Zhicheng Hu

Zhicheng Hu, Guanghan Li, Chunwu Wang et al.

A high-order Newton multigrid method is proposed for steady-state shallow water flows in open channels with regular and irregular geometries. The method integrates a finite volume discretization with third-order weighted essentially non-oscillatory (WENO) reconstruction and a Newton multigrid framework with an efficient approximation of the Jacobian matrix for solving the resulting discrete system. In high-order schemes, the computational cost of Jacobian construction becomes dominant due to the wide stencil. Meanwhile, only a small fraction of the non-zero Jacobian entries exhibit large magnitudes. Based on this observation, a simplified Jacobian approximation is introduced using reduced stencils, in which selected off-stencil contributions are neglected, thereby achieving a substantial reduction in computational cost. The proposed approach is verified numerically to show significant efficiency improvement while maintaining comparable convergence behavior to that obtained with the full Jacobian approach. To further enhance performance, a geometric multigrid method incorporating a successive over-relaxation iteration as the smoother is applied to solve the linear systems arising in each Newton step. A variety of numerical experiments, including a one-dimensional smooth subcritical flow, flows over a hump, and a two-dimensional hydraulic jump over a wedge, are carried out to illustrate the third-order accuracy, efficiency, and robustness of the proposed method.

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.

Zhicheng Hu, Guanghui Hu

In [Z. Hu, R. Li, and Z. Qiao. Acceleration for microflow simulations of high-order moment models by using lower-order model correction. J. Comput. Phys., 327:225-244, 2016], it has been successfully demonstrated that using lower-order moment model correction is a promising idea to accelerate the steady-state computation of high-order moment models of the Boltzmann equation. To develop the existing solver, the following aspects are studied in this paper. First, the finite volume method with linear reconstruction is employed for high-resolution spatial discretization so that the degrees of freedom in spatial space could be reduced remarkably without loss of accuracy. Second, by introducing an appropriate parameter $τ$ in the correction step, it is found that the performance of the solver can be improved significantly, i.e., more levels would be involved in the solver, which further accelerates the convergence of the method. Third, Heun's method is employed as the smoother in each level to enhance the robustness of the solver. Numerical experiments in microflows are carried out to demonstrate the efficiency and to investigate the behavior of the new solver. In addition, several order reduction strategies for the choice of the order sequence of the solver are tested, and the strategy $m_{l-1} = \lceil m_{l} / 2 \rceil$ is found to be most efficient.

Zhicheng Hu, Keiwei Liang

This paper studies the numerical solution of traveling singular sources problems. In such problems, a big challenge is the sources move with different speeds, which are described by some ordinary differential equations. A predictor-corrector algorithm is presented to simulate the position of singular sources. Then a moving mesh method in conjunction with domain decomposition is derived for the underlying PDE. According to the positions of the sources, the whole domain is splitted into several subdomains, where moving mesh equations are solved respectively. On the resulting mesh, the computation of jump $[\dot{u}]$ is avoided and the discretization of the underlying PDE is reduced into only two cases. In addition, the new method has a desired second-order of the spatial convergence. Numerical examples are presented to illustrate the convergence rates and the efficiency of the method. Blow-up phenomenon is also investigated for various motions of the sources.

Yaohua Tang, Zhicheng Hu, Kun Cheng et al.

The increasing context window size in large language models (LLMs) has improved their ability to handle complex, long-text tasks. However, as the conversation rounds continue, it is required to store a large amount of KV cache in GPU memory, which significantly affects the efficiency and even availability of the model serving systems. This paper analyzes dialogue data from real users on the granularity of round and discovers that the LLM inference manifests a watershed layer, after which the distribution of round-level attention shows notable similarity. Based on this, we propose Round Attention - a novel round-level attention mechanism that selectively processes the KV cache of top-k relevant rounds, where k is dynamically determined through the attention matrix in the watershed layer. Theoretical analysis demonstrates that our method reduces memory usage by 54\% to 82\%, while experimental results confirm that loading sparse critical-round KV cache maintains answer accuracy without performance degradation.

Zhicheng Hu, Ruo Li, Zhonghua Qiao

We study the acceleration of steady-state computation for microflow, which is modeled by the high-order moment models derived recently from the steady-state Boltzmann equation with BGK-type collision term. By using the lower-order model correction, a novel nonlinear multi-level moment solver is developed. Numerical examples verify that the resulting solver improves the convergence significantly thus is able to accelerate the steady-state computation greatly. The behavior of the solver is also numerically investigated. It is shown that the convergence rate increases, indicating the solver would be more efficient, as the total levels increases. Three order reduction strategies of the solver are considered. Numerical results show that the most efficient order reduction strategy would be $m_{l-1} = \lceil m_{l} / 2 \rceil$.