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One-sided difference schemes and transonic flow.

B Engquist1, S Osher

  • 1Department of Mathematics, University of California, Los Angeles, California 90024.

Proceedings of the National Academy of Sciences of the United States of America
|June 1, 1980
PubMed
Summary

New difference approximations for conservation laws ensure stable, accurate solutions for fluid dynamics problems. These methods guarantee entropy condition satisfaction and prevent nonphysical results in transonic flow simulations.

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Area of Science:

  • Numerical Analysis
  • Computational Fluid Dynamics
  • Partial Differential Equations

Background:

  • Conservation laws are fundamental in describing physical phenomena like fluid flow.
  • Developing accurate and stable numerical methods is crucial for solving these laws.
  • Existing methods may suffer from instability or produce nonphysical solutions.

Purpose of the Study:

  • To introduce novel, efficient difference approximations for scalar conservation laws.
  • To ensure these approximations are nonlinearly stable and satisfy the entropy condition.
  • To apply these methods to approximate transonic flow equations.

Main Methods:

  • Development of two one-sided, conservation form difference approximations (first- and second-order).
  • Analysis of nonlinear stability and convergence properties, focusing on entropy condition satisfaction.

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  • Construction and application of dimensional splitting algorithms for transonic flow approximation.

    • Main Results:

    • The proposed approximations exhibit minimum bandwidth and nonlinear stability.
    • Convergence is guaranteed only to entropy-satisfying solutions, yielding sharp monotone profiles.
    • No stable approximation of order higher than two is possible under these constraints.
    • Dimensional splitting algorithms applied to transonic flow yielded stable approximations without nonphysical limit solutions.

    Conclusions:

    • The developed difference approximations offer a robust and accurate approach for solving conservation laws.
    • These methods are particularly effective for simulating transonic flow phenomena.
    • The findings establish theoretical limits on the order of stable approximations for such problems.