Ziqing Xie

Xiaodan Zhao, Li-Lian Wang, Ziqing Xie

This paper provides a rigorous and delicate analysis for exponential decay of Jacobi polynomial expansions of analytic functions associated with the Bernstein ellipse. Using an argument that can recover the best estimate for the Chebyshev expansion, we derive various new and sharp bounds of the expansion coefficients, which are featured with explicit dependence of all related parameters and valid for degree $n\ge 1$. We demonstrate the sharpness of the estimates by comparing with existing ones, in particular, the very recent results in [38, SIAM J. Numer. Anal., 2012]. We also extend this argument to estimate the Gegenbauer-Gauss quadrature remainder of analytic functions, which leads to some new tight bounds for quadrature errors.

Ziqing Xie, Li-Lian Wang, Xiaodan Zhao

This paper is devoted to a rigorous analysis of exponential convergence of polynomial interpolation and spectral differentiation based on the Gegenbauer-Gauss and Gegenbauer-Gauss-Lobatto points, when the underlying function is analytic on and within an ellipse. Sharp error estimates in the maximum norm are derived.

Bo Wang, Li-Lian Wang, Ziqing Xie

We present in this paper a spectrally accurate numerical method for computing the spherical/vector spherical harmonic expansion of a function/vector field with given (elemental) nodal values on a spherical surface. Built upon suitable analytic formulas for dealing with the involved highly oscillatory integrands, the method is robust for high mode expansions. We apply the numerical method to the simulation of three-dimensional acoustic and electromagnetic multiple scattering problems. Various numerical evidences show that the high accuracy can be achieved within reasonable computational time. This also paves the way for spectral-element discretization of 3D scattering problems reduced by spherical transparent boundary conditions based on the Dirichlet-to-Neumann map.

Xin Liu, Ziqing Xie, Yongjun Yuan et al.

This paper investigates the dynamics of spin-2 Bose-Einstein condensates (BECs) with rotation and spin-orbit coupling (SOC). In order to better simulate the dynamics, we present an efficient high-order compact splitting Fourier spectral method. This method splits the Hamiltonian into a linear part, which consists of the Laplace, rotation and SOC terms, and a nonlinear part that includes all the remaining terms. The wave function is well approximated by the Fourier spectral method and is numerically accessed with discrete Fast Fourier transform (FFT). For linear subproblem, the handling of rotation term and SOC term poses a major challenge. Using a function mapping based on rotation, we can integrate the linear subproblem exactly and explicitly. This mapping we propose not only helps eliminate the rotation term, but also prevents the SOC term from evolving into a time-dependent form. The nonlinear subproblem is integrated analytically in physical space. Such "compact" splitting involves only two operators and facilitates the design of high-order splitting schemes. Our method is spectrally accurate in space and high order in time. It is efficient, explicit, unconditionally stable and simple to implement. In addition, we derive some dynamical properties and carry out a systematic study, including accuracy and efficiency tests, dynamical property verification, the SOC effects and dynamics of vortex lattice.

Zhaoxing Chen, Wei Liu, Ziqing Xie et al.

This paper presents a class of efficient manifold optimization algorithms for computing the ground state solutions of a semilinear elliptic system, which are unstable saddle points of the variational functional. Variational arguments show that these unstable saddle points can be characterized as the local minimizers of the variational functional constrained to the Nehari manifold $\mathcal{N}$. The Nehari manifold optimization method (NMOM) proposed in [Z. Chen, W. Liu, Z. Xie, and W. Yi. SIAM J. Sci. Comput., 47(4): A2098-A2126, 2025] provides a Riemannian gradient descent framework on $\mathcal{N}$ for such constrained minimization problems. To deal with both the intrinsic instability of the solutions and the increased computational complexity introduced by the coupling between components, we combine the ideas from the NMOM and the Nesterov-type acceleration to develop a new efficient Riemannian accelerated gradient algorithm on $\mathcal{N}$ (RAG-$\mathcal{N}$). The key idea is to perform an easy-to-implement nonlinear extrapolation step on $\mathcal{N}$, followed by a Riemannian steepest-descent update at the extrapolated point. To enhance the robustness, we further incorporate a nonmonotone step-size search strategy into the RAG-$\mathcal{N}$ algorithm, obtaining a variant with improved stability. Numerical experiments show that the RAG-$\mathcal{N}$ algorithms substantially reduce the number of iterations compared with the Riemannian steepest descent algorithm of NMOM. Finally, we apply the RAG-$\mathcal{N}$ algorithms to compute the ground state solutions of semilinear elliptic systems with two, three and four components, and investigate their behavior under different coupling coefficients and various settings, including Gaussian-type external potentials and singular diffusion coefficients.