SSOCs

                                                        ..........
                                                ....:::::^^^^^^^^^^^.
                                            .::^^^^^^^^^^^^^^^^^^^:::.
                                          .~~~~~~~^^^^^^^^^:::::::::.
                                           :^^^^^^^::::::::::::::::::
                                           .:^^^^::::::::::::::::::::
                                            :^^^^^::::::::::::::::::.
                            ::::^^^^^^^^^^^^^^^^^^::::::::::::::::::.
                            ::::^^^^^^^^^^^^:^^^^^:::::::::::::::::::
                            ::::^^^^^^^^^^^^^:~#####################:
                            .:::^^^^^^^^^^^^^^^######################.
                            .::^^^^^^^^^^^^^^^:^#####################.
                             ::::^^^^^^^^^^^::::####################:.
                             :####################~~^^:::::::::::::::.
                             :####################:^^::::::::::::::::.
                             :###################..^^^^::::::::::::::.
                             :###################. :^^^::::::::::::::.
                              !^^^^^^^^^^^^^^^      ^^^:::::SSOCs::::.
                              :::^^^^^^^^^^^^^^^.   .^^^::::::::::::::
                              .::^^^^^^^~^^^^^^^.    :^^::::::::::::::
                              .::^^^^^^~~^^^^^^^     .^^::::::::::::::
                              .::^^^^^SSOCs~^^^:      :^^:::::::::::::
                               :::^^^^~~~~~~~^^.      :^^:::::::::::::.
                               :::^^^~~~~~~~~~^       .^:::::::::::::::.
                               :::^^^~~~~~~~~~:       .:::::::::::::::::.
                               :::^^^~~~~~~~~~.        ::::::::::::::::::.
                               :::^^^~~~~~~~~^         ::::::::::::::::::::
                              .::^^^^~~~~~~~~^        .:::::::::::::::::::::
                              :^^^^^~~~~~~~!~^        :::::::::::::::::::::::
                            .:^^^^^~~~~~~~~!~^       .:::::..::::::::::::::::.
                          .:^^^^^~~~~~~~~~!!~~.      ::::::..::::::::::::::::
                       .::^^^^^^^~~~~~~~~!!!~~~     .:::::...::::::::::::::.
                     .::^:^^^^^^~~~~!!!!!!!!!~~^   .::::::...:::::::::::::.
                  .::::::::^^^^~~~~!!!!!!!!~~~~~.  ::::::....:::::::::::.
              ..::^^:::::::^^^~~~~~!!!!!!!~~~~~~^.::::::....::::::::::.
     ...::::::::::^!~:::::^^^~~~~~~!!!!!!~~~~~~^^:::::::::..:::::::::.
   .::::::::::::::::::::^^^^^~~~~~!!!!!!~~~~^^^:::::::::::::::::::::.
  .::::::.::::::::::::^^^^^~~~!!!!!!!~^^:::::::::::::::::::::::::::.
   .::::::::::::::::::^^^^^^^^^::::::::::..................:::::::.
        .....  .                    ::::::::................:::::.

Code for generating optimal Stability Polynomials by Second Order Cones in arbitrary precision.

Dependencies

Eigen3

In order to use EiCOS you need to have Eigen3 installed. If you are on Ubuntu or an other Debian based Linux distribution it is recommended to install Eigen3 via

sudo apt install libeigen3-dev

Boost Multiprecision Types

Within this project the multiprecision floating point types are based on the Boost Multiprecision types. See the docs for a detailed introduction.

On Debian-based systems it is again easiest to install Boost over the package manager via

sudo apt-get install libboost-all-dev

although this brings in the whole Boost library.

CMake

To build EiCOS you need CMake to be installed on your system, which is the case on most Linux distributions.

EiCOS

To be able to use SSOCs you need to compile the shared library libeicos_MP.so of my fork from the EiCOS repository. After installing Eigen3 and the Boost multiprecision library you need to cd to the directory where you downloaded or forked EiCOS. It is assumed that this is ~/git/EiCOS/ in the compile scripts, where ~ is synonymous with $HOME.

Then, execute

cmake -DCMAKE_BUILD_TYPE=Release .

to generate the make files with release flags. Then type

make

to compile the shared library libeicos_MP.so. A successful output should look like this:

[ 50%] Building CXX object CMakeFiles/eicos_MP.dir/src/eicos_MP.cpp.o
[100%] Linking CXX shared library libeicos_MP.so
[100%] Built target eicos_MP

Compiling SSOCs

If you installed EiCOS not in ~/git/EiCOS you need to exchange that path wherever it occurrs in the Makefile. Then, you can simply execute

make MP_TARGETS -j 2

which gives you both SSOCs_MP.exe and SSOCs_PERK4.exe. It might be required that you make these binaries executable, i.e.,

chmod +x *.exe

Usage

To generate the fourth order accurate stability polynomial for the fourth order Paired-Explicit Runge-Kutta schemes type

./SSOCs_PERK4_MP.exe <Degree> <dtMax> <path/to/spectrum/file> <OPTIONAL:Stages>

where

• <Degree> needs to be exchanged for the stability polynomial degree. For the PERK4 schemes (SSOCs_PERK4.cpp), this needs to be an integer > 5, while for the general version (SSOCs.cpp) also a polynomial with degreee 5 may be optimized.
• <dtMax> is the maximum timestep that may be possible. In principle this can be chosen arbitrarily large, which slows down the optimization. Supplying a too small value gives wrong results if the true admissible timestep is larger.
• <path/to/spectrum/file> is the path to the file with the eigenvalues used for constraining the stability polynomial. The file should contain the eigenvalues should that one eigenvalue is there per row with syntax Re(lambda)+Im(lambda)i to allow correct processing.
• The last argument is optional and may be supplied to obtain the Butcher array coefficients of a method with <Degree> stage-evaluations which is embedded into a <Stages> stage overall paired-explicit Runge-Kutta method.

Credit

If you use the implementations provided here, please also cite this repository as

@misc{doehring2024ssocs,
  title={{SSOCs}: Arbitrary Precision Stability Polynomials by Second-Order Cones},
  author={Doehring, Daniel},
  year={2024},
  howpublished={\url{https://github.com/DanielDoehring/SSOCs}},
  doi={https://doi.org/10.5281/zenodo.11184359}
}

Affiliation

This code was developed at the Institute of Applied and Computational Mathematics (ACoM) at RWTH Aachen University.

Acknowledgements

This code is a result of research performed in the research unit "Structure-Preserving Numerical Methods for Bulk- and Interface Coupling of Heterogeneous Models (SNuBIC)"

This project has benefited from funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the research unit FOR 5409 "Structure-Preserving Numerical Methods for Bulk- and Interface Coupling of Heterogeneous Models (SNuBIC)" (project number 463312734).