Numerical simulation of two-dimensional sine-Gordon solitons by differential quadrature method

During the past few decades, the idea of using differential quadrature methods for numerical solutions of partial differential equations (PDEs) has received much attention throughout the scientific community. In this article, we proposed a numerical technique based on polynomial differential quadrature method (PDQM) to find the numerical solutions of two-dimensional sine-Gordon equation with Neumann boundary conditions. The PDQM reduced the problem into a system of second-order linear differential equations. Then, the obtained system is changed into a system of ordinary differential equations and lastly, RK4 method is used to solve the obtained system. Numerical results are obtained for various cases involving line and ring solitons. The numerical results are found to be in good agreement with the exact solutions and the numerical solutions that exist in literature. It is shown that the technique is easy to apply for multidimensional problems.     

A study of implicit difference schemes for inviscid equations

The analysis presented in this paper relates to the numerical difficulties encountered in obtaining a solution of the Full Viscous Shock Layer (FVSL) equations for flow past blunt bodies. A simple model problem consisting of two first order inviscid equations are shown to exhibit all the numerical difficulties encountered by the FVSL Set. In an effort to alleviate these difficulties, several numerical schemes for simultaneously solving the two first order equations have been investigated. It has been found that a staggered coupled implicit difference scheme for solving these two equations eliminates the numerical problems.     

计算微分代数系统的实时仿真算法

§１．引言 对于由微分代数方程所表示的动力系统的数值算法,针对微分代数方程的一些特殊形式已经构造了一些有效算法如文献[1]-［４］．这些数值算法大部分都是基于常微分方程的一些隐式公式如隐式Ｒｕｎｇｅ－Ｋｕｔｔａ方法,向后微分公式（ＢＤＦ）等,因此这些算法都是非实时仿真算法．如果我们直接用求解常微分方程的显式公式如显式 Ｒｕｎｇｅ－Ｋｕｔｔａ方法,显式线性多步法等,虽然满足了实时仿真算法的一些特点,但是这些数值公式对微分代数方程的求解不甚理想．由于一个实时仿真算法具有实时性、周期性、可靠性等特性要求,因…     

Modal Hermite spectral collocation method for solving multi-dimensional hyperbolic telegraph equations

The present research is contemplated proposing a numerical solution of multi-dimensional hyperbolic telegraph equations with appropriate initial time and boundary space conditions. The truncated Hermite series with unknown coefficients are used for approximating the solution in both of the spatial and temporal variables. The basic idea for discretizing the considered one-dimensional (1D), two-dimensional (2D) and three-dimensional (3D) telegraph equations is based on the collocation method together with the Hermite operational matrices of derivatives. The resulted systems of linear algebraic equations are solved by some efficient methods such as LU factorization. The solution of the algebraic system contains the coefficients of the truncated Hermite series. Numerical experiments are provided to illustrate the accuracy and efficiency of the presented numerical scheme. Comparisons of numerical results associated to the proposed method with some of the existing numerical methods confirm that the method is accurate and fast experimentally.     

SDP-based approximation of stabilising solutions for periodic matrix Riccati differential equations

Numerically finding stabilising feedback control laws for linear systems of periodic differential equations is a nontrivial task with no known reliable solutions. The most successful method requires solving matrix differential Riccati equations with periodic coefficients. All previously proposed techniques for solving such equations involve numerical integration of unstable differential equations and consequently fail whenever the period is too large or the coefficients vary too much. Here, a new method for numerical computation of stabilising solutions for matrix differential Riccati equations with periodic coefficients is proposed. Our approach does not involve numerical solution of any differential equations. The approximation for a stabilising solution is found in the form of a trigonometric polynomial, matrix coefficients of which are found solving a specially constructed finite-dimensional semidefinite programming (SDP) problem. This problem is obtained using maximality property of the stabilising solution of the Riccati equation for the associated Riccati inequality and sampling technique. Our previously published numerical comparisons with other methods shows that for a class of problems only this technique provides a working solution. Asymptotic convergence of the computed approximations to the stabilising solution is proved below under the assumption that certain combinations of the key parameters are sufficiently large. Although the rate of convergence is not analysed, it appeared to be exponential in our numerical studies.     

Honeycomb Monolith微反应器中NH3还原NOx的一维数学模型

综述了最近燃料气中NOx和SOx的脱除,为了达到最好的设计和操作条件,对Honeycomb Monolith微反应器中进行了模拟,使用数学模型对一种壁面上涂有催化剂的方形HMMR管道进行了脱除NOx和SOx的模拟,并研究了质量和能量的传递问题。气相的一维方程是属于一阶常微分方程,本论文采用了四阶Runge-Kutta方法进行了数值求解。催化相方程属于一维二阶常微分方程,采用数值逼近法对其进行了求解,采用面向对象的C 对其进行了编程求解。在模拟中改变了HMMR反应器的操作条件,研究了气相中的温度和浓度的分布以及催化剂层的温度剃度和浓度的分布。     

Entropy Stable Numerical Schemes for Two-Fluid Plasma Equations

Two-fluid ideal plasma equations are a generalized form of the ideal MHD equations in which electrons and ions are considered as separate species. The design of efficient numerical schemes for the these equations is complicated on account of their non-linear nature and the presence of stiff source terms, especially for high charge to mass ratios and for low Larmor radii. In this article, we design entropy stable finite difference schemes for the two-fluid equations by combining entropy conservative fluxes and suitable numerical diffusion operators. Furthermore, to overcome the time step restrictions imposed by the stiff source terms, we devise time-stepping routines based on implicit-explicit (IMEX)-Runge Kutta (RK) schemes. The special structure of the two-fluid plasma equations is exploited by us to design IMEX schemes in which only local (in each cell) linear equations need to be solved at each time step. Benchmark numerical experiments are presented to illustrate the robustness and accuracy of these schemes.     

Adams methods for the efficient solution of stochastic differential equations with additive noise

The application of Adams methods for the numerical solution of stochastic differential equations is considered. Especially we discuss the path-wise (strong) solutions of stochastic differential equations with additive noise and their numerical computation. The special structure of these problems suggests the application of Adams methods, which are used for deterministic differential equations very successfully. Applications to circuit simulation are presented.     

On splitting-based numerical methods for nonlinear models of European options

We present a large class of nonlinear models of European options as parabolic equations with quasi-linear diffusion and fully nonlinear hyperbolic part. The main idea of the operator splitting method (OSM) is to couple known difference schemes for nonlinear hyperbolic equations with other ones for quasi-linear parabolic equations. We use flux limiter techniques, explicit–implicit difference schemes, Richardson extrapolation, etc. Theoretical analysis for illiquid market model is given. The numerical experiments show second-order accuracy for the numerical solution (the price) and Greeks Delta and Gamma, positivity and monotonicity preserving properties of the approximations.     

Manifold method coupled velocity and pressure for Navier-Stokes equations and direct numerical solution of unsteady incompressible viscous flow

Numerical manifold method (NMM) application to direct numerical solution for unsteady incompressible viscous flow Navier-Stokes (N-S) equations was discussed in this paper, and numerical manifold schemes for N-S equations were derived based on Galerkin weighted residuals method as well. Mixed covers with linear polynomial function for velocity and constant function for pressure was employed in finite element cover system. The patch test demonstrated that mixed covers manifold elements meet the stability conditions and can be applied to solve N-S equations coupled velocity and pressure variables directly. The numerical schemes with mixed covers have also been proved to be unconditionally stable. As applications, mixed cover 4-node rectangular manifold element has been used to simulate the unsteady incompressible viscous flow in typical driven cavity and flow around a square cylinder in a horizontal channel. High accurate results obtained from much less calculational variables and very large time steps are in very good agreement with the compact finite difference solutions from very fine element meshes and very less time steps in references. Numerical tests illustrate that NMM is an effective and high order accurate numerical method for unsteady incompressible viscous flow N-S equations.     

Convergence and stability of numerical methods with variable step size for stochastic pantograph differential equations

The stochastic pantograph equations (SPEs) are very special stochastic delay differential equations (SDDEs) with unbounded memory. When the numerical methods with a constant step size are applied to the pantograph equations, the most difficult problem is the limited computer memory. In this paper, we construct methods with variable step size to solve SPEs. The analysis is motivated by the example of a mean-square stable linear SPE for which the Euler–Maruyama (EM) method with variable step size fails to reproduce this behaviour for any nonzero timestep. Then we consider the Backward Euler (BE) method with variable step size and develop the fundamental numerical analysis concerning its strong convergence and mean-square linear stability. It is proved that the numerical solutions produced by the BE method with variable step size converge to the exact solution under the local Lipschitz condition and the Bounded condition. Furthermore, the order of convergence p=½ is given under the Lipschitz condition. The result of the mean-square linear stability is given. Some illustrative numerical examples are presented to demonstrate the order of strong convergence and the mean-square linear stability of the BE method.     

On solving hybrid optimal control problems with higher index DAEs

The paper deals with hybrid optimal control problems described by higher index differential–algebraic equations (DAEs). We introduce a numerical procedure for solving these problems. The procedure has the following features: it is based on the appropriately defined adjoint equations formulated for the discretized equations being the result of the numerical integration of systems equations by an implicit Runge–Kutta method; the consistent initialization procedure is applied whenever control functions jumps, or state variables transition occurs. The procedure can cope with hybrid optimal control problems which are defined by DAEs with the index not exceeding three. Our approach does not require differentiation of some system equations in order to transform higher index DAEs to the underlying ordinary differential equations (ODEs). The presented numerical examples show that the proposed approach can be used to solve efficiently hybrid optimal control problems with higher index DAEs.     

The new numerical method for solving the system of two-dimensional Burgers’ equations

In this paper, the system of two-dimensional Burgers’ equations are solved by local discontinuous Galerkin (LDG) finite element method. The new method is based on the two-dimensional Hopf–Cole transformations, which transform the system of two-dimensional Burgers’ equations into a linear heat equation. Then the linear heat equation is solved by the LDG finite element method. The numerical solution of the heat equation is used to derive the numerical solutions of Burgers’ equations directly. Such a LDG method can also be used to find the numerical solution of the two-dimensional Burgers’ equation by rewriting Burgers’ equation as a system of the two-dimensional Burgers’ equations. Three numerical examples are used to demonstrate the efficiency and accuracy of the method.     

Application of Sinc-collocation method for solving a class of nonlinear Fredholm integral equations

In this paper, the numerical solution of nonlinear Fredholm integral equations of the second kind is considered by two methods. The methods are developed by means of the Sinc approximation with the single exponential (SE) and double exponential (DE) transformations. These numerical methods combine a Sinc collocation method with the Newton iterative process that involves solving a nonlinear system of equations. We provide an error analysis for the methods. So far approximate solutions with polynomial convergence have been reported for this equation. These methods improve conventional results and achieve exponential convergence. Some numerical examples are given to confirm the accuracy and ease of implementation of the methods.     

薄膜体声波谐振器的FDTD数值分析

提出了基于时域有限差分方法对薄膜体声波谐振器进行数值分析的新方法。利用时域有限差分法理论对压电材料的控制方程,牛顿方程和电学方程在空间和时间进行了离散化,通过得到的差分方程直接得出了声场传播的时域数值解。使用该数值方法对薄膜体声波谐振器的电学特性阻抗进行了分析,并将结果与一维Mason模型的解析解进行了比较验证。     

Efficient numerical solution of constrained multibody dynamics systems

A numerical algorithm for conducting coupled system dynamical simulation is presented. The interconnected system, comprising numerous modules, is treated as a constrained multibody dynamics system. Of particular focus is the efficient solution of coupled system simulation without sacrificing the independence of the separate dynamical modules. The proposed algorithm, Maggi’s equations with perturbed iteration (MEPI) emanates from numerical methods for differential-algebraic equations. Separate treatment of the constraint equations from the resolution of subsystem dynamical responses marks MEPI’s main characteristic.     

Three dimensional discontinuous Galerkin methods for Euler equations on adaptive conforming meshes

In the numerical simulation of three dimensional fluid dynamical equations, the huge computational quantity is a main challenge. In this paper, the discontinuous Galerkin (DG) finite element method combined with the adaptive mesh refinement (AMR) is studied to solve the three dimensional Euler equations based on conforming unstructured tetrahedron meshes, that is according the equation solution variation to refine and coarsen grids so as to decrease total mesh number. The four space adaptive strategies are given and analyzed their advantages and disadvantages. The numerical examples show the validity of our methods.     

Convergence of the Euler–Maruyama method for stochastic differential equations with Markovian switching

Stochastic differential equations with Markovian switching (SDEwMSs), one of the important classes of hybrid systems, have been used to model many physical systems that are subject to frequent unpredictable structural changes. The research in this area has been both theoretical and applied. Most of SDEwMSs do not have explicit solutions so it is important to have numerical solutions. It is surprising that there are not any numerical methods established for SDEwMSs yet, although the numerical methods for stochastic differential equations (SDEs) have been well studied. The main aim of this paper is to develop a numerical scheme for SDEwMSs and estimate the error between the numerical and exact solutions. This is the first paper in this direction and the emphasis lies on the error analysis.     

CAS wavelet method for solving the fractional integro-differential equation with a weakly singular kernel

In this paper, a computational method for numerical solution of a class of integro-differential equations with a weakly singular kernel of fractional order which is based on Cos and Sin (CAS) wavelets and block pulse functions is introduced. Approximation of the arbitrary order weakly singular integral is also obtained. The fractional integro-differential equations with weakly singular kernel are transformed into a system of algebraic equations by using the operational matrix of fractional integration of CAS wavelets. The error analysis of CAS wavelets is given. Finally, the results of some numerical examples support the validity and applicability of the approach.     

A method for the numerical solution of singular integral equations with logarithmic singularities

A method of numerical solution of singular integral equations of the first kind with logarithmic singularities in their kernels along the integration interval is proposed. This method is based on the reduction of these equations to equivalent singular integral equations with Cauchy-type singularities in their kernels and the application to the latter of the methods of numerical solution, based on the use of an appropriate numerical integration rule for the reduction to a system of linear algebraic equations. The aforementioned method is presented in two forms giving slightly different numerical results. Furthermore, numerical applications of the proposed methods are made. Some further possibilities are finally investigated