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Numerical Approximations of the Dynamical System Generated byBurgers’ Equation with Neumann–Dirichlet Boundary Conditions

Published online by Cambridge University Press:  30 July 2013

Edward J. Allen ,
John A. Burns  and
David S. Gilliam
Edward J. Allen
Affiliation:
Supported in part by the National Science Foundation grant NSF-DMS 0718302. Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, USA.. edward.allen@ttu.edu
John A. Burns
Affiliation:
Supported by the Air Force Office of Scientific Research grants FA9550-07-1-0273 and FA9550-10-1-0201 and by the DOE contract DE-EE0004261 under subcontract # 4345-VT-DOE-4261 from Penn State University., Interdisciplinary Center for Applied Mathematics, Virginia Tech, Blacksburg, VA 24061, USA.; jaburns@vt.edu
David S. Gilliam
Affiliation:
Supported in part by Air Force Office of Scientific Research grant FA9550-12-1-0114. Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, USA.; david.gilliam@ttu.edu
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Abstract

Using Burgers’ equation with mixed Neumann–Dirichlet boundary conditions, we highlight aproblem that can arise in the numerical approximation of nonlinear dynamical systems oncomputers with a finite precision floating point number system. We describe the dynamicalsystem generated by Burgers’ equation with mixed boundary conditions, summarize some ofits properties and analyze the equilibrium states for finite dimensional dynamical systemsthat are generated by numerical approximations of this system. It is important to notethat there are two fundamental differences between Burgers’ equation with mixedNeumann–Dirichlet boundary conditions and Burgers’ equation with both Dirichlet boundaryconditions. First, Burgers’ equation with homogenous mixed boundary conditions on a finiteinterval cannot be linearized by the Cole–Hopf transformation. Thus, on finite intervalsBurgers’ equation with a homogenous Neumann boundary condition is truly nonlinear. Second,the nonlinear term in Burgers’ equation with a homogenous Neumann boundary condition isnot conservative. This structure plays a key role in understanding the complex dynamicsgenerated by Burgers’ equation with a Neumann boundary condition and how this structureimpacts numerical approximations. The key point is that, regardless of the particularnumerical scheme, finite precision arithmetic will always lead to numerically generatedequilibrium states that do not correspond to equilibrium states of the Burgers’ equation.In this paper we establish the existence and stability properties of these numericalstationary solutions and employ a bifurcation analysis to provide a detailed mathematicalexplanation of why numerical schemes fail to capture the correct asymptotic dynamics. Weextend the results in [E. Allen, J.A. Burns, D.S. Gilliam, J. Hill and V.I. Shubov,Math. Comput. Modelling 35 (2002) 1165–1195] and provethat the effect of finite precision arithmetic persists in generating a nonzero numericalfalse solution to the stationary Burgers’ problem. Thus, we show that the results obtainedin [E. Allen, J.A. Burns, D.S. Gilliam, J. Hill and V.I. Shubov, Math. Comput.Modelling 35 (2002) 1165–1195] are not dependent on a specifictime marching scheme, but are generic to all convergent numerical approximations ofBurgers’ equation.

Keywords

Nonlinear dynamical systemfinite precision arithmeticbifurcationasymptotic behaviornumerical approximationstabilitynonlinear partial differential equationboundary value problem
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 5 , September 2013 , pp. 1465 - 1492
Copyright
© EDP Sciences, SMAI, 2013

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