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

Edward J. Allen; John A. Burns; David S. Gilliam

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2013)

  • Volume: 47, Issue: 5, page 1465-1492
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Allen, Edward J., Burns, John A., and Gilliam, David S.. "Numerical Approximations of the Dynamical System Generated by Burgers’ Equation with Neumann–Dirichlet Boundary Conditions." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 47.5 (2013): 1465-1492. <http://eudml.org/doc/273133>.

@article{Allen2013,
abstract = {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.},
author = {Allen, Edward J., Burns, John A., Gilliam, David S.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {nonlinear dynamical system; finite precision arithmetic; bifurcation; asymptotic behavior; numerical approximation; stability; nonlinear partial differential equation; boundary value problem},
language = {eng},
number = {5},
pages = {1465-1492},
publisher = {EDP-Sciences},
title = {Numerical Approximations of the Dynamical System Generated by Burgers’ Equation with Neumann–Dirichlet Boundary Conditions},
url = {http://eudml.org/doc/273133},
volume = {47},
year = {2013},
}

TY - JOUR
AU - Allen, Edward J.
AU - Burns, John A.
AU - Gilliam, David S.
TI - Numerical Approximations of the Dynamical System Generated by Burgers’ Equation with Neumann–Dirichlet Boundary Conditions
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2013
PB - EDP-Sciences
VL - 47
IS - 5
SP - 1465
EP - 1492
AB - 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.
LA - eng
KW - nonlinear dynamical system; finite precision arithmetic; bifurcation; asymptotic behavior; numerical approximation; stability; nonlinear partial differential equation; boundary value problem
UR - http://eudml.org/doc/273133
ER -

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