A class of geometric asynchronous parallel algorithms for solving large-scale discrete PDE eigenvalues has been studied by the author (Sun in Sci China Math 41(8): 701–725, 2011; Sun in Math Numer Sin 34(1): 1–24, 2012; Sun in J Numer Methods Comput Appl 42(2): 104–125, 2021; Sun in Math Numer Sin 44(4): 433–465, 2022; Sun in Sci China Math 53(6): 859–894, 2023; Sun et al. in Chin Ann Math Ser B 44(5): 735–752, 2023). Different from traditional preconditioning algorithm with the discrete matrix directly, our geometric pre-processing algorithm (GPA) algorithm is based on so-called intrinsic geometric invariance, i.e., commutativity between the stiff matrix A and the grid mesh matrix $G{:}\,AG=GA$. Thus, the large-scale system solvers can be replaced with a much smaller block-solver as a pretreatment. In this paper, we study a sole PDE and assume G satisfies a periodic condition $G^m=I, m<< \textrm{dim}(G)$. Four special cases have been studied in this paper: two-point ODE eigen-problem, Laplace eigen-problems over L-shaped region, square ring, and 3D hexahedron. Two conclusions that “the parallelism of geometric mesh pre-transformation is mainly proportional to the number of faces of polyhedron” and “commutativity of grid mesh matrix and mass matrix is the essential condition for the GPA algorithm” have been obtained.

In this paper, we shall carry out the L$^2$-norm stability analysis of the Runge-Kutta discontinuous Galerkin (RKDG) methods on rectangle meshes when solving a linear constant-coefficient hyperbolic equation. The matrix transferring process based on temporal differences of stage solutions still plays an important role to achieve a nice energy equation for carrying out the energy analysis. This extension looks easy for most cases; however, there are a few troubles with obtaining good stability results under a standard CFL condition, especially, for those ${\mathcal {Q}}^k$-elements with lower degree k as stated in the one-dimensional case. To overcome this difficulty, we make full use of the commutative property of the spatial DG derivative operators along two directions and set up a new proof line to accomplish the purpose. In addition, an optimal error estimate on ${\mathcal {Q}}^k$-elements is also presented with a revalidation on the supercloseness property of generalized Gauss-Radau (GGR) projection.

A so-called grid-overlay finite difference method (GoFD) was proposed recently for the numerical solution of homogeneous Dirichlet boundary value problems (BVPs) of the fractional Laplacian on arbitrary bounded domains. It was shown to have advantages of both finite difference (FD) and finite element methods, including their efficient implementation through the fast Fourier transform (FFT) and the ability to work for complex domains and with mesh adaptation. The purpose of this work is to study GoFD in a meshfree setting, a key to which is to construct the data transfer matrix from a given point cloud to a uniform grid. Two approaches are proposed, one based on the moving least squares fitting and the other based on the Delaunay triangulation and piecewise linear interpolation. Numerical results obtained for examples with convex and concave domains and various types of point clouds are presented. They show that both approaches lead to comparable results. Moreover, the resulting meshfree GoFD converges in a similar order as GoFD with unstructured meshes and finite element approximation as the number of points in the cloud increases. Furthermore, numerical results show that the method is robust to random perturbations in the location of the points.

In this paper, we present a novel and efficient iterative approach for computing the polar decomposition of rectangular (or square) complex (or real) matrices. The method proposed herein entails four matrix multiplications in each iteration, effectively circumventing the need for matrix inversions. We substantiate that this method exhibits fourth-order convergence. To illustrate its efficacy relative to alternative techniques, we conduct numerical experiments using randomly generated matrices of dimensions $n\times n$, where n assumes values of 80, 90, 100, 120, 150, 180, and 200. Through two illustrative examples, we provide numerical results. We gauge the performance of different methods by calculating essential metrics based on ten matrices for each dimension. These metrics include the average iteration count, the average total matrix multiplication count, the average precision, and the average execution time. Through meticulous comparison, our newly devised method emerges as a proficient and rapid solution, boasting a reduced computational overhead.

In this paper, we investigate the global dynamics of a predator-prey model with a general growth rate function and carrying capacity. We prove that the origin is unstable using the blow-up method. Also, by constructing a new Lyapunov function and using LaSalle’s invariance principle, we obtain the global stability of the positive equilibrium state of the system. In addition, the system undergoes the Hopf bifurcation at the positive equilibrium point when the predator birth rate $\delta$ is used as the bifurcation parameter. Finally, two examples are given to verify the feasibility of the theoretical results. One example is given to reconsider the global stability of the positive equilibrium of a Leslie-Gower predator-prey model with prey cannibalism, and the obtained results confirm the conjecture proposed by Lin et al. (Adv Differ Equ 2020, 153, 2020). The other example is given to verify the occurrence of the Hopf bifurcation of a Leslie-Gower predator-prey model with a square root response function, and obtain the Hopf bifurcation diagram by the numerical simulation.

This paper deals with the numerical solution of the large-scale Stein and discrete-time Lyapunov matrix equations. Based on the global Arnoldi process and the squared Smith iteration, we propose a low-rank global Krylov squared Smith method for solving large-scale Stein and discrete-time Lyapunov matrix equations, and estimate the upper bound of the error and the residual of the approximate solutions for the matrix equations. Moreover, we discuss the restarting of the low-rank global Krylov squared Smith method and provide some numerical experiments to show the efficiency of the proposed method.

In this paper, the regularity and finite difference methods for the two-dimensional delay fractional equations are considered. The analytic solution is derived by eigenvalue expansions and Laplace transformation. However, due to the derivative discontinuities resulting from the delay effect, the traditional L1-ADI scheme fails to achieve the optimal convergence order. To overcome this issue and improve the convergence order, a simple and cost-effective decomposition technique is introduced and a fitted L1-ADI scheme is proposed. The numerical tests are conducted to verify the theoretical result.

Strong ${\mathcal {H}}$-tensors play a significant role in identifying the positive definiteness of an even-order real symmetric tensor. In this paper, first, an improved iterative algorithm is proposed to determine whether a given tensor is a strong ${\mathcal {H}}$-tensor, and the validity of the iterative algorithm is proved theoretically. Second, the iterative algorithm is employed to identify the positive definiteness of an even-order real symmetric tensor. Finally, numerical examples are presented to illustrate the advantages of the proposed algorithm.

We develop and investigate a new block preconditioner for a class of double saddle point (DSP) problems arising from liquid crystal directors modeling using a finite element scheme. We analyze the spectral properties of the preconditioned matrix. Numerical results are provided to evaluate the behavior of preconditioned iterative methods using the new preconditioner.

Higher order finite difference Weighted Essentially Non-Oscillatory (FD-WENO) schemes for conservation laws are extremely popular because, for multidimensional problems, they offer high order accuracy at a fraction of the cost of finite volume WENO or DG schemes. Such schemes come in two formulations. The very popular classical FD-WENO method (Shu and Osher J Comput Phys 83: 32–78, 1989) relies on two reconstruction steps applied to two split fluxes. However, the method cannot accommodate different types of Riemann solvers and cannot preserve free stream boundary conditions on curvilinear meshes. This limits its utility. The alternative FD-WENO (AFD-WENO) method can overcome these deficiencies, however, much less work has been done on this method. The reasons are three-fold. First, it is difficult for the casual reader to understand the intricate logic that requires higher order derivatives of the fluxes to be evaluated at zone boundaries. The analytical methods for deriving the update equation for AFD-WENO schemes are somewhat recondite. To overcome that difficulty, we provide an easily accessible script that is based on a computer algebra system in Appendix A of this paper. Second, the method relies on interpolation rather than reconstruction, and WENO interpolation formulae have not been documented in the literature as thoroughly as WENO reconstruction formulae. In this paper, we explicitly provide all necessary WENO interpolation formulae that are needed for implementing the AFD-WENO up to the ninth order. The third reason is that the AFD-WENO requires higher order derivatives of the fluxes to be available at zone boundaries. Since those derivatives are usually obtained by finite differencing the zone-centered fluxes, they become susceptible to a Gibbs phenomenon when the solution is non-smooth. The inclusion of those fluxes is also crucially important for preserving the order property when the solution is smooth. This has limited the utility of the AFD-WENO in the past even though the method per se has many desirable features. Some efforts to mitigate the effect of finite differencing of the fluxes have been tried, but so far they have been done on a case by case basis for the PDE being considered. In this paper we find a general-purpose strategy that is based on a different type of the WENO interpolation. This new WENO interpolation takes the first derivatives of the fluxes at zone centers as its inputs and returns the requisite non-linearly hybridized higher order derivatives of flux-like terms at the zone boundaries as its output. With these three advances, we find that the AFD-WENO becomes a robust and general-purpose solution strategy for large classes of conservation laws. It allows any Riemann solver to be used. The AFD-WENO has a computational complexity that is entirely comparable to the classical FD-WENO, because it relies on two interpolation steps which cost the same as the two reconstruction steps in the classical FD-WENO. We apply the method to several stringent test problems drawn from Euler flow, relativistic hydrodynamics (RHD), and ten-moment equations. The method meets its design accuracy for smooth flow and can handle stringent problems in one and multiple dimensions.