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Stable second-order scheme for integrating the Kuramoto-Sivanshinsky equation in polar coordinates using distributed

Peter Blomgren1, Scott Gasner, Antonio Palacios

  • 1Nonlinear Dynamical Systems Group, Department of Mathematics and Statistics, San Diego State University, San Diego, California 92182, USA. blomgren@mail.SDSU.EDU

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 26, 2005
PubMed
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A new algorithm accurately integrates nonlinear partial differential equations using distributed approximating functionals and a Crank-Nicolson scheme. This robust method reproduces complex patterns observed in combustion fronts, offering a generalizable approach for scientific simulations.

Area of Science:

  • Numerical Analysis
  • Computational Physics
  • Fluid Dynamics

Background:

  • Nonlinear partial differential equations (PDEs) model complex phenomena.
  • Accurate numerical integration is crucial for simulating these systems.
  • Existing methods face challenges with stiffness and spatial accuracy.

Purpose of the Study:

  • To develop a robust and accurate algorithm for time integration of nonlinear PDEs.
  • To apply the algorithm to the computationally challenging Kuramoto-Sivanshinsky equation in polar coordinates.
  • To validate the scheme by comparing numerical results with experimental observations.

Main Methods:

  • Utilized distributed approximating functionals (DAFs) for high-accuracy spatial derivatives.
  • Employed a second-order, unconditionally A-stable Crank-Nicolson scheme with a Newton solver for time integration.

Related Experiment Videos

  • Applied the combined scheme to the Kuramoto-Sivanshinsky equation in polar coordinates.

    • Main Results:

    • The algorithm accurately reproduced various stationary and nonstationary solutions of the Kuramoto-Sivanshinsky equation.
    • Numerical results matched patterns observed in circular burner combustion fronts, including rotating and stationary multi-cell structures.
    • The scheme demonstrated robustness and capability for long-term simulations.

    Conclusions:

    • The combination of DAFs and Crank-Nicolson time integration provides a powerful and generalizable method for nonlinear PDEs.
    • The developed scheme is highly effective for problems with stiffness, such as those involving spatial derivatives at singularities.
    • This approach offers a reliable tool for simulating complex physical phenomena with high fidelity.