Convergence of a time-stepping scheme to the free boundary in the supercooled Stefan problem
The supercooled Stefan problem and its variants describe the freezing of a supercooled liquid in physics, as well as the large system limits of systemic risk models in finance and of integrate-and-fire models in neuroscience. Adopting the physics terminology, the supercooled Stefan problem is known...
| Main Authors: | , , , |
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| Format: | Journal article |
| Language: | English |
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Institute of Mathematical Statistics
2023
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| _version_ | 1826309834142646272 |
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| author | Song, ZQ Reisinger, C Kaushansky, V Shkolnikov, M |
| author_facet | Song, ZQ Reisinger, C Kaushansky, V Shkolnikov, M |
| author_sort | Song, ZQ |
| collection | OXFORD |
| description | The supercooled Stefan problem and its variants describe the freezing of
a supercooled liquid in physics, as well as the large system limits of systemic risk
models in finance and of integrate-and-fire models in neuroscience. Adopting the physics
terminology, the supercooled Stefan problem is known to feature a finite-time blow-up of
the freezing rate for a wide range of initial temperature distributions in the liquid. Such
a blow-up can result in a discontinuity of the liquid-solid boundary. In this paper, we
prove that the natural Euler time-stepping scheme applied to a probabilistic formulation
of the supercooled Stefan problem converges to the liquid-solid boundary of its physical
solution globally in time, in the Skorokhod M1 topology. In the course of the proof, we
give an explicit bound on the rate of local convergence for the time-stepping scheme. We
also run numerical tests to compare our theoretical results to the practically observed
convergence behavior. |
| first_indexed | 2024-03-07T07:41:37Z |
| format | Journal article |
| id | oxford-uuid:e2e36e5c-2290-4ffe-92f1-74566090f468 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:41:37Z |
| publishDate | 2023 |
| publisher | Institute of Mathematical Statistics |
| record_format | dspace |
| spelling | oxford-uuid:e2e36e5c-2290-4ffe-92f1-74566090f4682023-04-20T09:45:40ZConvergence of a time-stepping scheme to the free boundary in the supercooled Stefan problemJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:e2e36e5c-2290-4ffe-92f1-74566090f468EnglishSymplectic ElementsInstitute of Mathematical Statistics2023Song, ZQReisinger, CKaushansky, VShkolnikov, MThe supercooled Stefan problem and its variants describe the freezing of a supercooled liquid in physics, as well as the large system limits of systemic risk models in finance and of integrate-and-fire models in neuroscience. Adopting the physics terminology, the supercooled Stefan problem is known to feature a finite-time blow-up of the freezing rate for a wide range of initial temperature distributions in the liquid. Such a blow-up can result in a discontinuity of the liquid-solid boundary. In this paper, we prove that the natural Euler time-stepping scheme applied to a probabilistic formulation of the supercooled Stefan problem converges to the liquid-solid boundary of its physical solution globally in time, in the Skorokhod M1 topology. In the course of the proof, we give an explicit bound on the rate of local convergence for the time-stepping scheme. We also run numerical tests to compare our theoretical results to the practically observed convergence behavior. |
| spellingShingle | Song, ZQ Reisinger, C Kaushansky, V Shkolnikov, M Convergence of a time-stepping scheme to the free boundary in the supercooled Stefan problem |
| title | Convergence of a time-stepping scheme to the free boundary in the supercooled Stefan problem |
| title_full | Convergence of a time-stepping scheme to the free boundary in the supercooled Stefan problem |
| title_fullStr | Convergence of a time-stepping scheme to the free boundary in the supercooled Stefan problem |
| title_full_unstemmed | Convergence of a time-stepping scheme to the free boundary in the supercooled Stefan problem |
| title_short | Convergence of a time-stepping scheme to the free boundary in the supercooled Stefan problem |
| title_sort | convergence of a time stepping scheme to the free boundary in the supercooled stefan problem |
| work_keys_str_mv | AT songzq convergenceofatimesteppingschemetothefreeboundaryinthesupercooledstefanproblem AT reisingerc convergenceofatimesteppingschemetothefreeboundaryinthesupercooledstefanproblem AT kaushanskyv convergenceofatimesteppingschemetothefreeboundaryinthesupercooledstefanproblem AT shkolnikovm convergenceofatimesteppingschemetothefreeboundaryinthesupercooledstefanproblem |