Zhenning Cai

Zhicheng Hu, Zhenning Cai, Yanli Wang

We propose a Hermite spectral method for the spatially inhomogeneous Boltzmann equation. For the inverse-power-law model, we generalize an approximate quadratic collision operator defined in the normalized and dimensionless setting to an operator for arbitrary distribution functions. An efficient algorithm with a fast transform is introduced to discretize this new collision operator. The method is tested for one-dimensional benchmark microflow problems.

Zhenning Cai, Yanli Wang

We study the unphysical recurrence phenomenon arising in the numerical simulation of the transport equations using Hermite-spectral method. From a mathematical point of view, the suppression of this numerical artifact with filters is theoretically analyzed for two types of transport equations. It is rigorously proven that all the non-constant modes are damped exponentially by the filters in both models, and formally shown that the filter does not affect the damping rate of the electric energy in the linear Landau damping problem. Numerical tests are performed to show the effect of the filters.

Zhenning Cai, Yuwei Fan, Lexing Ying

We propose an entropic Fourier method for the numerical discretization of the Boltzmann collision operator. The method, which is obtained by modifying a Fourier Galerkin method to match the form of the discrete velocity method, can be viewed both as a discrete velocity method and as a Fourier method. As a discrete velocity method, it preserves the positivity of the solution and satisfies a discrete version of the H-theorem. As a Fourier method, it allows one to readily apply the FFT-based fast algorithms. A second-order convergence rate is validated by numerical experiments

Zhenning Cai, Ruo Li, Yanli Wang

We first investigate the structure of the systems derived from the gPC based stochastic Galerkin method for the nonlinear hyperbolic systems with random inputs. This method adopts a generalized Polynomial Chaos (gPC) approximations in the stochastic Galerkin framework, but such approximations to the nonlinear hyperbolic systems do not necessarily yield hyperbolic systems \cite{Lucor2013}. Thus based on the work in \cite{framework}, we propose a framework to carry out the model reduction for the general nonlinear hyperbolic system to derive a final global system. Within this framework, the nonlinear hyperbolic system in one space dimension and the symmetric hyperbolic system in multiple space dimensions are reduced into a symmetric hyperbolic system based on the stochastic Galerkin method. We note that the basis functions in the expansion are not restricted to the random-dependent polynomials as that in gPC method and there is no restriction on the dimensions of the random variables neither.

Zhenning Cai, Jianfeng Lu

We propose a surface hopping Gaussian beam method to numerically solve a class of high frequency linear transport systems in high spatial dimensions, based on asymptotic analysis. The stochastic surface hopping is combined with Gaussian beam method to deal with the multiple characteristic directions of the transport system in high dimensions. The Monte Carlo nature of the proposed algorithm makes it easy for parallel implementations. We validate the performance of the algorithms for applications on the quantum-classical Liouville equations.

Zhenning Cai, Yana Di, Xiaoyu Dong

We study the mechanism of the gauge cooling technique to stabilize the complex Langevin method in the one-dimensional periodic setting. In this case, we find the exact solutions for the gauge transform which minimizes the Frobenius norm of link variables. Thereby, we derive the underlying stochastic differential equations by continuing the numerical method with gauge cooling, and thus provide a number of insights on the effects of gauge cooling. A specific case study is carried out for the Polyakov loop model in $SU(2)$ theory, in which we show that the gauge cooling may help form a localized distribution to guarantee there is no excursion too far away from the real axis.