Yana Di

Yana Di, Yuwei Fan, Zhenzhong Kou et al.

In this paper, we investigate the effect of the filter for the hyperbolic moment equations(HME) [15] of the Vlasov-Poisson equations and propose a novel quasi time-consistent filter to suppress the numerical recurrence effect. By taking properties of HME into consideration, the filter preserves a lot of physical properties of HME, including Galilean invariance and the conservation of mass, momentum and energy. We present two viewpoints, collisional viewpoint and dissipative viewpoint, to dissect the filter, and show that the filtered hyperbolic moment method can be treated as a solver of Vlasov equation. Numerical simulations of the linear Landau damping and two stream instability are tested to demonstrate the effectiveness of the filter in restraining recurrence arising from particle streaming. Both the analysis and the numerical results indicate that the filtered HME can capture the evolution of the Vlasov equation, even when phase mixing and filamentation are dominant.

Yana Di, Yuwei Fan, Ruo Li

We point out that the quantum Grad's 13-moment system [R. Yano, Physica A: Statistical Mechanics and its Applications, 416:231-241, 2014] is lack of global hyperbolicity, and even worse, the thermodynamic equilibrium is not an interior point of the hyperbolicity region of the system. To remedy this problem, by fully considering Grad's expansion, we split the expansion into the equilibrium part and the non-equilibrium part, and propose a regularization for the system with the help of the new theory developed in [Z. Cai et al., SIAM J. Appl. Math., 75(5):2001-2023, 2015, Y. Fan, J. Stat. Phys., 161(4), 2015]. This provides us a new model which is hyperbolic for all admissible thermodynamic states, and meanwhile preserves the approximate accuracy of the original system. It should be noted that this procedure is not a trivial application of the theory in [Z. Cai et al., SIAM J. Appl. Math., 75(5):2001-2023, 2015, Y. Fan, J. Stat. Phys., 161(4), 2015].

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.

Yana Di, Hehu Xie, Xiaobo Yin

We propose a numerical strategy to generate the anisotropic meshes and select the appropriate stabilized parameters simultaneously for two dimensional convection-dominated convection-diffusion equations by stabilized continuous linear finite elements. Since the discretized error in a suitable norm can be bounded by the sum of interpolation error and its variants in different norms, we replace them by some terms which contain the Hessian matrix of the true solution, convective fields, and the geometric properties such as directed edges and the area of the triangle. Based on this observation, the shape, size and equidistribution requirements are used to derive the corresponding metric tensor and the stabilized parameters. It is easily found from our derivation that the optimal stabilized parameter is coupled with the optimal metric tensor on each element. Some numerical results are also provided to validate the stability and efficiency of the proposed numerical strategy.