62 research outputs found
Numerical Regularized Moment Method of Arbitrary Order for Boltzmann-BGK Equation
We introduce a numerical method for solving Grad's moment equations or
regularized moment equations for arbitrary order of moments. In our algorithm,
we do not need explicitly the moment equations. As an instead, we directly
start from the Boltzmann equation and perform Grad's moment method \cite{Grad}
and the regularization technique \cite{Struchtrup2003} numerically. We define a
conservative projection operator and propose a fast implementation which makes
it convenient to add up two distributions and provides more efficient flux
calculations compared with the classic method using explicit expressions of
flux functions. For the collision term, the BGK model is adopted so that the
production step can be done trivially based on the Hermite expansion. Extensive
numerical examples for one- and two-dimensional problems are presented.
Convergence in moments can be validated by the numerical results for different
number of moments.Comment: 33 pages, 13 figure
Solving Vlasov Equations Using NRxx Method
In this paper, we propose a moment method to numerically solve the Vlasov
equations using the framework of the NRxx method developed in [6, 8, 7] for the
Boltzmann equation. Due to the same convection term of the Boltzmann equation
and the Vlasov equation, it is very convenient to use the moment expansion in
the NRxx method to approximate the distribution function in the Vlasov
equations. The moment closure recently presented in [5] is applied to achieve
the globally hyperbolicity so that the local well-posedness of the moment
system is attained. This makes our simulations using high order moment
expansion accessible in the case of the distribution far away from the
equilibrium which appears very often in the solution of the Vlasov equations.
With the moment expansion of the distribution function, the acceleration in the
velocity space results in an ordinary differential system of the macroscopic
velocity, thus is easy to be handled. The numerical method we developed can
keep both the mass and the momentum conserved. We carry out the simulations of
both the Vlasov-Poisson equations and the Vlasov-Poisson-BGK equations to study
the linear Landau damping. The numerical convergence is exhibited in terms of
the moment number and the spatial grid size, respectively. The variation of
discretized energy as well as the dependence of the recurrence time on moment
order is investigated. The linear Landau damping is well captured for different
wave numbers and collision frequencies. We find that the Landau damping rate
linearly and monotonically converges in the spatial grid size. The results are
in perfect agreement with the theoretic data in the collisionless case
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