---
_id: '60298'
abstract:
- lang: eng
  text: In this work, we introduce PHOENIX, a highly optimized explicit open-source
    solver for two-dimensional nonlinear Schrödinger equations with extensions. The
    nonlinear Schrödinger equation and its extensions (Gross-Pitaevskii equation)
    are widely studied to model and analyze complex phenomena in fields such as optics,
    condensed matter physics, fluid dynamics, and plasma physics. It serves as a powerful
    tool for understanding nonlinear wave dynamics, soliton formation, and the interplay
    between nonlinearity, dispersion, and diffraction. By extending the nonlinear
    Schrödinger equation, various physical effects such as non-Hermiticity, spin-orbit
    interaction, and quantum optical aspects can be incorporated. PHOENIX is designed
    to accommodate a wide range of applications by a straightforward extendability
    without the need for user knowledge of computing architectures or performance
    optimization. The high performance and power efficiency of PHOENIX are demonstrated
    on a wide range of entry-class to high-end consumer and high-performance computing
    GPUs and CPUs. Compared to a more conventional MATLAB implementation, a speedup
    of up to three orders of magnitude and energy savings of up to 99.8% are achieved.
    The performance is compared to a performance model showing that PHOENIX performs
    close to the relevant performance bounds in many situations. The possibilities
    of PHOENIX are demonstrated with a range of practical examples from the realm
    of nonlinear (quantum) photonics in planar microresonators with active media including
    exciton-polariton condensates. Examples range from solutions on very large grids,
    the use of local optimization algorithms, to Monte Carlo ensemble evolutions with
    quantum noise enabling the tomography of the system's quantum state.
article_number: '109689'
article_type: original
author:
- first_name: Jan
  full_name: Wingenbach, Jan
  id: '69187'
  last_name: Wingenbach
- first_name: David
  full_name: Bauch, David
  id: '44172'
  last_name: Bauch
- first_name: Xuekai
  full_name: Ma, Xuekai
  id: '59416'
  last_name: Ma
- first_name: Robert
  full_name: Schade, Robert
  id: '75963'
  last_name: Schade
  orcid: 0000-0002-6268-5397
- first_name: Christian
  full_name: Plessl, Christian
  id: '16153'
  last_name: Plessl
  orcid: 0000-0001-5728-9982
- first_name: Stefan
  full_name: Schumacher, Stefan
  id: '27271'
  last_name: Schumacher
  orcid: 0000-0003-4042-4951
citation:
  ama: Wingenbach J, Bauch D, Ma X, Schade R, Plessl C, Schumacher S. PHOENIX – Paderborn
    highly optimized and energy efficient solver for two-dimensional nonlinear Schrödinger
    equations with integrated extensions. <i>Computer Physics Communications</i>.
    2025;315. doi:<a href="https://doi.org/10.1016/j.cpc.2025.109689">10.1016/j.cpc.2025.109689</a>
  apa: Wingenbach, J., Bauch, D., Ma, X., Schade, R., Plessl, C., &#38; Schumacher,
    S. (2025). PHOENIX – Paderborn highly optimized and energy efficient solver for
    two-dimensional nonlinear Schrödinger equations with integrated extensions. <i>Computer
    Physics Communications</i>, <i>315</i>, Article 109689. <a href="https://doi.org/10.1016/j.cpc.2025.109689">https://doi.org/10.1016/j.cpc.2025.109689</a>
  bibtex: '@article{Wingenbach_Bauch_Ma_Schade_Plessl_Schumacher_2025, title={PHOENIX
    – Paderborn highly optimized and energy efficient solver for two-dimensional nonlinear
    Schrödinger equations with integrated extensions}, volume={315}, DOI={<a href="https://doi.org/10.1016/j.cpc.2025.109689">10.1016/j.cpc.2025.109689</a>},
    number={109689}, journal={Computer Physics Communications}, publisher={Elsevier
    BV}, author={Wingenbach, Jan and Bauch, David and Ma, Xuekai and Schade, Robert
    and Plessl, Christian and Schumacher, Stefan}, year={2025} }'
  chicago: Wingenbach, Jan, David Bauch, Xuekai Ma, Robert Schade, Christian Plessl,
    and Stefan Schumacher. “PHOENIX – Paderborn Highly Optimized and Energy Efficient
    Solver for Two-Dimensional Nonlinear Schrödinger Equations with Integrated Extensions.”
    <i>Computer Physics Communications</i> 315 (2025). <a href="https://doi.org/10.1016/j.cpc.2025.109689">https://doi.org/10.1016/j.cpc.2025.109689</a>.
  ieee: 'J. Wingenbach, D. Bauch, X. Ma, R. Schade, C. Plessl, and S. Schumacher,
    “PHOENIX – Paderborn highly optimized and energy efficient solver for two-dimensional
    nonlinear Schrödinger equations with integrated extensions,” <i>Computer Physics
    Communications</i>, vol. 315, Art. no. 109689, 2025, doi: <a href="https://doi.org/10.1016/j.cpc.2025.109689">10.1016/j.cpc.2025.109689</a>.'
  mla: Wingenbach, Jan, et al. “PHOENIX – Paderborn Highly Optimized and Energy Efficient
    Solver for Two-Dimensional Nonlinear Schrödinger Equations with Integrated Extensions.”
    <i>Computer Physics Communications</i>, vol. 315, 109689, Elsevier BV, 2025, doi:<a
    href="https://doi.org/10.1016/j.cpc.2025.109689">10.1016/j.cpc.2025.109689</a>.
  short: J. Wingenbach, D. Bauch, X. Ma, R. Schade, C. Plessl, S. Schumacher, Computer
    Physics Communications 315 (2025).
date_created: 2025-06-23T07:38:52Z
date_updated: 2025-06-29T12:00:36Z
department:
- _id: '27'
doi: 10.1016/j.cpc.2025.109689
intvolume: '       315'
language:
- iso: eng
project:
- _id: '52'
  name: 'PC2: Computing Resources Provided by the Paderborn Center for Parallel Computing'
publication: Computer Physics Communications
publication_identifier:
  issn:
  - 0010-4655
publication_status: published
publisher: Elsevier BV
status: public
title: PHOENIX – Paderborn highly optimized and energy efficient solver for two-dimensional
  nonlinear Schrödinger equations with integrated extensions
type: journal_article
user_id: '75963'
volume: 315
year: '2025'
...