{"date_published":"2023-09-01T00:00:00Z","volume":57,"quality_controlled":"1","scopus_import":"1","intvolume":"        57","title":"The regularised inertial Dean' Kawasaki equation: Discontinuous Galerkin approximation and modelling for low-density regime","related_material":{"link":[{"relation":"software","url":"https://github.com/tonyshardlow/RIDK-FD"}]},"citation":{"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>.","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.","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.","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>.","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>","short":"F. Cornalba, T. Shardlow, ESAIM: Mathematical Modelling and Numerical Analysis 57 (2023) 3061–3090."},"article_processing_charge":"Yes (in subscription journal)","publication_identifier":{"issn":["2822-7840"],"eissn":["2804-7214"]},"ddc":["510"],"has_accepted_license":"1","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"author":[{"full_name":"Cornalba, Federico","id":"2CEB641C-A400-11E9-A717-D712E6697425","first_name":"Federico","last_name":"Cornalba","orcid":"0000-0002-6269-5149"},{"last_name":"Shardlow","first_name":"Tony","full_name":"Shardlow, Tony"}],"day":"01","language":[{"iso":"eng"}],"page":"3061-3090","oa":1,"abstract":[{"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.","lang":"eng"}],"year":"2023","status":"public","article_type":"original","file":[{"success":1,"date_updated":"2023-11-20T08:34:57Z","access_level":"open_access","creator":"dernst","file_size":1508534,"date_created":"2023-11-20T08:34:57Z","file_name":"2023_ESAIM_Cornalba.pdf","checksum":"3aef1475b1882c8dec112df9a5167c39","content_type":"application/pdf","file_id":"14560","relation":"main_file"}],"_id":"14554","file_date_updated":"2023-11-20T08:34:57Z","ec_funded":1,"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).","project":[{"grant_number":"F6504","_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","name":"Taming Complexity in Partial Differential Systems"},{"grant_number":"754411","call_identifier":"H2020","name":"ISTplus - Postdoctoral Fellowships","_id":"260C2330-B435-11E9-9278-68D0E5697425"}],"doi":"10.1051/m2an/2023077","department":[{"_id":"JuFi"}],"oa_version":"Published Version","date_updated":"2023-11-20T08:38:47Z","type":"journal_article","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publication":"ESAIM: Mathematical Modelling and Numerical Analysis","issue":"5","date_created":"2023-11-19T23:00:55Z","month":"09","publication_status":"published","license":"https://creativecommons.org/licenses/by/4.0/","publisher":"EDP Sciences"}