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Code for generating Optimal Stability Polynomials in Roots for Explicit Time Integration.

• IpOpt is the core package (optimizer). Following the installation instructions should suffice, no special installation directory is required. For the linear solver only MUMPS has been applied both in combination with METIS and without.
• NAG dco/c++ is used to compute the necessary derivatives algorithmically. dco/c++ is proprietary software, but chances are that you can obtain an academic license (NAG Campus) if you are working in research. After obtaining dco/c++ and licensing it, you need to change the path in the Makefiles (line 7) accordingly, i.e., DCO_PATH=YOUR/PATH/TO/DCO.
• Optional: If you want to compute also the monomial coefficients of the stability polynomial the usage of higher precision datatypes is necessary, where I resort to the implementation by Boost.

After obtaining and licensing the dependencies, execute in both directories Feasibility_Problem and Optimization_Problem

make -j NUMTHREADS

where you can specify the NUMTHREADS according to your machine, e.g. 8. This builds object files and binaries in the corresponding directories obj and bin.

Best starting point are the examples. In order to carry out the optimization of a stability polynomial of degree $S$ and linear order of accuracy $p$ you require the spectrum and a reference timestep $\Delta t_\text{Ref}$ with a corresponding reference stage count $S_\text{Ref}$. The syntax of calling the feasible point searchers/optimizers is then

./Roots_Real(Imag).exe S p S_ref dt_ref Spectrum

If the spectrum itself does not form a convex hull, you need to supply the path to the files containing the real and imaginary part, respectively. A call would then look like this:

./Roots_Real(Imag).exe S p S_ref dt_ref Spectrum PathToHullPoints

Again, this is best seen in the examples.

Roots_Real.exe looks for the parameter file Roots_Real.opt and Roots_RealImag.exe accordingly for Roots_RealImag.opt in the working directory. If none of these files is present, default Ipopt options are used.

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

@misc{OSPREI,
  author={Doehring, Daniel and Torrilhon, Manuel},
  title={{OSPREI}: Optimal Stability Polynomials in Roots for Explicit Time Integration},
  year={2023},
  doi={10.5281/zenodo.8009493},
  publisher = {GitHub},
  journal = {GitHub Repository},
  howpublished = {\url{https://github.com/DanielDoehring/OSPREI}}
}

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

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).