The so-called "small denominator problem" was a fundamental problem of dynamics, as pointed out by Poincaré. Small denominators appear most commonly in perturbative theory. The Duffing equation is the simplest example of a non-integrable system exhibiting all problems due to small denominators. In this paper, using the forced Duffing equation as an example, we illustrate that the famous "small denominator problems" never appear if a non-perturbative approach based on the homotopy analysis method (HAM), namely "the method of directly defining inverse mapping" (MDDiM), is used. The HAM-based MDDiM provides us great freedom to directly define the inverse operator of an undetermined linear operator so that all small denominators can be completely avoided and besides the convergent series of multiple limit-cycles of the forced Duffing equation with high nonlinearity are successfully obtained. So, from the viewpoint of the HAM, the famous "small denominator problems" are only artifacts of perturbation methods. Therefore, completely abandoning perturbation methods but using the HAM-based MDDiM, one would be never troubled by "small denominators". The HAM-based MDDiM has general meanings in mathematics and thus can be used to attack many open problems related to the so-called "small denominators".

Multivariate time series segmentation is an important problem in data mining and it has arisen in more and more practical applications in recent years. The task of time series segmentation is to partition a time series into segments by detecting the abrupt changes or anomalies in the time series. Multivariate time series segmentation can provide meaningful information for further data analysis, prediction and policy decision. A time series can be considered as a piecewise continuous function, it is natural to take its total variation norm as a prior information of this time series. In this paper, by minimizing the negative log-likelihood function of a time series, we propose a total variation based model for multivariate time series segmentation. An iterative process is applied to solve the proposed model and a search combined the dynamic programming method is designed to determine the breakpoints. The experimental results show that the proposed method is efficient for multivariate time series segmentation and it is competitive to the existing methods for multivariate time series segmentation.

The main contents of this paper are to establish a finite element fully-discrete approximate scheme for multi-term time-fractional mixed sub-diffusion and diffusion-wave equation with spatial variable coefficient, which contains a time-space coupled derivative. The nonconforming $EQ_1^{rot}$ element and Raviart-Thomas element are employed for spatial discretization, and L1 time-stepping method combined with the Crank-Nicolson scheme are applied for temporal discretization. Firstly, based on some significant lemmas, the unconditional stability analysis of the fully-discrete scheme is acquired. With the assistance of the interpolation operator $I_h$ and projection operator $R_h$, superclose and convergence results of the variable $u$ in $H^1$-norm and the flux $\vec{p}=\kappa_5(\textbf{x})\nabla u(\textbf{x},t)$ in $L^2$-norm are obtained, respectively. Furthermore, the global superconvergence results are derived by applying the interpolation postprocessing technique. Finally, the availability and accuracy of the theoretical analysis are corroborated by experimental results of numerical examples on anisotropic meshes.

We have analyzed the kinetics of solid circular particles interacting with fluid, outer boundary and internal square shaped obstacles tilted at a $45^{\circ}$ angle. The effects on the motion of particle due to collision with obstacles and wall are inspected. An Eulerian approach is used to study the behavior of particle in the fixed computational mesh. The interactions between fluid, particles and obstacles have been carried out in the whole domain by using fictitious boundary method (FBM). In this work, the particulate flow simulations are computed by using finite element solver FEATFLOW. Numerical results are presented by assigning different alignments to the obstacles and varying their positions in the domain. Particle-wall, particle-particle and particle-obstacle collisions are treated by applying a modified collision model proposed by Glowinski et al. The rapid change in drag forces acting on obstacles due to nearby passing particles and its effect on the fluid motion has been investigated.

This study is focused on the forest edge flow by using numerical method. To model the effects of a forest canopy on airflow, source terms are introduced into the governing equations. The lattice Boltzmann method in conjunction with the standard $k−ε$ model is applied to solve the turbulent wind field. In order to perform the simulation on non-uniform grids, the Taylor series expansion and least square based lattice Boltzmann method (TLLBM) is adopted to improve the accuracy and computational efficiency. The present method and code are verified with an earlier forest edge simulation. A series of forest canopies are established to explore the impacts of canopy morphology on wind field. These canopies cover 3 canopy architectures and the Leaf Area Index (LAI) ranges from 2.0 to 4.0. The further study is carried out by adjusting the canopy foliage amount and the canopy architecture. The present study demonstrates the potential of lattice Boltzmann method to simulate the high Re number forest edge flow. The impacts of canopy morphology on zero plane displacement, aerodynamic roughness length, friction wind velocity, permeability coefficient, wall-shear stress are illustrated in detail. The results show that the canopy sub-layer wind field, especially the wind velocity profiles within and above the forest canopy, are mainly determined by canopy morphology.

In this paper, we construct a new cell-centered nonlinear finite volume scheme that preserves the extremum principle for heterogeneous anisotropic diffusion equation on distorted meshes. We introduce a new nonlinear approach to construct the conservative flux, that is, a linear second order flux is firstly given and a nonlinear conservative flux is then constructed by using an adaptive method and a nonlinear weighted method. Our new scheme does not need to use the convex combination of the cell-center unknowns to approximate the auxiliary unknowns, so it can deal with the problem with general discontinuous coefficients. Numerical results show that our new scheme performs more robust than some existing schemes on highly distorted meshes.

Imposing appropriate numerical boundary conditions at the symmetrical center $r=0$ is vital when computing compressible fluids with radial symmetry. Extrapolation and other traditional techniques are often employed, but spurious numerical oscillations or wall-heating phenomena can occur. In this paper, we emphasize that because of the conservation property, the updating formula of the boundary cell average can coincide with the one for interior cell averages. To achieve second-order accuracy both in time and space, we associate obtaining the inner boundary value at $r=0$ with the resolution of the corresponding one-sided generalized Riemann problem (GRP). Acoustic approximation is applied in this process. It creates conditions to avoid the singularity of type $1/r$ and aids in obtaining the value of the singular quantity using L'Hospital's rule. Several challenging scenarios are tested to demonstrate the effectiveness and robustness of our approach.

A discontinuous Galerkin (DG) scheme for solving semilinear elliptic problem is developed and analyzed in this paper. The DG finite element discretization is first established, then the corresponding well-posedness is provided by using Brouwer’s fixed point method. Some optimal priori error estimates under both DG norm and $L^2$ norm are presented, respectively. Numerical results are given to illustrate the efficiency of the proposed approach.

The main aim of this paper is to present an analytical approach on the postbuckling for torsionally loaded sandwich carbon nanotube (CNT) reinforced cylindrical shells with the auxetic core. The considered shells consist of three layers, external and internal CNT reinforced layers, and the auxetic lattice core made by isotropic material. The homogenization model for honeycomb auxetic lattice core is utilized, and the equilibrium equations are formulated based on the nonlinear Donnell's thin shell theory with von Kármán geometrical nonlinearities. The three terms of deflection are considered, the Airy's stress function and Galerkin's method are utilized, the explicit expression of critical buckling of torsionally loaded shells and load-deflection expression of postbuckling states are achieved. The effects of two carbon nanotube reinforced layers, the auxetic core layer, the volume fraction of carbon nanotube on the torsional buckling behavior are examined and remarked.

This paper presents an Eulerian diffuse-interface method using a high-order compact difference scheme for simulating elastic-plastic flows with the Mie–Grüneisen (MG) equation of state (EoS). For simulations of multimaterial problems, numerical errors were generated in the material discontinuities owing to inconsistent treatment of the convective terms. Based on the normal-stress-based mechanical equilibrium assumption for elastic-plastic solids, we introduce an improved form of the consistent localized artificial diffusivity (LAD) method to ensure an oscillation-free interface for velocity and normal stress. The proposed algorithm uses a hyperelastic model. A mixture type of the model system was formed by combining the conservation equations for the basic conserved variables, an equation of a unified deviatoric tensor describing solid deformation, and an additional set of equations for solving the material quantities in the MG EoS. Several one- and two-dimensional problems with various discontinuities, including the elastic-plastic Richtmyer–Meshkov instability, were considered for testing the proposed method.

In Zhu, Wang and Gao (SIAM J. Sci. Comput., 43 (2021), pp. A3009–A3031), we proposed a new framework of troubled-cell indicator (TCI) using K-means clustering and the numerical results demonstrate that it can detect the troubled cells accurately using the KXRCF indication variable. The main advantage of this TCI framework is its great potential of extensibility. In this follow-up work, we introduce three more indication variables, i.e., the TVB, Fu-Shu and cell-boundary jump indication variables, and show their good performance by numerical tests to demonstrate that the TCI framework offers great flexibility in the choice of indication variables. We also compare the three indication variables with the KXRCF one, and the numerical results favor the KXRCF and the cell-boundary jump indication variables.