# Browsing by Subject "boltzmann equation"

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Item Conservative Deterministic Spectral Boltzmann Solver Near The Grazing Collisions Limit(2012-07) Haack, J. R.; Gamba, I. M.; Haack, Jeffrey R.; Gamba, Irene M.Show more 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.Show more Item A Conservative Discontinuous Galerkin Scheme With O(N-2) Operations In Computing Boltzmann Collision Weight Matrix(2014-07) Gamba, I. M.; Zhang, C. L.; Gamba, Irene M.; Zhang, ChenglongShow more 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.Show more Item A Fast Conservative Spectral Solver For The Nonlinear Boltzmann Collision Operator(2014-07) Gamba, I. M.; Haack, J. R.; Hu, J. W.; Gamba, Irene M.; Haack, Jeffrey R.Show more 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.Show more Item High Performance Computing With A Conservative Spectral Boltzmann Solver(2012-07) Haack, J. R.; Gamba, I. M.; Haack, Jeffrey R.; Gamba, Irene M.Show more 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.Show more Item Improvement Of A Discrete Velocity Boltzmann Equation Solver With Arbitrary Post-Collision Velocities(2009-12) Morris, A. B.; Varghese, P. L.; Goldstein, D. B.; Morris, A. B.; Varghese, P. L.; Goldstein, D. B.Show more We present a discrete velocity scheme which solves the Boltzmann equation and show numerical results for homogeneous relaxation problems. Although direct simulation of the Boltzmann equation can be efficient for transient problems, computational costs have restricted its use. A velocity interpolation algorithm enables us to select post-collision velocity pairs not restricted to those that lie precisely on the grid. This allows efficient evaluation of the replenishing part of the collision integral with reasonable accuracy. In previous work [1] the scheme was demonstrated with the depleting terms evaluated exactly, which made the method of O(N(2)) where N is the number of grid points in the velocity space. In order to reduce the computational cost, we have developed an acceptance-rejection scheme to enable more efficient evaluation of the depleting term. We show that the total collision integral can be evaluated accurately in combination with the mapping scheme for the replenishing term. To improve our scheme, we study the error and computational time associated with the number of depleting and replenishing points. We predict the correct relaxation rate for the Bobylev-Krook-Wu distribution and obtain exact conservation of mass, momentum, and energy. Comparisons between computed and reference solutions are shown as well, demonstrating the correct relaxation rate and dependence of error on parameters in the computational scheme.Show more Item Molecular dynamics simulation of collision operator eigenvalues(2009-03) Gust, Erich D.; Reichl, L. E.; Gust, Erich D.; Reichl, L. E.Show more We simulate numerically the time evolution of 1000 interacting hard spheres in a finite box with periodic boundary conditions, and repeat the simulations many times for a selected random distribution of initial conditions. We use the resulting data to compute, directly, the smallest nonzero eigenvalue of the collision operator for this gas. We also give exact expressions for the transport coefficients and compare them to approximate expressions commonly seen in the literature.Show more Item A Novel Discrete Velocity Method For Solving The Boltzmann Equation Including Internal Energy And Non-Uniform Grids In Velocity Space(2012) Clarke, P.; Varghese, P.; Goldstein, D.; Morris, A.; Bauman, P.; Hegermiller, D.; Clarke, P.; Varghese, P.; Goldstein, D; Morris, A.; Bauman, P.; Hergermiller, D.Show more The discrete velocity method has been extended to include inelastic collisions with rotational-translational energy exchange. A single value of rotational energy per unit mass is assigned to every velocity in the velocity domain and inelastic collisions are modeled using the Larsen-Borgnakke method. The discrete velocity version of energy exchange is used to simulate both a homogeneous relaxation of a distribution with non-equilibrium rotational and translational temperatures and a 1D shock with rotational energy modes. The method has also been modified to allow for non-uniform grids in velocity space. Non-uniform grids permit computational effort to be focused on specific areas of interest within the velocity distribution function. The Bobylev-Krook-Wu solution to the Boltzmann equation (the only analytic solution known) is used to compare a non-uniform grid with a uniform grid.Show more Item Variance Reduction For A Discrete Velocity Gas(2011-07) Morris, A. B.; Varghese, P. L.; Goldstein, D. B.; Morris, A. B.; Varghese, P. L.; Goldstein, D. B.Show more We extend a variance reduction technique developed by Baker and Hadjiconstantinou [1] to a discrete velocity gas. In our previous work, the collision integral was evaluated by importance sampling of collision partners [2]. Significant computational effort may be wasted by evaluating the collision integral in regions where the flow is in equilibrium. In the current approach, substantial computational savings are obtained by only solving for the deviations from equilibrium. In the near continuum regime, the deviations from equilibrium are small and low noise evaluation of the collision integral can be achieved with very coarse statistical sampling. Spatially homogenous relaxation of the Bobylev-Krook-Wu distribution [3,4], was used as a test case to verify that the method predicts the correct evolution of a highly non-equilibrium distribution to equilibrium. When variance reduction is not used, the noise causes the entropy to undershoot, but the method with variance reduction matches the analytic curve for the same number of collisions. We then extend the work to travelling shock waves and compare the accuracy and computational savings of the variance reduction method to DSMC over Mach numbers ranging from 1.2 to 10.Show more