Abstract
In this paper we consider the impact of using “time marching” numerical schemes for computing asymptotic solutions of nonlinear differential equations. We show that stable and consistent approximating schemes can produce numerical solutions that do not correspond to the correct asymptotic solutions of the differential equation. In addition, we show that this problem cannot be avoided by placing additional side conditions on the boundary value problem, even if the numerical scheme preserves the side conditions at every step. Examples are given to illustrate the problems that can arise and the implications of using such methods in control design are discussed.
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[1] 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, Journal of Mathematical and Computer Modelling, to appear.
H. T. Banks and K. Kunisch, An Approximation Theory for Nonlinear Partial Differential Equations with Applications to Identication and Control, SIAM J. Control and Optimization, Vol. 20, (1982), 815–489.
J. M. Burgers, Application of a Model System to Illustrate Some Points of the Statistical Theory of Free Turbulence, Proc. Acad. Sci. Amsterdam, Vol. 43, (1940), 2–12.
J. A. Burns, A. Balogh, D. Gilliam and V. Shubov, Numerical Stationary Solutions for a Viscous Burgers’ Equation, Journal of Mathematical Systems, Estimation and Control, Vol. 8, (1998), 189–192.
C. I. Byrnes, D. S. Gilliam and V. I. Shubov, Boundary Control, Feedback Stabilization and the Existence of Attractors for a Viscous Burgers’ Equation, Preprint,(1994).
C. I. Byrnes, D. S. Gilliam, V. I. Shubov and Z. Xu, Steady State Response to Burgers’ Equation with Varying Viscosity, Progress in Systems and Control: Computation and Control IV,K. L. Bowers and J. Lund, eds., Birkhäuser, 75–98, 1995.
C. I. Byrnes, D. S. Gilliam and V. I. Shubov, On the Global Dynamics of a Controlled Viscous Burgers’ Equation, Journal of Dynamical and Control Systems, Vol. 4, (1998), 457–519.
C. I. Byrnes, D. S. Gilliam and V. I. Shubov, Boundary Control, Stabilization and Zero Pole Dynamics for a Nonlinear Distributed Parameter, International Journal of Robust and Nonlinear Control, Vol. 9, (1999), 737–768.
C. Cao and E. S. Titi, Asymptotic Behavior of Viscous Burgers’ Equations with Neumann Boundary Conditions, Private Communication.
P. Embid, J. Goodman and A. Majda, Multiple Steady States for 1-D Transonic Flow, SIAM J. Sci. Computing, Vol. 5, (1984), 21–41.
C.A.J. Fletcher, Burgers’ Equation: A Model For All Reasons, Numerical Solutions of Partial Differential Equations,J. Noye. ed., North-Holland Publishing, 139–225, 1982.
C.A.J. FletcherComputational Galerkin MethodsSpringer-Verlag, New York, 1984.
M. Garbey and H. G. Kaper, Asymptotic-Numerical Study of Supersensitivity for Generalized Burgers’ Equation, SIAM J. Sci. Computing, Vol. 22, (2000), 368–385.
A. Jameson, Airfoils Admitting Non-unique Solutions of the Euler Equations, AIAA 22nd Fluid Dynamics, Plasmadynamics Lasers Conference, Honolulu, (1991), Paper AIAA 91–1625.
T. Kato, Pertubation Theory for Linear Operators, Springer-Verlag, New York, 1966.
H. Van Ly, K. D. Mease and E. S. Titi, Some Remarks on Distributed and Boundary Control of the Viscous Burgers’ Equation, Numer. Funct. Anal. Optim., Vol 18, (1993), 143–188.
G. Kreiss and H. O. Kreiss, Convergence to Steady State of Solutions of Burgers’ Equation, Applied Numerical Mathematics, Vol. 2, (1986), 161–179.
H. O. Kreiss and J. Lorenz, Initial-Boundary Value Problems and the Navier-Stokes Equations, Academic Press, 1989.
O. A. Ladyzhenskaya and A. A. Kiselev, On the Existence and Uniqueness of the Solution of the Nonstationary Problem for a Viscous Incompressible Fluid, Izv. Akad. Nauk SSSR Ser. Mat., Vol. 21, (1957), 655–680.
H. Marrekchi, Dynamic Compensators for a Nonlinear Conservation Law, Ph.D. Thesis, Department of Mathematics, Virginia Polytechnic Institute and State University, 1993.
H. Matano, Convergence of Solutions of One Dimensional Semilinear Parabolic Equations, J. Math Kyoto Univ., Vol. 18, (1978), 221–227.
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
S. M. Pugh, Finite Element Approximations of Burgers’ Equation, Masters of Science Thesis, Department of Mathematics, Virginia Polytechnic Institute and State University, 1995.
L. G. Reyna and M. J. Ward, On the Exponentially Slow Motion of a Viscous Shock, Communications on Pure and Applied Math., Vol. XLVIII, (1995), 79–120.
M. D. Salas, S. Abarbanel and D. Gottlieb, Multiple Steady States for Characteristic Initial Value Problems, Appl. Numer. Math. Vol. 2, (1986), 193–210.
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Burns, J.A., Singler, J.R. (2003). On the Long Time Behavior of Approximating Dynamical Systems. In: Desch, W., Kappel, F., Kunisch, K. (eds) Control and Estimation of Distributed Parameter Systems. ISNM International Series of Numerical Mathematics, vol 143. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8001-5_5
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DOI: https://doi.org/10.1007/978-3-0348-8001-5_5
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