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Spectral element methods for nonlinear temporal dynamical systems

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.

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Communicated by S. N. Atluri, 11 March 1996

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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

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

Keywords

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