---
_id: '14554'
abstract:
- lang: eng
  text: 'The Regularised Inertial Dean–Kawasaki model (RIDK) – introduced by the authors
    and J. Zimmer in earlier works – is a nonlinear stochastic PDE capturing fluctuations
    around the meanfield limit for large-scale particle systems in both particle density
    and momentum density. We focus on the following two aspects. Firstly, we set up
    a Discontinuous Galerkin (DG) discretisation scheme for the RIDK model: we provide
    suitable definitions of numerical fluxes at the interface of the mesh elements
    which are consistent with the wave-type nature of the RIDK model and grant stability
    of the simulations, and we quantify the rate of convergence in mean square to
    the continuous RIDK model. Secondly, we introduce modifications of the RIDK model
    in order to preserve positivity of the density (such a feature only holds in a
    “high-probability sense” for the original RIDK model). By means of numerical simulations,
    we show that the modifications lead to physically realistic and positive density
    profiles. In one case, subject to additional regularity constraints, we also prove
    positivity. Finally, we present an application of our methodology to a system
    of diffusing and reacting particles. Our Python code is available in open-source
    format.'
acknowledgement: "The authors thank the anonymous referees for their careful reading
  of the manuscript and their\r\nvaluable suggestions. FC gratefully acknowledges
  funding from the Austrian Science Fund (FWF) through the project F65, and from the
  European Union’s Horizon 2020 research and innovation programme under the Marie
  Sk lodowska-Curie grant agreement No. 754411 (the latter funding source covered
  the first part of this project)."
article_processing_charge: Yes (in subscription journal)
article_type: original
author:
- first_name: Federico
  full_name: Cornalba, Federico
  id: 2CEB641C-A400-11E9-A717-D712E6697425
  last_name: Cornalba
  orcid: 0000-0002-6269-5149
- first_name: Tony
  full_name: Shardlow, Tony
  last_name: Shardlow
citation:
  ama: 'Cornalba F, Shardlow T. The regularised inertial Dean’ Kawasaki equation:
    Discontinuous Galerkin approximation and modelling for low-density regime. <i>ESAIM:
    Mathematical Modelling and Numerical Analysis</i>. 2023;57(5):3061-3090. doi:<a
    href="https://doi.org/10.1051/m2an/2023077">10.1051/m2an/2023077</a>'
  apa: 'Cornalba, F., &#38; Shardlow, T. (2023). The regularised inertial Dean’ Kawasaki
    equation: Discontinuous Galerkin approximation and modelling for low-density regime.
    <i>ESAIM: Mathematical Modelling and Numerical Analysis</i>. EDP Sciences. <a
    href="https://doi.org/10.1051/m2an/2023077">https://doi.org/10.1051/m2an/2023077</a>'
  chicago: 'Cornalba, Federico, and Tony Shardlow. “The Regularised Inertial Dean’
    Kawasaki Equation: Discontinuous Galerkin Approximation and Modelling for Low-Density
    Regime.” <i>ESAIM: Mathematical Modelling and Numerical Analysis</i>. EDP Sciences,
    2023. <a href="https://doi.org/10.1051/m2an/2023077">https://doi.org/10.1051/m2an/2023077</a>.'
  ieee: 'F. Cornalba and T. Shardlow, “The regularised inertial Dean’ Kawasaki equation:
    Discontinuous Galerkin approximation and modelling for low-density regime,” <i>ESAIM:
    Mathematical Modelling and Numerical Analysis</i>, vol. 57, no. 5. EDP Sciences,
    pp. 3061–3090, 2023.'
  ista: 'Cornalba F, Shardlow T. 2023. The regularised inertial Dean’ Kawasaki equation:
    Discontinuous Galerkin approximation and modelling for low-density regime. ESAIM:
    Mathematical Modelling and Numerical Analysis. 57(5), 3061–3090.'
  mla: 'Cornalba, Federico, and Tony Shardlow. “The Regularised Inertial Dean’ Kawasaki
    Equation: Discontinuous Galerkin Approximation and Modelling for Low-Density Regime.”
    <i>ESAIM: Mathematical Modelling and Numerical Analysis</i>, vol. 57, no. 5, EDP
    Sciences, 2023, pp. 3061–90, doi:<a href="https://doi.org/10.1051/m2an/2023077">10.1051/m2an/2023077</a>.'
  short: 'F. Cornalba, T. Shardlow, ESAIM: Mathematical Modelling and Numerical Analysis
    57 (2023) 3061–3090.'
date_created: 2023-11-19T23:00:55Z
date_published: 2023-09-01T00:00:00Z
date_updated: 2023-11-20T08:38:47Z
day: '01'
ddc:
- '510'
department:
- _id: JuFi
doi: 10.1051/m2an/2023077
ec_funded: 1
file:
- access_level: open_access
  checksum: 3aef1475b1882c8dec112df9a5167c39
  content_type: application/pdf
  creator: dernst
  date_created: 2023-11-20T08:34:57Z
  date_updated: 2023-11-20T08:34:57Z
  file_id: '14560'
  file_name: 2023_ESAIM_Cornalba.pdf
  file_size: 1508534
  relation: main_file
  success: 1
file_date_updated: 2023-11-20T08:34:57Z
has_accepted_license: '1'
intvolume: '        57'
issue: '5'
language:
- iso: eng
month: '09'
oa: 1
oa_version: Published Version
page: 3061-3090
project:
- _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2
  grant_number: F6504
  name: Taming Complexity in Partial Differential Systems
- _id: 260C2330-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '754411'
  name: ISTplus - Postdoctoral Fellowships
publication: 'ESAIM: Mathematical Modelling and Numerical Analysis'
publication_identifier:
  eissn:
  - 2804-7214
  issn:
  - 2822-7840
publication_status: published
publisher: EDP Sciences
quality_controlled: '1'
related_material:
  link:
  - relation: software
    url: https://github.com/tonyshardlow/RIDK-FD
scopus_import: '1'
status: public
title: 'The regularised inertial Dean'' Kawasaki equation: Discontinuous Galerkin
  approximation and modelling for low-density regime'
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 57
year: '2023'
...