In this work we improve and extend a technique named recursive doubling procedure developed by Yuan and Lu [J. Lightwave Technology 25 (2007), 3649-3656] for solving periodic array problems. It turns out that when the periodic array contains an infinite number of periodic cells, our method gives a fast evaluation of the exact boundary Robin-to-Robin mapping if the wave number is complex, or real but in the stop bands. This technique is also used to solve the time-dependent Schrödinger equation in both one and two dimensions, when the periodic potential functions have some local defects. 

Stochastic well-stirred chemically reacting systems can be accurately modeled by a continuous-time Markov-chain. The corresponding master equation evolves the system's probability density function in time but can only rarely be explicitly solved. We investigate a numerical solution strategy in the form of a spectral method with an inherent natural adaptivity and a very favorable choice of basis functions. Theoretical results related to convergence have been developed previously and are briefly summarized while implementation issues, including how to adapt the basis functions to follow the solution they represent, are covered in more detail here. The method is first applied to a model problem where the convergence can easily be studied. Then we take on two more realistic systems from molecular biology where stochastic descriptions are often necessary to explain experimental data. The conclusion is that, for sufficient accuracy demands and not too high dimensionality, the method indeed provides an alternative to other methods.

We considered the point source identification problems for heat equations from noisy observation data taken at the minimum number of spatially fixed measurement points. We aim to identify the unknown number of sources and their locations along with their strengths. In our previous work, we proved that minimum measurement points needed under the noise-free setting. In this paper, we extend the proof to cover the noisy cases over a border class of source functions. We show that if the regularization parameter is chosen properly, the problem can be transformed into a poles identification problem. A reconstruction scheme is proposed on the basis of the developed theoretical results. Numerical demonstrations in 2D and 3D conclude the paper. 

A general finite element solution of the Schrödinger equation for a one-dimensional problem is presented. The solver is applicable to both stationary and time-dependent cases with a general user-selected potential term. Furthermore, it is possible to include external magnetic or electric fields, as well as spin-orbital and spin-magnetic interactions. We use analytically soluble problems to validate the solver. The predicted numerical auto-states are compared with the analytical ones, and selected mean values are used to validate the auto-functions. In order to analyze the performance of the time-dependent Schrödinger equation, a traveling wave package benchmark was reproduced. In addition, a problem involving the scattering of a wave packet over a double potential barrier shows the performance of the solver in cases of transmission and reflection of packages. Other general problems, related to periodic potentials, are treated with the same general solver and a Lagrange multiplier method to introduce periodic boundary conditions. Some simple cases of known periodic potential solutions are reported. 

We study the multiscale finite element method for solving multiscale elliptic problems with highly oscillating coefficients, which is designed to accurately capture the large scale behaviors of the solution without resolving the small scale characters. The key idea is to construct the multiscale base functions in the local partial differential equation with proper boundary conditions. The boundary conditions are chosen to extract more accurate boundary information in the local problem. We consider periodic and non-periodic coefficients with linear and oscillatory boundary conditions for the base functions. Numerical examples will be provided to demonstrate the effectiveness of the proposed multiscale finite element method.

In this paper we develop an efficient meshless method for solving inhomogeneous elliptic partial differential equations. We first approximate the source function by Chebyshev polynomials. We then focus on how to find a polynomial particular solution when the source function is a polynomial. Through the principle of the method of undetermined coefficients and a proper arrangement of the terms for the polynomial particular solution to be determined, the coefficients of the particular solution satisfy a triangular system of linear algebraic equations. Explicit recursive formulas for the coefficients of the particular solutions are derived for different types of elliptic PDEs. The method is further incorporated into the method of fundamental solutions for solving inhomogeneous elliptic PDEs. Numerical results show that our approach is efficient and accurate.

The theory of domain decomposition is described and used to divide the variable domain of a diatomic molecule into separate regions which are solved independently. This approach makes it possible to use fast Krylov methods in the broad interior of the region while using explicit methods such as Gaussian elimination on the boundaries. As is demonstrated by solving a number of model problems, these methods enable one to obtain solutions of the relevant partial differential equations and eigenvalue equations accurate to six significant figures with a small amount of computational time. Since the numerical approach described in this article decomposes the variable space into separate regions where the equations are solved independently, our approach is very well-suited to parallel computing and offers the long term possibility of studying complex molecules by dividing them into smaller fragments that are calculated separately. 

In [J. Comput. Phys. 192(1), pp.325-354 (2003)], we have developed a multi-domain spectral method with stable and conservative penalty interface conditions for the numerical simulation of supersonic reactive recessed cavity flows with homogeneous grid. In this work, the previously developed methodology is generalized to inhomogeneous grid to simulate the two dimensional supersonic injector-cavity system. Non-physical modes in the solution generated at the domain interfaces due to the spatial grid inhomogeneity are minimized using the new weighted multi-domain spectral penalty method. The proposed method yields accurate and stable solutions of the injector-cavity system which agree well with experiments qualitatively. Through the direct numerical simulation of the injector-cavity system using the weighted method, the geometric effect of the cavity wall on pressure fluctuations is investigated. It is shown that the recessed slanted cavity attenuates pressure fluctuations inside cavity enabling the cavity to act potentially as a stable flameholder for scramjet engine.

In this work, we propose an efficient multi-mesh adaptive finite element method for simulating the dendritic growth in two- and three-dimensions. The governing equations used are the phase field model, where the regularity behaviors of the relevant dependent variables, namely the thermal field function and the phase field function, can be very different. To enhance the computational efficiency, we approximate these variables on different h-adaptive meshes. The coupled terms in the system are calculated based on the implementation of the multi-mesh h-adaptive algorithm proposed by Li (J. Sci. Comput., pp. 321-341, 24 (2005)). It is illustrated numerically that the multi-mesh technique is useful in solving phase field models and can save storage and the CPU time significantly.

An arbitrarily small positive value of the gravitational aberration slightly increases the angular momentum of a two-body and multiple body system. This could potentially contribute to the accelerated expansion of the whole Universe. We present some geometrical, physical, geophysical, heliophysical, climatological, cosmological, and astronomical observational arguments, and also numerical tests to support this conjecture. We found a remarkable coincidence between the Hubble constant and the increasing distance of the Moon from the Earth, that is not only due to tidal forces. Numerical examples illustrating the expansion caused by the gravitational aberration are given. This will be modeled by a nonautonomous system of ordinary differential equations with delay.

Many realistic social networks share some universal characteristic properties, such as the small-world effects and the heterogeneous distribution of connectivity degree, which affect the dynamics in society system, especially the opinion dynamics in society. To see this, we study the opinion dynamics of the Improved Deffuant Model (IDM) in complex networks. When the two opinions differ by less than the confidence parameter ϵ (0<ϵ<1), each opinion moves partly in the direction of the other with the convergence parameter µ, which is a function of the opposite's degree k; otherwise, the two refuse to discuss and no opinion is changed. We analyze the evolution of the steady opinion s_∗ as a function of the confidence parameter ϵ, the relation between the minority steady opinion $s_∗^{min}$ and the individual connectivity k, and find some interesting results that show the dependence of the opinion dynamics on the confidence parameter and on the system topology. This study provides a new perspective and tools to understand the effects of complex system topology on opinion dynamics.