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UBC Theses and Dissertations

Numerical method for solving the Boltzmann equation using cubic B-splines Khurana , Saheba


A numerical method for solving a one-dimensional linear Boltzmann equation is developed using cubic B-splines. Collision kernels are derived for smooth and rough hard spheres. A complete velocity kernel for spherical particles is shown that is reduced to the smooth, rigid sphere case. Similarly, a collision kernel for the rough hard sphere is derived that depends upon velocity and angular velocity. The exact expression for the rough sphere collision kernel is reduced to an approximate expression that averages over the rotational degrees of freedom in the system. The rough sphere collision kernel tends to the smooth sphere collision kernel in the limit when translational-rotational energy exchange is attenuated. Comparisons between the smooth sphere and approximate rough sphere kernel are made. Four different representations for the distribution function are presented. The eigenvalues and eigenfunctions of the collision matrix are obtained for various mass ratios and compared with known values. The distribution functions, first and second moments are also evaluated for different mass and temperature ratios. This is done to validate the numerical method and it is shown that this method is accurate and well-behaved. In addition to smooth and rough hard spheres, the collision kernels are used to model the Maxwell molecule. Here, a variety of mass ratios and initial energies are used to test the capability of the numerical method. Massive tracers are set to high initial energies, representing kinetic energy loss experiments with biomolecules in experimental mass spectrometry. The validity of the Fokker-Planck expression for the Rayleigh gas is also tested. Drag coefficients are calculated and compared to analytic expressions. It is shown that these values are well predicted for massive tracers but show a more complex behaviour for small mass ratios especially at higher energies. The numerical method produced well converged values, even when the tracers were initialized far from equilibrium. In general this numerical method produces sparse matrices and can be easily generalized to higher dimensions that can be cast into efficient parallel algorithms. Future work has been planned that involves the use of this numerical method for a multi-dimension linear Boltzmann equation.

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