Discontinuous Galerkin method coupled with an integral representation for solving the three-dimensional time-harmonic Maxwell equations

In this paper, we present a mathematical and numerical studies of the three-dimensional time-harmonic Maxwell equations. The problem is solved by a discontinuous Galerkin DG method coupled with an integral representation. This study was completed by some numerical tests to justify the effectiveness of the proposed approach. The numerical simulation was done by an iterative solver implemented in FORTRAN.     

A domain decomposition method for solving the three-dimensional time-harmonic Maxwell equations discretized by discontinuous Galerkin methods

We present here a domain decomposition method for solving the three-dimensional time-harmonic Maxwell equations discretized by a discontinuous Galerkin method. In order to allow the treatment of irregularly shaped geometries, the discontinuous Galerkin method is formulated on unstructured tetrahedral meshes. The domain decomposition strategy takes the form of a Schwarz-type algorithm where a continuity condition on the incoming characteristic variables is imposed at the interfaces between neighboring subdomains. A multifrontal sparse direct solver is used at the subdomain level. The resulting domain decomposition strategy can be viewed as a hybrid iterative/direct solution method for the large, sparse and complex coefficients algebraic system resulting from the discretization of the time-harmonic Maxwell equations by a discontinuous Galerkin method.     

非线性波动方程的新数值迭代方法

提出了一种新的求解非线性波动方程的数值迭代法,它是一种半解析的方法.与完全的数值计算方法扰法相比,它能够考虑各阶谐波的相互作用,且能够满足能量守恒定律.用它研究了非线性声波在液体中的传播性质,结果表明,在微扰法适用的声强范围内迭代法也适用,在微扰法不适用的一个较宽的声强范围内迭代法依然适用.     

New method of solving singularly perturbed equations

A system of singularly perturbed equations is considered. It is shown that the representation of the system in Hamiltonian form allows an approximate solution to be obtained on the basis of the theory of canonical transformations.Translated from Izvestiya Vysshikh Uchebnyh Zavedenii, Fizika, No. 7, pp. 36–40, July, 1984.     

Symplectic and multisymplectic numerical methods for Maxwell’s equations

In this paper, we compare the behaviour of one symplectic and three multisymplectic methods for Maxwell’s equations in a simple medium. This is a system of PDEs with symplectic and multisymplectic structures. We give a theoretical discussion of how some numerical methods preserve the discrete versions of the local and global conservation laws and verify this behaviour in numerical experiments. We also show that these numerical methods preserve the divergence. Furthermore, we extend the discussion on dispersion for (multi)symplectic methods applied to PDEs with one spatial dimension, to include anisotropy when applying (multi)symplectic methods to Maxwell’s equations in two spatial dimensions. Lastly, we demonstrate how varying the Courant–Friedrichs–Lewy (CFL) number can cause the (multi)symplectic methods in our comparison to behave differently, which can be explained by the study of backward error analysis for PDEs.     

The Huygens subgridding for the numerical solution of the Maxwell equations

The Huygens subgridding (HSG) is a subgridding technique developed for the numerical solution of the Maxwell equations. It relies on the theoretical equivalence of any physical volume with two or more fictitious volumes connected by equivalent currents. The application of this concept to the finite-difference time-domain (FDTD) method has been previously published in the one dimensional and two dimensional cases. In this paper the HSG is extended to the general three dimensional case, the exchange of the electromagnetic energy between the two FDTD grids is investigated theoretically, and some modifications to the HSG algorithm are presented with the objective of simplifying its implementation.     

求解微分方程的Hojman方法

首先，将Hojman用于求解二阶微分方程组守恒量的方法推广并应用于一阶微分方程组，特别是奇数维微分方程组的积分问题.然后，证明 Hojman定理是本文定理的特殊情形.最后，举例说明结果的应用. 关键词： 微分方程 Hojman定理 守恒量 积分     

A method for solving nonlinear equations

A direct method for solving constant coefficient nonlinear evolutional problems of mathematical physics extending the Hirota method to the case of degeneracy of k-parameter is described. An application of this method to the problem of finding the solutions describing coupled states in φ⁴-theory with damping is considered.     

一个新的求解点堆中子动力学方程组的数值方法

使用中子密度一阶泰劳多项式分段近似技术,给出一个新的求解点堆中子动力学方程组的数值方法并采用全隐格式以克服方程组的刚性,同时确保解的必要精度。数值结果表明:在隐式一阶多项式近似下,对合适的反应性输入能够取得足够精确的结果。当反应性给定时,对于求解反应堆动力学问题,能给出一个简法的计算过程。     

Higher-order implicit strong numerical schemes for stochastic differential equations

Higher-order implicit numerical methods which are suitable for stiff stochastic differential equations are proposed. These are based on a stochastic Taylor expansion and converge strongly to the corresponding solution of the stochastic differential equation as the time step size converges to zero. The regions of absolute stability of these implicit and related explicit methods are also examined.     

电磁场2.5维时域间断有限元方法

介质沿空间固定方向均匀分布的结构在电磁导波器件中有十分广泛的应用,对这类器件的分析通常被称为2.5D电磁问题。利用器件在固定方向介质分布均匀的特点,将电磁场量沿该方向进行空间傅里叶变换,可以把对三维问题的分析转化为两维问题求解,从而极大地减小计算开销。针对传统基于差分的2.5D电磁场算法在弯曲形状逼近上有阶梯误差的缺陷,本文提出了基于三角形网格的2.5D时域间断有限元方法（DGTD）,并用它模拟了电偶极子与光纤的耦合效率和光子晶体光纤的色散特性。与基于规则网格的2.5D差分方法进行对比。结果表明,文中建立的2.5D DGTD方法对弯曲形状的模拟更加逼真,计算内存占用最大减少10.4%,计算精度最大相差0.011%,计算时间缩短74.9%,计算效率提高。     

Functional method for solving gravitational field equations

A new method for solving the equations of a gravitational field (G field), which is based on functional transformations (FT), which replace holonomic transformations (HT), is discussed. The method is equivalent to nonholonomic transformations (NT) of the field functions and makes it possible to use a known solution to construct a new set of solutions in the same coordination, which contains a fixed number (up to 12) of arbitrary functions. The FT method can be used to solve the Cauchy problem for a non-steady-state G field, to eliminate singularities in the field, etc.This article is published as a matter for discussion.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 50–55, February, 1979.     

Whittaker方程的场方法

用场方法来求解Whittaker方程.将一个场变量取作为其余场变量和时间的函数并对这个函数建立基本偏微方程.如能求得它的完全积分，那么Whittaker方程的解可由解代数方程来得到. 关键词： 场变量 基本偏微分方程 场方法 积分     

A new method for solving perturbation equations

A new, generally applicable technique is described for constructing solutions of a coupled linear system of partial differential equations when a decoupled equation has been derived. This method has already yielded an extremely simple derivation of perturbation formulas given by Cohen and Kegeles and by Chrzanowski as well as new formulas for the complete solutions of the coupled Einstein-Maxwell equations describing Reissner-Nordström perturbations. It is hoped that it will prove valuable for many other applications.This essay received the third award from the Gravity Research Foundation for the year 1979-Ed.     

Variational iteration method for solving biharmonic equations

By means of the variational iteration method the numerical solution of biharmonic equations are obtained. Biharmonic equation has significant applications in physics and engineering. There is a difficulty to solve the biharmonic equation due to the existence of fourth order derivatives. For this reason, we use the variational iteration method to solve this equation. Test problems, are used to validate this algorithm which is found to be more accurate and efficient than previous ones.     

Hamilton--Jacobi method for solving ordinary differential equations

The Hamilton--Jacobi method for solving ordinary differential equations is presented in this paper. A system of ordinary differential equations of first order or second order can be expressed as a Hamilton system under certain conditions. Then the Hamilton--Jacobi method is used in the integration of the Hamilton system and the solution of the original ordinary differential equations can be found. Finally, an example is given to illustrate the application of the result.     

A method for solving phase transition equations

The kinetics of phase transitions is calculated on the assumption of mechanical and temperature equilibrium between mixture components. The simulation of kinetics is based on solving relaxation-type kinetic equations. The deviator components of the constituents of a mixture of phases are assumed to be equal. When a material is deformed, several phase transitions can occur simultaneously. The determination of the thermodynamic parameters of mixture components is based on solving linearized equations for component pressure and temperature increments at known specific volume and mixture energy increments. We suggest an effective algorithm for solving these equations. The algorithm is based on an explicit Lagrangian difference scheme and takes into account pressure and temperature discrepancies. The algorithm is implemented in the SPRUT two-dimensional program. We present the results of calculations for one-and two-dimensional problems with and without taking into account the α-? transition in iron.     

Variational iteration method for solving integro-differential equations

In this Letter, the variational iteration method is applied to solve integro-differential equations. Some examples are given to illustrate the effectiveness of the method, the results show that the method provides a straightforward and powerful mathematical tool for solving various integro-differential equations.     

Lagrange--Noether method for solving second-order differential equations

The purpose of this paper is to provide a new method called the Lagrange--Noether method for solving second-order differential equations. The method is, firstly, to write the second-order differential equations completely or partially in the form of Lagrange equations, and secondly, to obtain the integrals of the equations by using the Noether theory of the Lagrange system. An example is given to illustrate the application of the result.