Skip to main content

Spectral element methods for nonlinear temporal dynamical systems

Computational Mechanics volume 18, pages 302–313 (1996)Cite this article

Abstract

The time spectral element (TSE) method is a high order accurate method for solving nonlinear and chaotic temporal dynamical systems. It's performance on the Hamiltonian Duffing equation and a bilinear oscillator is examined and compared with standard numerical schemes. These systems were chosen for analysis due to the availability of accurate solutions independent of spurious discretization effects. The implicit form of the TSE, is unconditionally stable in the linear case. For both systems p-convergence is exponential and the h-convergence rate of the end points is of order 2p≦α≦2p+1, (where α is the convergence rate and p the polynomial order).

The explicit form of the TSE is conditionally stable for the linear system. The TSE method competes successfully with the other numerical schemes examined where high accuracy is desired and therefore proves an attractive numerical method for simulation of chaotic dynamical systems.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price includes VAT (USA)
Tax calculation will be finalised during checkout.

References

  • AharoniD. Bar-YosephP. 1992: Mixed finite element formulations in the time domain for solution of dynamic problems. Comput. Mech. 9, 359–374

    Google Scholar 

  • Bar-YosephP. 1989: Space-time discontinuous finite element approximations for multi-dimensional nonlinear hyperbolic systems. Comput. Mech. 5, 149–160

    Google Scholar 

  • Bar-YosephP.; MosesE; ZrahiaU.; YarinL. 1995: Space-time spectral element methods for one dimensional nonlinear advection-diffusion problems. J. Comput. Phys. 118, 62–74

    Google Scholar 

  • BaruchM.; RiffR. 1984: Time finite element discretization of Hamilton's law of varying action. AIAA J. 22, 1310–1318

    Google Scholar 

  • BoriM.; MelloF. J.; AtluriS. N. 1990: Variational approaches for dynamics and time-finite-elements: numerical studies. Comput. Mech. 7, 49–76

    Google Scholar 

  • BorriM.; MelloF. J.; AtluriS. N. 1991: Primal and mixed forms of Hamilton's principle for constrained rigid body systems: numerical studies. Comput. Mech 7, 205–220

    Google Scholar 

  • CorlessR. M. 1992: Defect-controlled numerical methods and shadowing for chaotic differential equations. Physica D, 60, 323–334

    Google Scholar 

  • CorlessR. M. 1994: What good are numerical simulations of chaotic dynamical systems?. Comput. Math. Appl. 28, 107–121

    Google Scholar 

  • CorlessR. M.; EssexC.; NerenbergM. A. H. 1991: Numerical methods can suppress chaos. Phys. Letters A 157, 27–36

    Google Scholar 

  • Fletcher, C. A. J. 1991: Computational techniques for fluid dynamics. Springer-Verlag, 2nd edition, Vol. 1, 355–360

  • GriffithsD. F.; SwebyP. K.; YeeH. C. 1992: On spurious asymptotic numerical solutions of explicit Runge-Kutta methods. IMA J. Numer. Analysis 12, 319–338

    Google Scholar 

  • HiraiK.; AdachiT. 1994: Chaos and bifurcation in numerical computation by the Runge-Kutta method. Int. J. Systems Sci. 11, 1695–1706

    Google Scholar 

  • HulbertG. M.; HughesT. J. R. 1990: Space-time finite element methods for second-order hyperbolic equations. Comput. Methods Appl. Mech. Eng. 84, 327–348

    Google Scholar 

  • HumphriesA. R. 1993: Spurious solutions of numerical methods for initial value problems. IMA J. Numer. Analysis 13, 263–290

    Google Scholar 

  • JohnsonC. 1993: Discontinuous Galerkin finite element methods for second order hyperbolic problems. Comput. Methods Appl. Mech. Eng. 107, 117–129

    Google Scholar 

  • KohC. G.; LiawC. Y. 1991: Effects of time step size on the response of a bilinear system, I: Numerical study. J. Sound & Vibration 144, 17–29

    Google Scholar 

  • LorenzE. N. 1989: Computational chaos — a prelude to computational instability. Physica D 35, 299–317

    Google Scholar 

  • MelloF. J.; BorriM.; AtluriS. N. 1990: Time finite element methods for large rotational dynamics of multibody systems. Comput. & Struct. 37, 231–240

    Google Scholar 

  • MoorthyR. I. K.; KakodkarA.; SrirangarajanH. R.; SuryanarayanS. 1993: An assessment of the Newmark method for solving chaotic vibrations of impacting oscillators. Comput. & Struct. 49, 597–603

    Google Scholar 

  • NatsiavasS. 1990: On the dynamics of oscillators with bi-linear damping and stiffness. Int. J. Non-Linear Mechanics 25, 535–554

    Google Scholar 

  • PateraA. T. 1984: A spectral element method for fluid dynamics; laminar flow in a channel expansion. J. Comput. Phys. 54, 468–488

    Google Scholar 

  • ReinhallP. J.; CaugheyT. K.; StortiD. W. 1989: Order and chaos in a discrete Duffing oscillator-implications on numerical integration. J. Appl. Mech. 56, Trans. ASME, 162–167

    Google Scholar 

  • ShawS. W.; HolmesP. 1983: A periodically forced piecewise linear oscillator. J. Sound & Vibration 90, 129–155

    Google Scholar 

  • ThompsonJ. M. T. 1983: Complex dynamics of compliant off-shore structures. Proc. R. Soc. Lond. 387, 407–427

    Google Scholar 

  • TongueB. H. 1987: Characteristics of numerical simulations of chaotic systems. J. Appl. Mech. 54, Trans. ASME, 695–699

    Google Scholar 

  • YeeH. C.; SwebyP. K.; GriffithsD. F. 1991: Dynamical approach study of spurious steady state numerical solutions of nonlinear differential equations. I. The dynamics of time discretization and its implications for algorithm development in computational fluid dynamics. J. Comput. Phys. 97, 249–310

    Google Scholar 

  • ZrahiaU.; Bar-YosephP. 1994: Space-time spectral element method for solution of second-order hyperbolic equations. Comput. Methods Appl. Mech. Eng. 116, 135–146

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computational Mechanics Laboratory, Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, 32000, Haifa, Israel

    P. Z. Bar-Yoseph, D. Fisher & O. Gottlieb

Authors
  1. P. Z. Bar-Yoseph
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. Fisher
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. O. Gottlieb
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by S. N. Atluri, 11 March 1996

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Bar-Yoseph, P.Z., Fisher, D. & Gottlieb, O. Spectral element methods for nonlinear temporal dynamical systems. Computational Mechanics 18, 302–313 (1996). https://doi.org/10.1007/BF00364145

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00364145

Keywords

  • Dynamical System
  • Linear System
  • Information Theory
  • Convergence Rate
  • Explicit Form