WIAS Preprint No. 771, (2002)

Numerical algorithms to calculate periodic solutions of the Sivashinsky equation


  • Karlin, Vladimir
  • Maz'ya, Vladimir
  • Schmidt, Gunther

2010 Mathematics Subject Classification

  • 65M70 76E17 65G50


  • saturated asymptotic approximations, Sivashinsky equation, flame fronts, stability, round-off errors




The primary aim of this work is the accurate calculation of periodic solutions to the Sivashinsky equation, which models dynamics of the long wave flame instability. A highly accurate computational algorithm has been developed in both one and two spatial dimensions and its crucial implementation details has been presented. The algorithm is based on the concept of "approximate approximations" which also can be referred to as saturated asymptotic approximations. The given computations support the idea of the instability of steady solutions to the Sivashinsky equation in large domains through huge linear amplification of nonmodal perturbations. Unlike the presentation of the algorithm is given for a particular equation, the evaluations have been carried out in a very general manner and the algorithm can be straightforwardly applied to a wide variety of nonlinear integro-differential equations.

Appeared in

  • J. Comput. Phys. 188 (2003), pp. 209--231; under the new title: High accuracy periodic solutions to the Sivashinsky equation

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