High Performance Computing With A Conservative Spectral Boltzmann Solver

We present new results building on the conservative deterministic spectral method for the space inhomogeneous Boltzmann equation developed by Gamba and Tharkabhushaman. This approach is a two-step process that acts on the weak form of the Boltzmann equation, and uses the machinery of the Fourier transform to reformulate the collisional integral into a weighted convolution in Fourier space. A constrained optimization problem is solved to preserve the mass, momentum, and energy of the resulting distribution. We extend this method to second order accuracy in space and time, and explore how to leverage the structure of the collisional formulation for high performance computing environments. The locality in space of the collisional term provides a straightforward memory decomposition, and we perform some initial scaling tests on high performance computing resources. We also use the improved computational power of this method to investigate a boundary-layer generated shock problem that cannot be described by classical hydrodynamics.Mathematic

A Conservative Discontinuous Galerkin Scheme With O(N-2) Operations In Computing Boltzmann Collision Weight Matrix

In the present work, we propose a deterministic numerical solver for the homogeneous Boltzmann equation based on Discontinuous Galerkin (DG) methods. The weak form of the collision operator is approximated by a quadratic form in linear algebra setting. We employ the property of >shifting symmetry> in the weight matrix to reduce the computing complexity from theoretical O(N-3) down to O(N-2), with N the total number of freedom for d-dimensional velocity space. In addition, the sparsity is also explored to further reduce the storage complexity. To apply lower order polynomials and resolve loss of conserved quantities, we invoke the conservation routine at every time step to enforce the conservation of desired moments (mass, momentum and/or energy), with only linear complexity. Due to the locality of the DG schemes, the whole computing process is well parallelized using hybrid OpetiMP and MPI. The current work only considers integrable angular cross-sections under elastic and/or inelastic interaction laws. Numerical results on 2-D and 3-D problems are shown.Mathematic

Mean Field Approach to the Giant Wormhole Problem

We introduce a gaussian probability density for the space-time distribution of wormholes, thus taking effectively into account wormhole interaction. Using a mean-field approximation for the free energy, we show that giant wormholes are probabilistically suppressed in a homogenous isotropic ``large'' universe.Comment: 10 pages, Late

Conservative Deterministic Spectral Boltzmann Solver Near The Grazing Collisions Limit

We present new results building on the conservative deterministic spectral method for the space homogeneous Boltzmann equation developed by Gamba and Tharkabhushaman. This approach is a two-step process that acts on the weak form of the Boltzmann equation, and uses the machinery of the Fourier transform to reformulate the collisional integral into a weighted convolution in Fourier space. A constrained optimization problem is solved to preserve the mass, momentum, and energy of the resulting distribution. Within this framework we have extended the formulation to the case of more general case of collision operators with anisotropic scattering mechanisms, which requires a new formulation of the convolution weights. We also derive the grazing collisions limit for the method, and show that it is consistent with the Fokker-Planck-Landau equations as the grazing collisions parameter goes to zero.Mathematic

A Fast Conservative Spectral Solver For The Nonlinear Boltzmann Collision Operator

We present a conservative spectral method for the fully nonlinear Boltzmann collision operator based on the weighted convolution structure in Fourier space developed by Gamba and Tharkabhushnanam.. This method can simulate a broad class of collisions, including both elastic and inelastic collisions as well as angularly dependent cross sections in which grazing collisions play a major role. The extension presented in this paper consists of factorizing the convolution weight on quadrature points by exploiting the symmetric nature of the particle interaction law, which reduces the computational cost and memory requirements of the method to O(M(2)N(4)logN) from the O(N-6) complexity of the original spectral method, where N is the number of velocity grid points in each velocity dimension and M is the number of quadrature points in the factorization, which can be taken to be much smaller than N. We present preliminary numerical results.Mathematic

Masculinity and feminity measurement in physical education teachers

Esta investigación tuvo por objetivo analizar las mediciones de masculinidad, feminidad, machismo y sumisión, características asociadas a la personalidad, de un grupo de docentes de Educación Física. Participaron en el estudio 53 docentes de nivel básico que laboran en un programa implementado por una institución gubernamental en la Ciudad de México. El muestreo fue de tipo no probabilístico. Se empleó como instrumento el Inventario de Masculinidad y Feminidad (IMAFE), instrumento confiable y válido en México, sujeto a prueba en otros países, en él se incluyen aspectos de los papeles de género tradicionales: machismo y sumisión. El análisis de los datos se efectúo mediante la prueba “t-Student” y el análisis de varianza de una clasificación, así como la comparación de medias de los resultados arrojados. Se concluye que no hay diferencias estadísticamente significativas en las cuatro escalas propuestas por el IMAFE y las variables de trabajo, sexo, edad y estado civil, en el grupo de docentes de Educación Física, en lo que respecta a las características asociadas a la personalidadThis research aimed to analyze the measurements of masculinity, femininity, machismo and submission features associated with the personality characteristics of a group of physical education teachers. Participated in the study53 basic level teachers working in a program implemented by a government institution in Mexico City. The sampling was not probabilistic type. As a tool for data collection was used the Inventory of Masculinity and femininity (IMAFE), reliable and valid instrument in Mexico, subject to testing in other countries, there aspects of traditional gender roles: machismo and submission. Data analysis undertaken using the “t-student” test and analysis of variance classification and comparison of the results obtained. It is concluded that no statistically significant differences in the four scales proposed by IMAFE and work variables sex, age and marital status in the group of physical education teachers in regard to the characteristics associated with personalit

The Lyapunov Spectrum of a Continuous Product of Random Matrices

We expose a functional integration method for the averaging of continuous products $\hat{P}_t$ of $N\times N$ random matrices. As an application, we compute exactly the statistics of the Lyapunov spectrum of $\hat{P}_t$. This problem is relevant to the study of the statistical properties of various disordered physical systems, and specifically to the computation of the multipoint correlators of a passive scalar advected by a random velocity field. Apart from these applications, our method provides a general setting for computing statistical properties of linear evolutionary systems subjected to a white noise force field.Comment: Latex, 9 page