• Abstract

Test-particle simulations provide a useful complement to the kinetic simulations of many-body systems and their approximate treatment with multiple moments. In a kinetic approach, systems are described at a microscopic level in terms of a large number of degrees of freedom. Fluid or multiple moment approaches, however, provide a description at the macroscopic level, in terms of relatively few physical parameters involving averages or moments of particle distribution functions. Ideally, fully kinetic descriptions should be done whenever possible. Due to their complexity, the use of these approaches is often not practical in many cases of interest. In comparison, the fluid approximation is much simpler to implement and solve. It can be used to describe complex phenomena in multi-dimensional geometry with realistic boundary conditions. Its main drawback is its inability to account for many phenomena taking place on fine space or time scales, or phenomena involving nonlocal transport. Macroscopic approaches are also not adapted to describe large deviations from local equilibrium, such as the occurrence of particle beams or otherwise strong anisotropy. With the test-particle method, particle trajectories are calculated using approximated fields obtained from a low level approach, such as multiple moments. Approximate fields can also be obtained from experiments or observations. Assuming that these fields are representative of actual systems, various kinetic and statistical properties of the system can then be calculated, such as particle distribution functions and moments thereof. In this paper, the test-particle method is discussed in the context of classical statistical physics of many-body interacting point particles. Four different formulations of the method are presented, which correspond to four broad categories of the application encountered in the field of plasma physics and astronomy.

• Abstract

In this work, we are concerned with numerical approximation of the gyroaverage operators arising in plasma physics to take into account the effects of the finite Larmor radius corrections. Several methods are proposed in the space configuration and compared to the reference spectral method. We then investigate the influence of the different approximations considering the coupling with some guiding-center models available in the literature.

• Abstract

Based on an error estimate in terms of element edge vectors on arbitrary unstructured simplex meshes, we propose a new edge-based anisotropic mesh refinement algorithm. As the mesh adaptation indicator, the error estimate involves only the gradient of error rather than higher order derivatives. The preferred refinement edge is chosen to reduce the maximal term in the error estimate. The algorithm is implemented in both two- and three-dimensional cases, and applied to the singular function interpolation and the elliptic interface problem. The numerical results demonstrate that the convergence order obtained by using the proposed anisotropic mesh refinement algorithm can be higher than that given by the isotropic one.

• Abstract

Based on a novel numerical flux involving jumps of even order derivatives of the numerical solution, a direct discontinuous Galerkin (DDG) method for diffusion problems was introduced in [H. Liu and J. Yan, SIAM J. Numer. Anal. 47(1) (2009), 475-698]. In this work, we show that higher order (k≥4) derivatives in the numerical flux can be avoided if some interface corrections are included in the weak formulation of the DDG method; still the jump of 2nd order derivatives is shown to be important for the method to be efficient with a fixed penalty parameter for all p^k elements. The refined DDG method with such numerical fluxes enjoys the optimal (k+1)th order of accuracy. The developed method is also extended to solve convection diffusion problems in both one- and two-dimensional settings. A series of numerical tests are presented to demonstrate the high order accuracy of the method.

• Abstract

Many problems in engineering shape design involve eigenvalue optimizations. The relevant difficulty is that the eigenvalues are not continuously differentiable with respect to the density. In this paper, we are interested in the case of multi-density inhomogeneous materials which minimizes the least eigenvalue. With the finite element discretization, we propose a monotonically decreasing algorithm to solve the minimization problem. Some numerical examples are provided to illustrate the efficiency of the present algorithm as well as to demonstrate its availability for the case of more than two densities. As the computations are sensitive to the choice of the discretization mesh sizes, we adopt the refined mesh strategy, whose mesh grids are 25-times of the amount used in [S. Osher and F. Santosa, J. Comput. Phys., 171 (2001), pp. 272-288]. We also show the significant reduction in computational cost with the fast convergence of this algorithm.

• Abstract

The paper extends weighted essentially non-oscillatory (WENO) schemes to two-dimensional quadrilateral and mixed-element unstructured meshes. The key element of the proposed methods is a reconstruction procedure suitable for arbitrarily-shaped cells. The resulting schemes achieve the designed uniformly high-order of accuracy and compute discontinuous solutions without spurious oscillations at interfaces between cells of two different types.

• Abstract

Spiral waves appear in many different natural contexts: excitable biological tissues, fungi and amoebae colonies, chemical reactions, growing crystals, fluids and gas eddies as well as in galaxies. While the existing theories explain the presence of spirals in terms of nonlinear parabolic equations, it is explored here the fact that self-sustained spiral wave regime is already present in the linear heat operator, in terms of integer Bessel functions of complex argument. Such solutions, even if commonly not discussed in the literature because diverging at spatial infinity, play a central role in the understanding of the universality of spiral process. In particular, we have studied how in nonlinear reaction-diffusion models the linear part of the equations determines the wave front appearance while nonlinearities are mandatory to cancel out the blowup of solutions. The spiral wave pattern still requires however at least two cross-reacting species to be physically realized. Biological implications of such a results are discussed.

• Abstract

In this paper, we propose a multiphase fuzzy region competition model for texture image segmentation. In the functional, each region is represented by a fuzzy membership function and a probability density function that is estimated by a nonparametric kernel density estimation. The overall algorithm is very efficient as both the fuzzy membership function and the probability density function can be implemented easily. We apply the proposed method to synthetic and natural texture images, and synthetic aperture radar images. Our experimental results have shown that the proposed method is competitive with the other state-of-the-art segmentation methods.

• Abstract

This paper is concerned with the elliptic problems with nonlinear Stefan-Boltzmann boundary condition. By combining with the monotone method, the Robin-Robin domain decomposition methods are proposed to decouple the nonlinear interface and boundary condition. The monotone properties are verified for both the multiplicative and the additive domain decomposition methods. The numerical results confirm the theoretical analysis.

• Abstract

A primitive variable spectral method for simulating incompressible viscous flows inside a finite cylinder is presented. One element of originality of the proposed method is that the radial discretization of the Fourier coefficients depends on the Fourier mode, its dimension decreasing with the increase of the azimuthal modal number. This principle was introduced independently by Matsushima and Marcus and by Verkley in polar coordinates and is adopted here for the first time to formulate a 3D cylindrical Galerkin projection method. A second element of originality is the use of a special basis of Jacobi polynomials introduced recently for the radial dependence in the solution of Dirichlet problems. In this basis the radial operators are represented by matrices of minimal sparsity — diagonal stiffness and tridiagonal mass — provided here in closed form for the first time, and lead to a Helmholtz operator characterized by a favorable condition number. Finally, a new method is presented for eliminating the singular behaviour of the solution originated by the rotation of the lid with respect to the cylindrical wall. Thanks to these elements, the resulting Navier-Stokes spectral solver guarantees the differentiability to any order of the solution in the entire computational domain and does not suffer from the time-step stability restriction occurring in spectral methods with a point clustering close to the axis. Several test examples are offered that demonstrate the spectral accuracy of the solution method under different representative conditions.

• Abstract

With words as nodes, and a link exits between two neighboring words, the weighted directed English and Chinese written human language networks are constructed from one English novel and two Chinese ones. We hereby analyze in detail the topological structure of them, in order to clarify their similar and different statistical properties. The empirical results show that the English and Chinese language networks all possess the shifted power law (SPL) degree distribution, the small-world property and the hierarchical structure, the connections among the words have positive assortativity coefficient and reciprocal characteristics. We also investigate the features of the strength and the centrality, which describe the importance of a specific word. Furthermore, considering the growth properties of the language networks and part of topological property, we find that the English written human language network grows slower than the Chinese one, which implies different mechanisms of the English and Chinese languages.