Numerical Approximations of the Dynamical System Generated by Burgers' Equation with Neumann-Dirichlet Boundary Conditions

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dc.contributorVirginia Techen
dc.contributor.authorAllen, Edward J.en
dc.contributor.authorBurns, John A.en
dc.contributor.authorGilliam, David S.en
dc.contributor.departmentMathematicsen
dc.date.accessed2013-12-06en
dc.date.accessioned2013-12-06T18:58:35Zen
dc.date.available2013-12-06T18:58:35Zen
dc.date.issued2013-09en
dc.description.abstractUsing Burgers' equation with mixed Neumann-Dirichlet boundary conditions, we highlight a problem that can arise in the numerical approximation of nonlinear dynamical systems on computers with a finite precision floating point number system. We describe the dynamical system generated by Burgers' equation with mixed boundary conditions, summarize some of its properties and analyze the equilibrium states for finite dimensional dynamical systems that are generated by numerical approximations of this system. It is important to note that there are two fundamental differences between Burgers' equation with mixed Neumann-Dirichlet boundary conditions and Burgers' equation with both Dirichlet boundary conditions. First, Burgers' equation with homogenous mixed boundary conditions on a finite interval cannot be linearized by the Cole-Hopf transformation. Thus, on finite intervals Burgers' equation with a homogenous Neumann boundary condition is truly nonlinear. Second, the nonlinear term in Burgers' equation with a homogenous Neumann boundary condition is not conservative. This structure plays a key role in understanding the complex dynamics generated by Burgers' equation with a Neumann boundary condition and how this structure impacts numerical approximations. The key point is that, regardless of the particular numerical scheme, finite precision arithmetic will always lead to numerically generated equilibrium states that do not correspond to equilibrium states of the Burgers' equation. In this paper we establish the existence and stability properties of these numerical stationary solutions and employ a bifurcation analysis to provide a detailed mathematical explanation of why numerical schemes fail to capture the correct asymptotic dynamics. We extend the results in [E. Allen, J.A. Burns, D. S. Gilliam, J. Hill and V. I. Shubov, Math. Comput. Modelling 35 92002) 1165-1195] and prove that the effect of finite precision arithmetic persists in generating a nonzero numerical false solution to the stationary Burgers' problem. Thus, we show that the results obtained in [E. Allen, J. A. Burns, D. S. Gilliam, J. Hill and V. I. Shubov, Math. Comput. Modelling 35 92002) 1165-1195] are not dependent on a specific time marching scheme, but are generic to all convergent numerical approximations of Burgers' equation.en
dc.description.sponsorshipNational Science Foundation NSF-DMS 0718302en
dc.description.sponsorshipAir Force Office of Scientific Research FA9550-07-1-0273, FA9550-10-1-0201, FA9550-12-1-0114en
dc.description.sponsorshipDOE under Penn State University., Interdisciplinary Center for Applied Mathematics, Virginia Tech, Blacksburg, VA, USA DE-EE0004261, 4345-VT-DOE-4261en
dc.identifier.citationAllen, Edward J. ; Burns, John A. ; Gilliam, David S., Sep 2013. “Numerical Approximations of the Dynamical System Generated by Burgers’ Equation with Neumann–Dirichlet Boundary Conditions,” ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE 47(5):1465-1492. DOI: 10.1051/m2an/2013084en
dc.identifier.doihttps://doi.org/10.1051/m2an/2013084en
dc.identifier.issn0764-583Xen
dc.identifier.urihttp://hdl.handle.net/10919/24447en
dc.language.isoen_USen
dc.publisherEDP Sciencesen
dc.rightsIn Copyrighten
dc.rights.urihttp://rightsstatements.org/vocab/InC/1.0/en
dc.subjectNonlinear dynamical systemen
dc.subjectfinite precision arithmeticen
dc.subjectbifurcationen
dc.subjectasymptotic behavioren
dc.subjectnumerical approximationen
dc.subjectstabilityen
dc.subjectnonlinear partial differential equationen
dc.subjectboundary value problemen
dc.subjectnavier-stokesen
dc.subjectinvarient-manifoldsen
dc.subjectsensitivityen
dc.subjectstabilizationen
dc.subjectmotionen
dc.titleNumerical Approximations of the Dynamical System Generated by Burgers' Equation with Neumann-Dirichlet Boundary Conditionsen
dc.title.serialEsaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique Et Analyse Numeriqueen
dc.typeArticle - Refereeden

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