Using a Separable Mathematical Entropy to Construct Entropy-Stable Schemes for a Reduced Blood Flow Model
The aim of this paper is to derive a separable entropy for a one-dimensional reduced blood flow model, which will be used to treat the symmetrizability of the model in full generality and for constructing entropy conservative fluxes, which are one of the essential building blocks of designing entrop...
- Autores:
-
Valbuena, Sonia
- Tipo de recurso:
- Fecha de publicación:
- 2022
- Institución:
- Universidad del Atlántico
- Repositorio:
- Repositorio Uniatlantico
- Idioma:
- eng
- OAI Identifier:
- oai:repositorio.uniatlantico.edu.co:20.500.12834/766
- Acceso en línea:
- https://hdl.handle.net/20.500.12834/766
- Palabra clave:
- blood flow model
entropy pair
symmetrizability
entropy conservative flux
IMEX schemes
- Rights
- openAccess
- License
- http://creativecommons.org/licenses/by-nc/4.0/
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Using a Separable Mathematical Entropy to Construct Entropy-Stable Schemes for a Reduced Blood Flow Model |
| title |
Using a Separable Mathematical Entropy to Construct Entropy-Stable Schemes for a Reduced Blood Flow Model |
| spellingShingle |
Using a Separable Mathematical Entropy to Construct Entropy-Stable Schemes for a Reduced Blood Flow Model blood flow model entropy pair symmetrizability entropy conservative flux IMEX schemes |
| title_short |
Using a Separable Mathematical Entropy to Construct Entropy-Stable Schemes for a Reduced Blood Flow Model |
| title_full |
Using a Separable Mathematical Entropy to Construct Entropy-Stable Schemes for a Reduced Blood Flow Model |
| title_fullStr |
Using a Separable Mathematical Entropy to Construct Entropy-Stable Schemes for a Reduced Blood Flow Model |
| title_full_unstemmed |
Using a Separable Mathematical Entropy to Construct Entropy-Stable Schemes for a Reduced Blood Flow Model |
| title_sort |
Using a Separable Mathematical Entropy to Construct Entropy-Stable Schemes for a Reduced Blood Flow Model |
| dc.creator.fl_str_mv |
Valbuena, Sonia |
| dc.contributor.author.none.fl_str_mv |
Valbuena, Sonia |
| dc.contributor.other.none.fl_str_mv |
Vega, Carlos A. |
| dc.subject.keywords.spa.fl_str_mv |
blood flow model entropy pair symmetrizability entropy conservative flux IMEX schemes |
| topic |
blood flow model entropy pair symmetrizability entropy conservative flux IMEX schemes |
| description |
The aim of this paper is to derive a separable entropy for a one-dimensional reduced blood flow model, which will be used to treat the symmetrizability of the model in full generality and for constructing entropy conservative fluxes, which are one of the essential building blocks of designing entropy-stable schemes. Time discretization is conducted by implicit–explicit (IMEX) Runge–Kutta schemes, but solutions for nonlinear systems will not be required due to the particular form of the source term. To validate the numerical schemes obtained, some numerical tests are presented. |
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2022 |
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2022-11-15T19:11:11Z |
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2022-11-15T19:11:11Z |
| dc.date.issued.none.fl_str_mv |
2022-09-13 |
| dc.date.submitted.none.fl_str_mv |
2022-07-19 |
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http://purl.org/coar/version/c_b1a7d7d4d402bcce |
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http://purl.org/coar/resource_type/c_2df8fbb1 |
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Artículo |
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draft |
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Valbuena, S.; Vega, C.A. Using a Separable Mathematical Entropy to Construct Entropy-Stable Schemes for a Reduced Blood Flow Model. Mathematics 2022, 10, 3314. https://doi.org/10.3390/ math10183314 |
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https://hdl.handle.net/20.500.12834/766 |
| dc.identifier.doi.none.fl_str_mv |
10.3390/ math10183314 |
| dc.identifier.instname.spa.fl_str_mv |
Universidad del Atlántico |
| dc.identifier.reponame.spa.fl_str_mv |
Repositorio Universidad del Atlántico |
| identifier_str_mv |
Valbuena, S.; Vega, C.A. Using a Separable Mathematical Entropy to Construct Entropy-Stable Schemes for a Reduced Blood Flow Model. Mathematics 2022, 10, 3314. https://doi.org/10.3390/ math10183314 10.3390/ math10183314 Universidad del Atlántico Repositorio Universidad del Atlántico |
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https://hdl.handle.net/20.500.12834/766 |
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eng |
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eng |
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http://creativecommons.org/licenses/by-nc/4.0/ |
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Attribution-NonCommercial 4.0 International |
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info:eu-repo/semantics/openAccess |
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application/pdf |
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Barranquilla |
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Licenciatura en Matemáticas |
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Sede Norte |
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Valbuena, Sonia7e360e06-c139-4a0a-948c-39c12b8cdab4Vega, Carlos A.2022-11-15T19:11:11Z2022-11-15T19:11:11Z2022-09-132022-07-19Valbuena, S.; Vega, C.A. Using a Separable Mathematical Entropy to Construct Entropy-Stable Schemes for a Reduced Blood Flow Model. Mathematics 2022, 10, 3314. https://doi.org/10.3390/ math10183314https://hdl.handle.net/20.500.12834/76610.3390/ math10183314Universidad del AtlánticoRepositorio Universidad del AtlánticoThe aim of this paper is to derive a separable entropy for a one-dimensional reduced blood flow model, which will be used to treat the symmetrizability of the model in full generality and for constructing entropy conservative fluxes, which are one of the essential building blocks of designing entropy-stable schemes. Time discretization is conducted by implicit–explicit (IMEX) Runge–Kutta schemes, but solutions for nonlinear systems will not be required due to the particular form of the source term. To validate the numerical schemes obtained, some numerical tests are presented.Universidad del Atlánticoapplication/pdfenghttp://creativecommons.org/licenses/by-nc/4.0/Attribution-NonCommercial 4.0 Internationalinfo:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2MathematicsUsing a Separable Mathematical Entropy to Construct Entropy-Stable Schemes for a Reduced Blood Flow ModelPúblico generalblood flow modelentropy pairsymmetrizabilityentropy conservative fluxIMEX schemesinfo:eu-repo/semantics/articleinfo:eu-repo/semantics/draftArtículohttp://purl.org/coar/version/c_b1a7d7d4d402bccehttp://purl.org/coar/resource_type/c_2df8fbb1BarranquillaLicenciatura en MatemáticasSede NorteSherwin, S.J.; Franke, V.; Peiró, J.; Parker, K. One-dimensional modelling of a vascular network in space—Time variables. J. Eng. Math. 2003, 47, 217–250. [CrossRef]Formaggia, L.; Nobile, F.; Quarteroni, A.; Veneziani, A. Multiscale modelling of the circulatory system: A preliminar analysis. Comput. Vis. Sci. 1999, 2, 75–83. [CrossRef]Delestre, O.; Lagrée, P.Y. A ’well-balanced’ finite volume scheme for blood flow simulation. Int. J. Numer. Methods Fluids 2013, 72, 177–205. [CrossRef]Shapiro, A.H. Steady flow in collapsible tubes. J. Biomech. Eng. 1977, 99, 126–147. [CrossRef]Guigo, A.R.; Delestre, O.; Fullana, J.M.; Lagrée, P.Y. Low-Shapiro hydrostatic reconstruction tecnique for blood flow simulation in large arteries with varying geometrical and mechanical properties. J. Comput. Phys. 2017, 331, 108–136. [CrossRef]Formaggia, L.; Gerbeau, J.F.; Nobile, F.; Quarteroni, A. On the coupling of 3D and 1D Navier-Stokes equations for flow problems in compliant vessel. Comput. Methods Appl. Mech. Eng. 2001, 191, 561–582. [CrossRef]Mock, M.S. Systems of conservation laws of mixed type. J. Differ. Equ. 1980, 37, 70–88. [CrossRef]Bürger, R.; Valbuena, S.; Vega, C. A well-balanced and entropy stable scheme for a reduced blood flow model. Numer. Meth. Part Differ. Equ. 2021, submitted.Fjordholm, U.S.; Mishra, S.; Tadmor, E. Arbitrary high-order essentially non-oscillatory entropy stable schemes for systems of conservation laws. SIAM J. Numer. Anal. 2012, 50, 544–573. [CrossRef]Harten, A. On the symmetric form of systems of conservation laws with entropy. J. Comput. Phys. 1983, 49, 151–164. [CrossRef]Puelz, C.; Cˇ anic´, S.; Rivière, B.; Rusin, C. Comparison of reduced models for blood flow using Runge-Kutta discontinuos Galerkin methods. Appl. Numer. Math. 2017, 115, 114–141. [CrossRef]Vega, C.; Valbuena, S. Numerical approximations of the Keyfitz-Kranzer type models by using entropy stable schemes. J. Numer. Anal. Ind. Appl. Math. 2020, 3–4, 1–15.Luo, J.; Shu, C.W.; Zhang, Q. A priori error estimates to smooth solutions of the third order Runge–Kutta discontinuous Galerkin method for symmetrizable systems of conservation laws. ESAIM: Math. Model. Numer. Anal. 2015, 49, 991–1018. [CrossRef]Zhang, Q.; Shu, C.W. Error estimates to smooth solutions of Runge–Kutta discontinuous Galerkin method for symmetrizable systems of conservation laws. SIAM J. Numer. Anal. 2006, 44, 1703–1720. [CrossRef]Barth, T.J. Numerical methods for gas-dynamics systems on unstructured meshes. In An Introduction to Recent Developments in Theory and Numerics of Conservation Laws; Kroner, D., Ohlberger, M., Rohde, C., Eds.; Springer: Berlin, Germany, 1999; Volume 5, pp. 195–285. [CrossRef]Tadmor, E. The numerical viscosity of entropy stable schemes for systems of conservation laws, I. Math. Comput. 1987, 49, 91–103. [CrossRef]Fjordholm, U.S.; Mishra, S.; Tadmor, E. Energy preserving and energy stable schemes for the shallow water equations. In Foundations of Computational Mathematics, Hong Kong 2008; London Mathematical Society Lecture Note Series; Cucker, F., Pinkus, A., Todd, M., Eds.; Cambridge University Press: Cambridge, UK, 2009; Volume 363, pp. 93–139. [CrossRef]Tadmor, E. Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numer. 2003, 12, 451–512. [CrossRef]Lefloch, P.G.; Mercier, J.M.; Rohde, C. Fully discrete entropy conservative schemes of arbitrary order. SIAM J. Numer. Anal. 2002, 40, 1968–1992. [CrossRef]Fjordholm, U.S.; Mishra, S.; Tadmor, E. ENO reconstruction and ENO interpolation are stable. Found. Comput. Math. 2013, 13, 139–159. [CrossRef]Pareschi, L.; Russo, G. Implicit-Explicit Runge-Kutta Schemes and Applications to Hyperbolic Systems with Relaxation. J. Sci. Comput. 2005, 25, 129–155. [CrossRef]Boscarino, S.; Filbet, F.; Russo, G. High Order Semi-implicit Schemes for Time Dependent Partial Differential Equations. J. Sci. Comput. 2016, 68, 975–1001. [CrossRef]Boscarino, S.; Bürger, R.; Mulet, P.; Russo, G.; Villada, L.M. Linearly implicit IMEX Runge-Kutta methods for a class of degenerate convection-diffusion problems. SIAM J. Sci. Comput. 2015, 37, B305–B331. [CrossRef]Gottlieb, S.; Shu, C.W.; Tadmor, E. Strong stability-preserving high-order time discretization methods. SIAM Rev. 2001, 43, 89–112. [CrossRef]Womersley, J. On the oscillatory motion of a viscous liquid in thin-walled elastic tube: I. Philos. Mag. 1955, 46, 199–221. [CrossRef]http://purl.org/coar/resource_type/c_2df8fbb1ORIGINALmathematics-10-03314.pdfmathematics-10-03314.pdfapplication/pdf999344https://repositorio.uniatlantico.edu.co/bitstream/20.500.12834/766/1/mathematics-10-03314.pdffec45edf16952283462a4726d89f0398MD51CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; charset=utf-8914https://repositorio.uniatlantico.edu.co/bitstream/20.500.12834/766/2/license_rdf24013099e9e6abb1575dc6ce0855efd5MD52LICENSElicense.txtlicense.txttext/plain; charset=utf-81306https://repositorio.uniatlantico.edu.co/bitstream/20.500.12834/766/3/license.txt67e239713705720ef0b79c50b2ececcaMD5320.500.12834/766oai:repositorio.uniatlantico.edu.co:20.500.12834/7662022-11-15 14:11:12.225DSpace de la Universidad de Atlánticosysadmin@mail.uniatlantico.edu.coVMOpcm1pbm9zIGdlbmVyYWxlcyBkZWwgUmVwb3NpdG9yaW8gSW5zdGl0dWNpb25hbCBkZSBsYSBVbml2ZXJzaWRhZCBkZWwgQXRsw6FudGljbwoKRWwgKGxvcykgYXV0b3IgKGVzKSBoYW4gYXNlZ3VyYWRvIChuKSBsbyBzaWd1aWVudGUgc29icmUgbGEgb2JyYSBhIGludGVncmFyIGVuIGVsIFJlcG9zaXRvcmlvIEluc3RpdHVjaW9uYWwsIHF1ZToKCuKXjwlFcyBvcmlnaW5hbCwgZGUgc3UgZXhjbHVzaXZhIGF1dG9yw61hLCBzZSByZWFsaXrDsyBzaW4gdmlvbGFyIG8gdXN1cnBhciBkZXJlY2hvcyBkZSBhdXRvciBkZSB0ZXJjZXJvcyB5IHBvc2VlIGxhIHRpdHVsYXJpZGFkLgril48JQXN1bWlyw6FuIGxhIHJlc3BvbnNhYmlsaWRhZCB0b3RhbCBwb3IgZWwgY29udGVuaWRvIGEgbGEgb2JyYSBhbnRlIGxhIEluc3RpdHVjacOzbiB5IHRlcmNlcm9zLgril48JQXV0b3JpemFuIGEgdMOtdHVsbyBncmF0dWl0byB5IHJlbnVuY2lhcyBhIHJlY2liaXIgZW1vbHVtZW50b3MgcG9yIGxhcyBhY3RpdmlkYWRlcyBxdWUgc2UgcmVhbGljZW4gY29uIGVsbGEsIHNlZ8O6biBzdSBsaWNlbmNpYS4KCgpMYSBVbml2ZXJzaWRhZCBkZWwgQXRsw6FudGljbywgcG9yIHN1IHBhcnRlLCBzZSBjb21wcm9tZXRlIGEgYWN0dWFyIGVuIGxvcyB0w6lybWlub3MgZXN0YWJsZWNpZG9zIGVuIGxhIExleSAyMyBkZSAxOTgyIHkgbGEgRGVjaXNpw7NuIEFuZGluYSAzNTEgZGUgMTk5MywgZGVtw6FzIG5vcm1hcyBnZW5lcmFsZXMgc29icmUgbGEgbWF0ZXJpYSB5IGVsIEFjdWVyZG8gU3VwZXJpb3IgMDAxIGRlIDE3IGRlIG1hcnpvIGRlIDIwMTEsIHBvciBtZWRpbyBkZWwgY3VhbCBzZSBleHBpZGUgZWwgRXN0YXR1dG8gZGUgUHJvcGllZGFkIEludGVsZWN0dWFsIGRlIGxhIFVuaXZlcnNpZGFkIGRlbCBBdGzDoW50aWNvLgoKUG9yIMO6bHRpbW8sIGhhbiBzaWRvIGluZm9ybWFkb3Mgc29icmUgZWwgdHJhdGFtaWVudG8gZGUgZGF0b3MgcGVyc29uYWxlcyBwYXJhIGZpbmVzIGFjYWTDqW1pY29zIHkgZW4gYXBsaWNhY2nDs24gZGUgY29udmVuaW9zIGNvbiB0ZXJjZXJvcyBvIHNlcnZpY2lvcyBjb25leG9zIGNvbiBhY3RpdmlkYWRlcyBwcm9waWFzIGRlIGxhIGFjYWRlbWlhLCBiYWpvIGVsIGVzdHJpY3RvIGN1bXBsaW1pZW50byBkZSBsb3MgcHJpbmNpcGlvcyBkZSBsZXkuCgpMYXMgY29uc3VsdGFzLCBjb3JyZWNjaW9uZXMgeSBzdXByZXNpb25lcyBkZSBkYXRvcyBwZXJzb25hbGVzIHB1ZWRlbiBwcmVzZW50YXJzZSBhbCBjb3JyZW8gZWxlY3Ryw7NuaWNvIGhhYmVhc2RhdGFAbWFpbC51bmlhdGxhbnRpY28uZWR1LmNvCg== |