## Numerical Approximations of the Dynamical System Generated by Burgers’ Equation with Neumann–Dirichlet Boundary Conditions

^{1} 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
^{2} 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
^{3} 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

Received:
2
January
2012

Using 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 **(2002) 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 **(2002) 1165–1195] are not dependent on a specific
time marching scheme, but are generic to all convergent numerical approximations of
Burgers’ equation.

Mathematics Subject Classification: 37L05 / 65P30 / 35B32 / 35B41 / 65M99 / 65P40 / 34B15

Key words: Nonlinear dynamical system / finite precision arithmetic / bifurcation / asymptotic behavior / numerical approximation / stability / nonlinear partial differential equation / boundary value problem

*© EDP Sciences, SMAI, 2013*