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

Numerical Simulation of Microflows with Moment Method

This paper was presented at the 4th Micro and Nano Flows Conference (MNF2014), which was held at University College, London, UK. The conference was organised by Brunel University and supported by the Italian Union of Thermofluiddynamics, IPEM, the Process Intensification Network, the Institution of Mechanical Engineers, the Heat Transfer Society, HEXAG - the Heat Exchange Action Group, and the Energy Institute, ASME Press, LCN London Centre for Nanotechnology, UCL University College London, UCL Engineering, the International NanoScience Community, www.nanopaprika.eu.A series of hyperbolic moment equations is derived for the Boltzmann equation with ES-BGK collision term. These systems can be obtained through a slight modification in the deduction of Grad’s moment equations, and such a method is suitable for deriving systems with moments up to any order. The systems are equipped with proper wall boundary conditions so that the number of equations in the boundary conditions is consistent with the hyperbolic structure of the moment system. Our numerical scheme for solving the hyperbolic moment systems is of second order, and a special mapping method is introduced so that the numerical efficiency is highly enhanced. Our numerical results are validated by comparison with the DSMC results. Through the numerical solutions of systems with increasing number of moments, the convergence of the moment method is clearly observed

Quantum Hydrodynamic Model by Moment Closure of Wigner Equation

In this paper, we derive the quantum hydrodynamics models based on the moment closure of the Wigner equation. The moment expansion adopted is of the Grad type firstly proposed in \cite{Grad}. The Grad's moment method was originally developed for the Boltzmann equation. In \cite{Fan_new}, a regularization method for the Grad's moment system of the Boltzmann equation was proposed to achieve the globally hyperbolicity so that the local well-posedness of the moment system is attained. With the moment expansion of the Wigner function, the drift term in the Wigner equation has exactly the same moment representation as in the Boltzmann equation, thus the regularization in \cite{Fan_new} applies. The moment expansion of the nonlocal Wigner potential term in the Wigner equation is turned to be a linear source term, which can only induce very mild growth of the solution. As the result, the local well-posedness of the regularized moment system for the Wigner equation remains as for the Boltzmann equation

Projective integration for nonlinear BGK kinetic equations

Proceedings FVCA 8International audienceWe present a high-order, fully explicit, asymptotic-preserving projective integration scheme for the nonlinear BGK equation. The method first takes a few small (inner) steps with a simple, explicit method (such as direct forward Euler) to damp out the stiff components of the solution. Then, the time derivative is estimated and used in an (outer) Runge-Kutta method of arbitrary order. Based on the spectrum of the linearized BGK operator, we deduce that, with an appropriate choice of inner step size, the time step restriction on the outer time step as well as the number of inner time steps is independent of the stiffness of the BGK source term. We illustrate the method with numerical results in one and two spatial dimensions

A Nonlinear Multigrid Steady-State Solver for Microflow

We develop a nonlinear multigrid method to solve the steady state of microflow, which is modeled by the high order moment system derived recently for the steady-state Boltzmann equation with ES-BGK collision term. The solver adopts a symmetric Gauss-Seidel iterative scheme nested by a local Newton iteration on grid cell level as its smoother. Numerical examples show that the solver is insensitive to the parameters in the implementation thus is quite robust. It is demonstrated that expected efficiency improvement is achieved by the proposed method in comparison with the direct time-stepping scheme

A Framework on Moment Model Reduction for Kinetic Equation

By a further investigation on the structure of the coefficient matrix of the globally hyperbolic regularized moment equations for Boltzmann equation in [Z. Cai, Y. Fan and R. Li, Comm. Math. Sci., 11 (2013), pp. 547-571], we propose a uniform framework to carry out model reduction to general kinetic equations, to achieve certain moment system. With this framework, the underlying reason why the globally hyperbolic regularization in [Z. Cai, Y. Fan and R. Li, Comm. Math. Sci., 11 (2013), pp. 547-571] works is revealed. The even fascinating point is, with only routine calculation, existing models are represented and brand new models are discovered. Even if the study is restricted in the scope of the classical Grad's 13-moment system, new model with global hyperbolicity can be deduced.Comment: 22 page

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

NRxx Simulation of Microflows with Shakhov Model

In this paper, we propose a method to simulate the microflows with Shakhov model using the NRxx method developed in [4, 5, 6]. The equation under consideration is the Boltzmann equation with force terms and the Shakhov model is adopted to achieve the correct Prandtl number. As the focus of this paper, we derive a uniform framework for different order moment systems on the wall boundary conditions, which is a major difficulty in the moment methods. Numerical examples for both steady and unsteady problems are presented to show the convergence in the number of moments.Comment: 31 pages, 10 figure