Numerical Approximations of the Dynamical System Generated by Burgers' Equation with Neumann-Dirichlet Boundary Conditions
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 (2013) no. 5, pp. 1465-1492.

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.

DOI: 10.1051/m2an/2013084
Classification: 37L05, 65P30, 35B32, 35B41, 65M99, 65P40, 34B15
Mots-clés : nonlinear dynamical system, finite precision arithmetic, bifurcation, asymptotic behavior, numerical approximation, stability, nonlinear partial differential equation, boundary value problem
@article{M2AN_2013__47_5_1465_0,
     author = {Allen, Edward J. and Burns, John A. and Gilliam, David S.},
     title = {Numerical {Approximations} of the {Dynamical} {System} {Generated} by {Burgers'} {Equation} with {Neumann-Dirichlet} {Boundary} {Conditions}},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1465--1492},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {5},
     year = {2013},
     doi = {10.1051/m2an/2013084},
     mrnumber = {3100771},
     zbl = {1283.37072},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2013084/}
}
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 
PY  - 2013
SP  - 1465
EP  - 1492
VL  - 47
IS  - 5
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2013084/
DO  - 10.1051/m2an/2013084
LA  - en
ID  - M2AN_2013__47_5_1465_0
ER  - 
%0 Journal Article
%A Allen, Edward J.
%A Burns, John A.
%A Gilliam, David S.
%T Numerical Approximations of the Dynamical System Generated by Burgers' Equation with Neumann-Dirichlet Boundary Conditions
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2013
%P 1465-1492
%V 47
%N 5
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2013084/
%R 10.1051/m2an/2013084
%G en
%F M2AN_2013__47_5_1465_0
Allen, Edward J.; Burns, John A.; Gilliam, David S. Numerical Approximations of the Dynamical System Generated by Burgers' Equation with Neumann-Dirichlet Boundary Conditions. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 (2013) no. 5, pp. 1465-1492. doi : 10.1051/m2an/2013084. http://www.numdam.org/articles/10.1051/m2an/2013084/

[1] V.S. Afraimovich, M.K. Muezzinoglu and M.I. Rabinovich, Metastability and Transients in Brain Dynamics: Problems and Rigorous Results, in Long-range Interactions, Stochasticity and Fractional Dynamics; Nonlinear Physical Science, edited by Albert C.J. Luo and Valentin Afraimovich. Springer-Verlag (2010) 133-175. | MR | Zbl

[2] E. Allen, J.A. Burns, D.S. Gilliam, J. Hill and V.I. Shubov, The impact of finite precision arithmetic and sensitivity on the numerical solution of partial differential equations. Math. Comput. Model. 35 (2002) 1165-1195. | MR | Zbl

[3] E. Allen, J.A. Burns and D.S. Gilliam, On the use of numerical methods for analysis and control of nonlinear convective systems, in Proc. of 47th IEEE Conference on Decision and Control (2008) 197-202.

[4] J.A. Atwell and B.B. King, Stabilized Finite Element Methods and Feedback Control for Burgers' Equation, in Proc. of the 2000 American Control Conference (2000) 2745-2749.

[5] D.H. Bailey and J.M. Borwein, Exploratory Experimentation and Computation, Notices AMS 58 (2011) 1410-1419. | MR | Zbl

[6] A. Balogh, D.S. Gilliam and V.I. Shubov, Stationary solutions for a boundary controlled Burgers' equation. Math. Comput. Model. 33 (2001) 21-37. | MR | Zbl

[7] M. Beck and C.E. Wayne, Using Global Invariant Manifolds to Understand Metastability in the Burgers Equation With Small Viscosity. SIAM Review 53 (2011) 129-153 [Published originally SIAM J. Appl. Dyn. Syst. 8 (2009) 1043-1065]. | MR | Zbl

[8] T.R. Bewley, P. Moin and R. Temam, Control of Turbulent Flows, in Systems Modelling and Optimization, Chapman and Hall CRC, Boca Raton, FL (1999) 3-11. | MR | Zbl

[9] J.T. Borggaard and J.A. Burns, A PDE Sensitivity Equation Method for Optimal Aerodynamic Design. J. Comput. Phys. 136 (1997) 366-384. | MR | Zbl

[10] J. Burns, A. Balogh, D. Gilliam and V. Shubov, Numerical stationary solutions for a viscous Burgers' equation. J. Math. Syst. Estim. Control 8 (1998) 1-16. | MR | Zbl

[11] J.A. Burns and S. Kang, A control problem for Burgers' equation with bounded input/output. Nonlinear Dyn. 2 (1991) 235-262.

[12] J.A. Burns and S. Kang, A Stabilization problem for Burgers' equation with unbounded control and observation, in Estimation and Control of Distributed Parameter Systems. Int. Ser. Numer. Math. vol. 100, edited by W. Desch, F. Rappel, K. Kunisch. Springer-Verlag (1991) 51-72. | MR | Zbl

[13] J.A. Burns and H. Marrekchi, Optimal fixed-finite-dimensional compensator for Burgers' Equation with unbounded input/output operators. ICASE Report No. 93-19. Institute for Comput. Appl. Sci. Engrg., Hampton, VA. (1993). | MR | Zbl

[14] J.A. Burns and J.R. Singler, On the Long Time Behavior of Approximating Dynamical Systems, in Distributed Parameter Control, edited by F. Kappel, K. Kunisch and W. Schappacher. Springer-Verlag (2001) 73-86. | Zbl

[15] C.I. Byrnes and D.S. Gilliam, Boundary control and stabilization for a viscous Burgers' equation. Computation and Control, Progress in Systems Control Theory, vol. 15. Birkhäuser Boston, Boston, MA (1993) 105-120. | MR | Zbl

[16] C.I. Byrnes, D.S. Gilliam and V.I. Shubov, Convergence of trajectories for a controlled viscous Burgers' equation, Control and Estimation of Distributed Parameter Systems: Nonlinear Phenomena. Int. Ser. Numer. Math., vol. 118, edited by W. Desch, F. Rappel, K. Kunisch. Birkhäuser, Basel (1994) 61-77. | MR | Zbl

[17] C.I. Byrnes, D. Gilliam, V. Shubov and Z. Xu, Steady state response to Burgers' equation with varying viscosity, in Progress in Systems and Control: Computation and Control IV, edited by K. L.Bowers and J. Lund. Birkhäuser, Basel (1995) 75-98. | MR | Zbl

[18] C.I. Byrnes, D.S. Gilliam and V.I. Shubov, High gain limits of trajectories and attractors for a boundary controlled viscous Burgers' equation. J. Math. Syst. Estim. Control 6 (1996) 40. | MR | Zbl

[19] C.I. Byrnes, A. Balogh, D.S. Gilliam and V.I. Shubov, Numerical stationary solutions for a viscous Burgers' equation. J. Math. Syst. Estim. Control 8 (1998) 16 (electronic). | MR | Zbl

[20] C.I. Byrnes, D.S. Gilliam and V.I. Shubov, On the Global Dynamics of a Controlled Viscous Burgers' Equation. J. Dyn. Control Syst. 4 (1998) 457-519. | MR | Zbl

[21] C.I. Byrnes, D.S. Gilliam and V.I. Shubov, Boundary Control, Stabilization and Zero-Pole Dynamics for a Nonlinear Distributed Parameter System. Int. J. Robust Nonlinear Control 9 (1999) 737-768. | MR | Zbl

[22] C. Cao and E. Titi, Asymptotic Behavior of Viscous Burgers' Equations with Neumann Boundary Conditions, Third Palestinian Mathematics Conference, Bethlehem University, West Bank. Mathematics and Mathematics Education, edited by S. Elaydi, E. S. Titi, M. Saleh, S. K. Jain and R. Abu Saris. World Scientific (2002) 1-19. | Zbl

[23] M.H. Carpenter, J. Nordström and D. Gottlieb, Revisiting and extending interface penalties for multi-domain summation-by-parts operators. J. Sci. Comput. 45 (2010) 118-150. | MR | Zbl

[24] J. Carr and J.L. Pego, Metastable patterns in solutions of ut = ϵ2uxx − f(u). Comm. Pure Appl. Math. 42 (1989) 523-576. | MR | Zbl

[25] J. Carr, D.B. Duncan and C.H. Walshaw, Numerical approximation of a metastable system. IMA J. Numer. Anal. 15 (1995) 505-521. | MR | Zbl

[26] C.A.J. Fletcher, Burgers' equation: A model for all reasons, in Numerical Solutions of J. Partial Differ. Eqns., edited by J. Noye. North-Holland Publ. Co. Amsterdam (1982) 139-225. | MR | Zbl

[27] A.V. Fursikov and R. Rannacher, Optimal Neumann Control for the 2D Steady-State Navier-Stokes equations, in New Directions in Math. Fluid Mech. The Alexander. V. Kazhikhov Memorial Volume. Advances in Mathematical Fluid Mechanics, Birkhauser, Berlin (2009) 193-222. | MR | Zbl

[28] G. Fusco, G. and J. K. Hale, Slow-motion manifolds, dormant instability, and singular perturbations. J. Dyn. Differ. Eqns. 1 (1989) 75-94. | MR | Zbl

[29] T. Gallay and C.E. Wayne, Invariant manifolds and the long-time asymptotics of the navier-stokes and vorticity equations on R2. Arch. Rational Mech. Anal. 163 (2002) 209-258. | MR | Zbl

[30] T. Gallay and C.E. Wayne, Global stability of vortex solutions of the two-dimensional Navier-Stokes equation. Commun. Math. Phys. 255 (2005) 97-129. | MR | Zbl

[31] M. Garbey and H.G. Kaper, Asymptotic-Numerical Study of Supersensitivity for Generalized Burgers' Equation. SIAM J. Sci. Comput. 22 (2000) 368-385. | MR | Zbl

[32] S. Gottlieb, D. Gottlieb and C.-W. Shu, Recovering High-Order Accuracy in WENO Computations of Steady-State Hyperbolic Systems. J. Sci. Comput. 28 (2006) 307-318. | MR | Zbl

[33] M. Gunzburger, L. Hou and T. Svobodny, Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with distributed and Neumann controls. Math. Comput. 57 (1991) 123-151. | MR | Zbl

[34] M.D. Gunzburger, H.C. Lee and J. Lee, Error estimates of stochastic optimal Neumann boundary control problems. SIAM J. Numer. Anal. 49 (2011) 1532-1552. | MR | Zbl

[35] J.S. Hesthaven, S. Gottlieb and D. Gottlieb, Spectral Methods for Time Dependent Problems, Cambridge Monographs on Applied and Computational Mathematics, vol. 21. Cambridge University Press (2006). | Zbl

[36] IEEE Computer Society, IEEE Standard for Binary Floating-Point Arithmetic, IEEE Std 754-1985 (1985).

[37] A. Kanevsky, M.H. Carpenter, D. Gottlieb and J. S. Hesthaven, Application of implicit-explicit high order Runge-Kutta methods to discontinuous-Galerkin schemes. J. Comput. Phys. 225 (2007) 1753-1781. | MR | Zbl

[38] R. Kannan and Z.J. Wang, A high order spectral volume solution to the Burgers' equation using the Hopf-Cole transformation. Int. J. Numer. Meth. Fluids (2011). Available on wileyonlinelibrary.com. DOI: 10.1002/fld.2612. | Zbl

[39] O.A. Ladyzhenskaya, V.A. Solonnikov and N.N. Ural'Ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of the AMS, vol. 23 (1968). | Zbl

[40] J.G.L. Laforgue and R.E. O'Malley, Supersensitive Boundary Value Problems, Asymptotic and Numerical Methods for Partial Differential Equations with Critical Parameters, edited by H.G. Kaper and M. Garbey. Kluwer Publishers (1993) 215-224. | MR | Zbl

[41] H.V. Ly, K.D. Mease and E.S. Titi, Distributed and boundary control of the viscous Burgers' equation. Numer. Funct. Anal. Optim. 18 (1997) 143-188. | MR | Zbl

[42] H. Marrekchi, Dynamic Compensators for a Nonlinear Conservation Law, Ph.D. Thesis, Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA, 24061 (1993). | MR

[43] V.Q. Nguyen, A Numerical Study of Burgers' Equation With Robin Boundary Conditions, M.S. Thesis. Department of Mathematics, Polytechnic Institute and State University, Blacksburg, VA, 24061 (2001). | Zbl

[44] P. Pettersson, J. Nordström and G. Laccarino, Boundary procedures for the time-dependent Burgers' equation under uncertainty. Acta Math. Sci. 30 (2010) 539-550. | MR | Zbl

[45] J.T. Pinto, Slow motion manifolds far from the attractor in multistable reaction-diffusion equations. J. Differ. Eqns. 174 (2001) 101-132. | MR | Zbl

[46] S.M. Pugh, Finite element approximations of Burgers' Equation, M.S. Thesis. Departmant of Mathematics, Polytechnic Institute and State University, Blacksburg, VA, 24061 (1995).

[47] G.R. Sell and Y. You, Dynamics of Evolutionary Equations, vol. 143. Springer-Verlag (2002). | MR | Zbl

[48] Z.-H. Teng, Exact boundary conditions for the initial value problem of convex conservation laws. J. Comput. Phys. 229 (2010) 3792-3801. | MR | Zbl

[49] M.J. Ward and L.G. Reyna, Internal layers, small eigenvalues, and the sensitivity of metastable motion. SIAM J. Appl. Math. 55 (1995) 425-445. | MR | Zbl

[50] T.I. Zelenjak, Stabilization of solutions of boundary value problems for a second-order parabolic equation with one space variable, Differentsial'nye Uravneniya 4 (1968) 34D45. | MR | Zbl

[51] T.I. Zelenyak, M.M. Lavrentiev Jr. and M.P. Vishnevskii, Qualitative Theory of Parabolic Equations, Part 1, VSP, Utrecht, The Netherlands (1997). | MR | Zbl

Cited by Sources: