arXiv subjects

Chunyan Niu

Publications and source records attributed to Chunyan Niu.

3 recordsLinked to original sources

arXiv2025

On the rotating nonlinear Klein-Gordon equation with multiscale effects: structure-preserving methods and applications to vortex dynamics

We study numerical methods for the rotating nonlinear Klein-Gordon (RKG) equation, a fundamental model in relativistic quantum physics, which exhibits highly oscillatory multiscale behavior due to the presence of a small parameter {\epsilon}. The RKG equation models rotating galaxies under the Minkowski metric and also provides an effective description of phenomena such as cosmic superfluids. This work focuses on the development and rigorous analysis of structure-preserving Galerkin finite element methods (FEMs) for the RKG equation. A central challenge is that the rotational terms prevent traditional nonconforming FEMs from simultaneously conserving energy and charge. By employing a conservation-adjusting technique, we construct a consistent structure-preserving algorithm applicable to both conforming and nonconforming FEMs. Moreover, we provide a comprehensive convergence analysis, establishing unconditional optimal and high-order accuracy error estimates. These theoretical results are further validated through extensive numerical experiments, which demonstrate the accuracy, efficiency, and robustness of the structure-preserving schemes. Finally, simulations of vortex dynamics, ranging from the relativistic to the nonrelativistic regimes, are presented to illustrate vortex creation, relativistic effects on bound states, and interactions of vortex pairs.

math.NA↗

arXiv2020

Rejuvenating AMLI-Cycle: From Chebyshev Polynomials to Momentum Acceleration

In this paper, we investigate the AMLI-cycle method and make two contributions. First, we revisit the AMLI-cycle using the Chebyshev polynomials and establish a theory for its uniform convergence, assuming the two-grid method converges uniformly. This removes the need for estimating extreme eigenvalues at all coarse levels. Only an estimation of the two-grid convergence rate is needed, which could be done on the second coarsest level, simplifying implementation and reducing computational costs for large-scale problems. Second, we introduce a momentum-accelerated AMLI-cycle using polynomials from momentum accelerations. This novel approach ensures a uniform condition number without requiring extreme eigenvalue or two-grid convergence rate estimations, making its implementation as straightforward as standard multigrid methods. We prove that it is asymptotically as good as the AMLI-cycle using the Chebyshev polynomials when the quadratic momentum-accelerated polynomials is used. Numerical experiments confirm the robustness and efficiency of the momentum-accelerated AMLI-cycle across various problems, demonstrating performance comparable to the Chebyshev-based AMLI-cycle. These findings validate the theoretical advantages and practical efficacy of the momentum-accelerated AMLI-cycle.

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arXiv2020

A Stabilized Hybrid Mixed Finite Element Method for Poroelasticity

In this work, we consider a hybrid mixed finite element method for Biot's model. The hybrid P1-RT0-P0 discretization of the displacement-pressure-Darcy's velocity system of Biot's model presented in \cite{C. Niu} is not uniformly stable with respect to the physical parameters, resulting in some issues in numerical simulations. To alleviate such problems, following \cite{V. Girault}, we stabilize the hybrid scheme with face bubble functions and show the well-posedness with respect to physical and discretization parameters, which provide optimal error estimates of the stabilized method. We introduce a perturbation of the bilinear form of the displacement which allows for the elimination of the bubble functions. Together with eliminating Darcy's velocity by hybridization, we obtain an eliminated system whose size is the same as the classical P1-RT0-P0 discretization. Based on the well-posedness of the eliminated system, we design block preconditioners that are parameter-robust. Numerical experiments are presented to confirm the theoretical results of the stabilized scheme as well as the block preconditioners.

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