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Optimal renormalization of multiscale systems.

Jacob Price1, Brek Meuris2, Madelyn Shapiro3,4

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This summary is machine-generated.

This study introduces a new time-dependent renormalization method to stabilize reduced order models for complex systems. The approach ensures long-term accuracy and stability, even when dealing with singularities in fluid dynamics simulations.

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Area of Science:

  • Computational fluid dynamics
  • Numerical analysis
  • Applied mathematics

Background:

  • Model order reduction (MOR) is crucial for simulating large, time-dependent systems efficiently.
  • Standard reduced order models often exhibit instabilities, limiting their long-term predictive capabilities.
  • Existing time-dependent renormalization approaches offer stabilization but require further refinement.

Purpose of the Study:

  • To extend the time-dependent renormalization framework for enhanced stability of reduced order models.
  • To introduce and optimize a parameter controlling the memory decay in these models.
  • To validate the approach on challenging problems like the inviscid Burgers equation and 3D Euler equations.

Main Methods:

  • Developed a time-dependent renormalization approach with a tunable memory decay parameter.
  • Optimized the memory decay parameter using limited data from fully resolved simulations.
  • Applied and validated the framework on the inviscid Burgers equation and the 3D Euler equations.

Main Results:

  • Renormalized reduced order models demonstrated stability and accuracy for long simulation times.
  • The method successfully predicted behavior before singularity formation using pre-singularity data.
  • The framework yielded a perturbatively renormalizable and stable model for 3D Euler dynamics.

Conclusions:

  • The extended time-dependent renormalization approach effectively stabilizes reduced order models.
  • The optimized memory decay parameter is problem-dependent and crucial for model performance.
  • This method enables long-term, stable simulations of complex multiscale systems, including fluid flows.