IMI Interdisciplinary Mathematics InstituteCollege of Arts and Sciences

Mathematical Theories of Existence for 3D Conservation Laws and Stochastic Convergence

  • April 26, 2012
  • 3:30 p.m.
  • LeConte 405

Abstract

The Euler equations are fundamental as a model for simulations of Navier-Stokes turbulence in the high Reynolds number regime. Practical large eddy simulations (LES) for this regime resolve some but not all of the turbulent length scales. Existence theories, of necessity incomplete for the Euler equations, suggest that solutions will not be worse than Young measures.

Here we take this possibility seriously, and examine how this point of view will impact convergence for numerical simulations. The essential point of a Young measure is that the solution is a probability, i.e. described by a probability density function (PDF), dependent on space and time. We interpret the numerical solutions as space time dependent PDFs also, through sampling in a neighborhood of the space time point. Thus the LES interpretation requires some coarsening of the spatial resolution, in order to achieve enhanced statistical resolution of the fluctuating quantities defined by the turbulent flow.

Convergence of a PDF is assessed through an L_1 norm applied to the cumulative distribution function, which is the indefinite integral of the PDF.

Numerical methods require verification (mathematical correctness of the simulation), validation (agreement with relevant physical experiments) and uncertainty quantification (overall assessment of total possible errors of all types), known as V&V+UQ. For conventional simulation ideas, V&V+UQ is a large and ongoing effort. Its need for a novel simulation concept is still more important. Here we present partial results in the direction of V&V+UQ for LES simulations in a PDF/Young measure framework. Validation, as it involves laboratory experiments or observations, requires in practice involvement with serious application questions, in order to justify the large expenses involved. We present V&V+UQ results that span a range of application areas.

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