponential, with an application to exponential integrators" SIAM J. Sci. Comput., 33(2):488--511, 2011.

For the exponential integrators we use the EXPINT package for comparisons:

• H. Berland, B. Skaflestad, and W. M. Wright. EXPINT—a MATLAB package for exponential integrators. ACM Trans. Math. Softw., 33(1):4–es, mar 2007. ISSN 0098-3500. doi: 10.1145/1206040.1206044. URL https://doi.org/10.1145/1206040.1206044.

Contributors

• Lidia Aceto
• Fabio Durastante

Cite as

The associated paper is published in Advances in Computational Mathematics can be cited as:

@Article{Aceto2024,
      author={Aceto, Lidia and Durastante, Fabio},
      title={Efficient computation of the sinc matrix function for the integration of second-order differential equations},
      journal={Advances in Computational Mathematics},
      year={2024},
      month={Oct},
      day={28},
      volume={50},
      number={6},
      pages={109},
      abstract={This work deals with the numerical solution of systems of oscillatory second-order differential equations
      which often arise from the semi-discretization in space of partial differential equations. Since these differential
      equations exhibit (pronounced or highly) oscillatory behavior, standard numerical methods are known to perform poorly.
      Our approach consists in directly discretizing the problem by means of Gautschi-type integrators based on sinc matrix
      functions. The novelty contained here is that of using a suitable rational approximation formula for the sinc matrix
      function to apply a rational Krylov-like approximation method with suitable choices of poles. In particular, we
      discuss the application of the whole strategy to a finite element discretization of the wave equation.},
      issn={1572-9044},
      doi={10.1007/s10444-024-10202-y},
      url={https://doi.org/10.1007/s10444-024-10202-y}
}

If you decide to use the code in a scientific publication, also cite the related works for the codes described in the previous section.