3:00 pm Friday, November 20, 2009
Math/ICES Center of Numerical Analysis Seminar: Goal-oriented adaptivity for a 1D prototype of the Boltzmann equation by Harald van Brummelen (Eindhoven University of Technology ) in ACES, 6.304,
With the perpetual trend towards smaller and smaller scales in science and engineering, fluid-flow problems in the transitional molecular/continuum regime have rapidly gained prominence over the past years. Accurate numerical simulation of flow problems in the transitional regime poses a fundamental challenge, on account of the large difference between the molecular free path and a typical length scale of observation. To bridge the gap between the molecular length scale and the continuum length scale in numerical simulations, many heuristic approaches have recently appeared in the literature to couple molecular-dynamics models to continuum models, such as the Navier-Stokes equations. The appropriateness of such a direct connection between a molecular model and a continuum model is arguable, however, because the range of validity of the models is highly disparate. A suitable model for transitional molecular/continuum flows is provided by the Boltzmann equation. In the Boltzmann equation, the flow is characterized by a one-particle probability-density function, which measures the probability that a molecule resides in a certain subset of the position/momentum space. The Boltzmann equation itself is an integro-differential equation that governs the evolution of the one-particle probability-density. The Boltzmann equation is in principle valid downto the molecular scale, while on the other hand it encapsulates all conventional continuum models, such as the compressible and incompressible Navier-Stokes equations, in the sense that with appropriate scalings of the macroscopic length and time scales, limit solutions of the Boltzmann equation correspond to solutions of these continuum equations. Essentially, the Boltzmann equation is connected to the continuum equations by the fact that solutions of the Boltzmann equation converge to a particular class of solutions, the so-called Maxwell-Boltzmann equilibrium distributions. Direct numerical simulation of the Boltzmann equation is prohibited by its high-dimensional setting: for a problem in d spatial dimensions, the corresponding position/momentum domain is 2d dimensional. However, it is anticipated that in many cases the computational complexity can be significantly reduced by means of adaptive low (d) dimensional approximations based on Boltzmann moment closures. In many applications, interest is in fact restricted to one particular goal functional. For instance, in micro-scale heat-transfer problems, it is ultimately only the heat flux across a certain part of the boundary that is of interest. This class of problems provides fertile ground for goal-oriented adaptive-refinement strategies. An essential impediment in the development of goal-oriented adaptive-refinement techniques for the Boltzmann equation, is the fact that the convergence-to-equilibrium property which forms the basis of the hierarchical modeling process only occurs for d=2,3. Hence, to test our ideas, we would have to consider problems in 4 or 6 dimensions. To bypass this complication, we have developed a 1D prototype of the Boltzmann equation, which exhibits all the characteristic features of the Boltzmann equation, including the weak-convergence-to-equilibrium property. The underlying molecular model is based on random collisions, which conserve energy but not momentum. In the presentation, I will give an overview of transitional molecular/continuum flows, from the perspective of hierarchical modeling and model adaptivity. I will then elaborate the 1D prototype of the Boltzmann equation that we have developed, and derive its characteristic properties, such as an entropy inequality. Finally, I will present numerical result obtained by a discontinuous Galerkin finite-element discretization of the prototype, including recent results for goal-oriented error estimation. Submitted by
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