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High order regularization of Dirac-delta sources in two space dimensions.

Vries, W. de (2015) High order regularization of Dirac-delta sources in two space dimensions.

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Abstract:In this report the regularization of singular sources appearing in hyperbolic conservation laws is studied. Dirac-delta sources represent small particles which appear as source-terms in for example Particle-laden flow. The Dirac-delta sources are regularized with a high order accurate polynomial based regularization technique. Instead of regularizing a single isolated particle, the aim is to employ the regularization technique for a cloud of particles distributed on a nonuniform grid. By assuming that the resulting force exerted by a cloud of particles can be represented by a continuous function, the influence of the source term can be approximated by the convolution of the continuous function and the regularized Dirac-delta function. It is shown that in one dimension the distribution of the singular sources, represented by the convolution of the continuous function and Dirac-delta function, can be approximated with high accuracy using the regularization technique. The method is extended to two dimensions and high order approximations of the source-term are obtained as well. The resulting approximation of the singular sources are interpolated using a polynomial interpolation in order to regain a continuous distribution representing the force exerted by the cloud of particles. Finally the high-order approximation of the singular sources are used to solve the two-dimensional singular advection equation. A Chebyshev spectral collocation method and a third order Total Variation Diminishing Runge-Kutta method are employed to approximate the spatial and temporal derivative. For a single isolated moving source and a stationary cloud of particles high-order accuracy is obtained outside the regularized zones. Inside the regularization zones the results are less satisfying and deserve more attention. The validation of these results and additional experiments to study the accuracy of the results, remain as future work.
Item Type:Internship Report (Master)
San Diego State University, United States of America
Faculty:ET: Engineering Technology
Subject:52 mechanical engineering
Programme:Mechanical Engineering MSc (60439)
Keywords:Hyperbolic conservation laws, two-dimensional advection equation, regularization, Dirac-delta, singular sources, Chebyshev spectral collocation
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