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Runge-Kutta Discontinuous Galerkin method for the Boltzmann equation

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dc.contributor.advisor Nicholas G. Hadjiconstantinou. en_US Lui, Ho Man en_US
dc.contributor.other Massachusetts Institute of Technology. Computation for Design and Optimization Program. en_US 2007-10-19T20:32:12Z 2007-10-19T20:32:12Z 2006 en_US 2006 en_US
dc.description Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2006. en_US
dc.description Includes bibliographical references (p. 85-87). en_US
dc.description.abstract In this thesis we investigate the ability of the Runge-Kutta Discontinuous Galerkin (RKDG) method to provide accurate and efficient solutions of the Boltzmann equation. Solutions of the Boltzmann equation are desirable in connection to small scale science and technology because when characteristic flow length scales become of the order of, or smaller than, the molecular mean free path, the Navier-Stokes description fails. The prevalent Boltzmann solution method is a stochastic particle simulation scheme known as Direct Simulation Monte Carlo (DSMC). Unfortunately, DSMC is not very effective in low speed flows (typical of small scale devices of interest) because of the high statistical uncertainty associated with the statistical sampling of macroscopic quantities employed by this method. This work complements the recent development of an efficient low noise method for calculating the collision integral of the Boltzmann equation, by providing a high-order discretization method for the advection operator balancing the collision integral in the Boltzmann equation. One of the most attractive features of the RKDG method is its ability to combine high-order accuracy, both in physical space and time, with the ability to capture discontinuous solutions. en_US
dc.description.abstract (cont.) The validity of this claim is thoroughly investigated in this thesis. It is shown that, for a model collisionless Boltzmann equation, high-order accuracy can be achieved for continuous solutions; whereas for discontinuous solutions, the RKDG method, with or without the application of a slope limiter such as a viscosity limiter, displays high-order accuracy away from the vicinity of the discontinuity. Given these results, we developed a RKDG solution method for the Boltzmann equation by formulating the collision integral as a source term in the advection equation. Solutions of the Boltzmann equation, in the form of mean velocity and shear stress, are obtained for a number of characteristic flow length scales and compared to DSMC solutions. With a small number of elements and a low order of approximation in physical space, the RKDG method achieves similar results to the DSMC method. When the characteristic flow length scale is small compared to the mean free path (i.e. when the effect of collisions is small), oscillations are present in the mean velocity and shear stress profiles when a coarse velocity space discretization is used. With a finer velocity space discretization, the oscillations are reduced, but the method becomes approximately five times more computationally expensive. en_US
dc.description.abstract (cont.) We show that these oscillations (due to the presence of propagating discontinuities in the distribution function) can be removed using a viscosity limiter at significantly smaller computational cost. en_US
dc.description.statementofresponsibility by Ho Man Lui. en_US
dc.format.extent 87 p. en_US
dc.language.iso eng en_US
dc.publisher Massachusetts Institute of Technology en_US
dc.rights M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission. en_US
dc.subject Computation for Design and Optimization Program. en_US
dc.title Runge-Kutta Discontinuous Galerkin method for the Boltzmann equation en_US
dc.title.alternative RKDG method for the Boltzmann equation en_US
dc.type Thesis en_US S.M. en_US
dc.contributor.department Massachusetts Institute of Technology. Computation for Design and Optimization Program. en_US
dc.identifier.oclc 85844214 en_US

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