Asymptotic numerical hypocoercivity of the space-time discontinuous Galerkin method for the Kolmogorov equation

Mathematics > Numerical Analysis [Submitted on 25 May 2025 (v1), last revised 16 Jun 2026 (this version, v3)] Title:Asymptotic numerical hypocoercivity of the space-time discontinuous Galerkin method for the Kolmogorov equation View PDF HTML (experimental)Abstract:We are concerned with discretisations of the classical Kolmogorov equation by a standard space-time discontinuous Galerkin method. {The} Kolmogorov equation serves as simple, yet rich enough in the present context, model problem for a wide range of kinetic-type equations: although it involves diffusion in one of the two spatial dimensions only, the combined nature of the first order transport/drift term and the degenerate diffusion are sufficient to `propagate dissipation' across the spatial domain in its entirety. This is a manifestation of the celebrated concept of hypocoercivity, a term coined and studied extensively by Villani in \cite{villani}. We show that the {classical} space-time discontinuous Galerkin method {admits} a corresponding hypocoercivity property at the discrete level, asymptotically for large times. To the best of our knowledge, this is the first result of this kind for any standard Galerkin scheme. This property is shown by proving one part of a discrete inf-sup-type stability result for the method in a family of norms dictated by a modified scalar product motivated by the theory in \cite{villani}. This family of norms contains the full gradient of the numerical solution, thereby allowing for a full spectral gap/Poincaré-type inequality at the discrete level, thus, showcasing a subtle, discretisation-parameter-dependent, numerical hypocoercivity property. Further, we show that the space-time discontinuous Galerkin method is inf-sup stable in the family of norms containing the full gradient of the numerical solution, which may be a result of independent interest. Submission history From: Zhaonan Dong [view email][v1] Sun, 25 May 2025 16:36:09 UTC (34 KB) [v2] Wed, 18 Feb 2026 15:56:38 UTC (36 KB) [v3] Tue, 16 Jun 2026 17:39:27 UTC (1,013 KB) Current browse context: math.NA References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.