Numerical and theoretical study of weak Galerkin finite element solutions of Turing patterns in reaction–diffusion systems

In this paper, we introduce numerical schemes and their analysis based on weak Galerkin finite element framework for solving 2‐D reaction–diffusion systems. Weak Galerkin finite element method (WGFEM) for partial differential equations relies on the concept of weak functions and weak gradients, in which differential operators are approximated by weak forms through the Green's theorem. This method allows the use of totally discontinuous functions in the approximation space. In the current work, the WGFEM solves reaction–diffusion systems to find unknown concentrations (u, v) in element interiors and boundaries in the weak Galerkin finite element space WG(P₀, P₀, RT₀) . The WGFEM is used to approximate the spatial variables and the time discretization is made by the backward Euler method. For reaction–diffusion systems, stability analysis and error bounds for semi‐discrete and fully discrete schemes are proved. Accuracy and efficiency of the proposed method successfully tested on several numerical examples and obtained results satisfy the well‐known result that for small values of diffusion coefficient, the steady state solution converges to equilibrium point. Acquired numerical results asserted the efficiency of the proposed scheme.     

An adaptive weak Galerkin finite element method with hierarchical bases for the elliptic problem

Based on a posteriori error estimator with hierarchical bases, an adaptive weak Galerkin finite element method (WGFEM) is proposed for the elliptic problem with mixed boundary conditions. For the posteriori error estimator, we are only required to solve a linear algebraic system with diagonal entries corresponding to the degree of freedoms, which significantly reduces the computational cost. The upper and lower bounds of the error estimator are shown to addresses the reliability and efficiency of the adaptive approach. Numerical simulations are provided to demonstrate the effectiveness and robustness of the proposed method.     

Indefinite integrals of quotients of special functions

A new method is presented for deriving indefinite integrals involving quotients of special functions. The method combines an integration formula given previously with the recursion relations obeyed by the function. Some additional results are presented using an elementary method, here called reciprocation, which can also be used in combination with the new method to obtain additional quotient integrals. Sample results are given here for Bessel functions, Airy functions, associated Legendre functions and the three complete elliptic integrals. All results given have been numerically checked with Mathematica.     

A modified SQP-filter method for nonlinear complementarity problem

The nonlinear complementarity problem can be reformulated as a nonlinear programming. For solving nonlinear programming, sequential quadratic programming (SQP) type method is very effective. Moreover, filter method, for its good numerical results, are extensively studied to handle nonlinear programming problems recently. In this paper, a modified quadratic subproblem is proposed. Based on it, we employ filter technique to tackle nonlinear complementarity problem. This method has no demand on initial point. The restoration phase, which is always used in traditional filter method, is not needed. Global convergence results of the proposed algorithm are established under suitable conditions. Some numerical results are reported in this paper.     

Modeling of two-phase flows with surface tension by finite pointset method (FPM)

A meshfree method for two-phase immiscible incompressible flows including surface tension is presented. The continuum surface force (CSF) model is used to include the surface tension force. The incompressible Navier–Stokes equation is considered as the mathematical model. Application of implicit projection method results in linear second-order partial differential equations for velocities and pressure. These equations are then solved by the finite pointset method (FPM), which is a meshfree and Lagrangian method. The fluid is represented as finite number of particles and the immiscible fluids are distinguished by the color of each particle. The interface is tracked automatically by advecting the color functions for each particle. Two test cases, Laplace's law and the Rayleigh–Taylor instability in 2D have been presented. The results are found to be consistent with the theoretical results.     

一维不定参数结构系统振动特征问题的摄动传递矩阵法

基于Riccati传递矩阵法，给出了一维不确定参数结构系统振动特征问题的二阶摄动计算方法，该方法适用于一般的一维结构系统的实数和复数特征问题的分析，并给出了结构振动特征的灵敏度计算公式．算例对转子的陀螺特征值问题进行了摄动分析，摄动结果和精确计算结果吻合良好．     

A mixed method for free and forced vibration of rectangular plates

This paper presents a very first combined application of Ritz method and differential quadrature (DQ) method to vibration problem of rectangular plates. In this study, the spatial partial derivatives with respect to a coordinate direction are first discretized using the Ritz method. The resulting system of partial differential equations and the related boundary conditions are then discretized in strong form using the DQ method. The mixed method combines the simplicity of the Ritz method and high accuracy and efficiency of the DQ method. The results are obtained for various types of boundary conditions. Comparisons are made with existing analytical and numerical solutions in the literature. Numerical results prove that the present method is very suitable for the problem considered due to its simplicity, efficiency, and high accuracy.     

IMGS方法对于,H -矩阵的若干令人满意的改进

该文给出线性方程组改进的Gauss-Seidel迭代法（被称之为IMGS方法）对于H -矩阵的收敛性定理,并且对其参数a_i与SOR迭代法的参数ω的取值范围进行了比较. 所用方法及所得结论优于近年来相关结论,并且表明这种IMGS方法对,H -矩阵是有效的.     

A Fourier spectral-discontinuous Galerkin method for time-dependent 3-D Schrödinger–Poisson equations with discontinuous potentials

In this paper, we propose a high order Fourier spectral-discontinuous Galerkin method for time-dependent Schrödinger–Poisson equations in 3-D spaces. The Fourier spectral Galerkin method is used for the two periodic transverse directions and a high order discontinuous Galerkin method for the longitudinal propagation direction. Such a combination results in a diagonal form for the differential operators along the transverse directions and a flexible method to handle the discontinuous potentials present in quantum heterojunction and supperlattice structures. As the derivative matrices are required for various time integration schemes such as the exponential time differencing and Crank Nicholson methods, explicit derivative matrices of the discontinuous Galerkin method of various orders are derived. Numerical results, using the proposed method with various time integration schemes, are provided to validate the method.     

Algorithms for Unconstrained Optimization Problems via Control Theory

Existing algorithms for solving unconstrained optimization problems are generally only optimal in the short term. It is desirable to have algorithms which are long-term optimal. To achieve this, the problem of computing the minimum point of an unconstrained function is formulated as a sequence of optimal control problems. Some qualitative results are obtained from the optimal control analysis. These qualitative results are then used to construct a theoretical iterative method and a new continuous-time method for computing the minimum point of a nonlinear unconstrained function. New iterative algorithms which approximate the theoretical iterative method and the proposed continuous-time method are then established. For convergence analysis, it is useful to note that the numerical solution of an unconstrained optimization problem is none other than an inverse Lyapunov function problem. Convergence conditions for the proposed continuous-time method and iterative algorithms are established by using the Lyapunov function theorem.     

Two-level Galerkin–Lagrange multipliers method for the stationary Navier–Stokes equations

In this paper, we combine the Galerkin–Lagrange multiplier (GLM) method with the two-level method to solve the stationary Navier–Stokes equations in order to avoid the time-consuming process and the construction of zero-divergence elements. Different quadrilateral partitions are used for approximating the velocity and the pressure. Then some error estimates are obtained and some numerical results of the GLM method and the two-level GLM method are given. The results show that the two-level method based on the GLM method is more efficient than the GLM method under the convergence rate of same order.     

On a new class of inversive pseudorandom numbers for parallelized simulation methods

Inversive methods are attractive alternatives to the linear method for pseudorandom number generation. A particularly attractive method is the digital explicit inversive method recently introduced by the authors. We establish some new results on the statistical properties of parallel streams of pseudorandom numbers generated by this method. In particular, we extend the results of the first author on the statistical properties of pseudorandom numbers generated by the explicit inversive congruential method introduced by Eichenauer-Herrmann. These results demonstrate that the new method is eminently suitable for the generation of parallel streams of pseudorandom numbers with desirable properties.     

Numerical solution of the coupled viscous Burgers’ equation

In the present paper, a numerical method is proposed for the numerical solution of a coupled system of viscous Burgers’ equation with appropriate initial and boundary conditions, by using the cubic B-spline collocation scheme on the uniform mesh points. The scheme is based on Crank–Nicolson formulation for time integration and cubic B-spline functions for space integration. The method is shown to be unconditionally stable using von-Neumann method. The accuracy of the proposed method is demonstrated by applying it on three test problems. Computed results are depicted graphically and are compared with those already available in the literature. The obtained numerical solutions indicate that the method is reliable and yields results compatible with the exact solutions.     

On the convergence of a new trust region algorithm

Summary. In this paper we present a new trust region algorithm for general nonlinear constrained optimization problems. The algorithm is based on the exact penalty function. Under very mild conditions, global convergence results for the algorithm are given. Local convergence properties are also studied. It is shown that the penalty parameter generated by the algorithm will be eventually not less than the norm of the Lagrange multipliers at the accumulation point. It is proved that the method is equivalent to the sequential quadratic programming method for all large , hence superlinearly convergent results of the SQP method can be applied. Numerical results are also reported. Received March 21, 1993     

The numerical solution of a nonlinear system of second-order boundary value problems using the sinc-collocation method

The sinc-collocation method is presented for solving a nonlinear system of second-order boundary value problems. Some properties of the sinc-collocation method required for our subsequent development are given and are utilized to reduce the computation of solution of the system of second-order boundary value problems to some algebraic equations. Numerical examples are included to demonstrate the validity and applicability of the technique and a comparison is made with the existing results. The method is easy to implement and yields very accurate results.     

Exact solutions for the generalized Klein–Gordon equation via a transformation and Exp-function method and comparison with Adomian’s method

In this paper, a suitable transformation and a so-called Exp-function method are used to obtain different types of exact solutions for the generalized Klein–Gordon equation. These exact solutions are in full agreement with the previous results obtained in Refs. [Sirendaoreji, Auxiliary equation method and new solutions of Klein–Gordon equations, Chaos, Solitons & Fractals 31 (4) (2007) 943–950; Huiqun Zhang, Extended Jacobi elliptic function expansion method and its applications, Communications in Nonlinear Science and Numerical Simulation, 12 (5) (2007) 627–635]. One of these exact solutions is compared with the approximate solutions obtained by the modified decomposition method. Accurate numerical results for a wider range of time are obtained after using different types of ADM-Padè approximation. Our results show that the Exp-function method is very effective in finding exact solutions for the problem considered while the modified decomposition method is very powerful in finding numerical solutions with good accuracy for nonlinear PDE without any need for a transformation or perturbation.     

On the compact difference scheme for the two-dimensional coupled nonlinear Schrödinger equations

A high-order finite difference method for the two-dimensional coupled nonlinear Schrödinger equations is considered. The proposed scheme is proved to preserve the total mass and energy in a discrete sense and the solvability of the scheme is shown by using a fixed point theorem. By converting the scheme in the point-wise form into a matrix–vector form, we use the standard energy method to establish the optimal error estimate of the proposed scheme in the discrete L²-norm. The convergence order is proved to be of a fourth-order in space and a second-order in time, respectively. Finally, some numerical examples are given in order to confirm our theoretical results for the numerical method. The numerical results are compared with exact solutions and other existing method. The comparison between our numerical results and those of Sun and Wangreveals that our method improves the accuracy of space and time directions.     

三阶微分方程的多区域Legendre-Petrov-Galerkin谱方法

提出三阶微分方程初边值问题的多区域Legendre-Petrov-Galerkin谱方法.对于三阶线性微分方程,证明该方法全离散格式的稳定性,并给出L~2-误差估计.进而将该方法和Legendre配置方法相结合,应用于某些非线性问题.数值算例对单区域和多区域方法的结果进行比较.     

Convergence of Newton-like-iterative methods

Summary Newton-like methods in which the intermediate systems of linear equations are solved by iterative techniques are examined. By applying the theory of inexact Newton methods radius of convergence and rate of convergence results are easily obtained. The analysis is carried out in affine invariant terms. The results are applicable to cases where the underlying Newton-like method is, for example, a difference Newton-like or update-Newton method.     

Dominant eigenvalue problem for positive integral operators and its solution by Monte Carlo method

In this paper, a method of numerical solution to the dominant eigenvalue problem for positive integral operators is presented. This method is based on results of the theory of positive operators developed by Krein and Rutman. The problem is solved by Monte Carlo method constructing random variables in such a way that differences between results obtained and the exact ones would be arbitrarily small. Some numerical results are shown.