任意阶隐式指数时程差分多步法及其在非线性系统中的应用

对非线性系统提出了任意阶隐式指数时程差分多步法，实现了任意阶次指数时程差分预测 校正算法.发展完善了指数时程差分法.将新算法应用于非线性系统，取得了较好的效果.数值结果表明隐式指数时程差分多步法很好地修正了显式指数时程差分多步法，隐式指数时程差分多步法是一种高精度、高效率的方法. 关键词： 非线性系统 任意阶隐式指数时程差分多步法 混沌     

非线性系统的任意项精细积分外插多步法及其在混沌数值分析中的应用

对非线性系统提出了高精度的精细积分任意项外插多步法的计算公式.本方法只需增加插值 项数即可提高计算精度，同时不会增加过大的计算量，发展完善了精细积分法.将本方法应 用于混沌方程中，取得了较好的效果.数值计算结果表明，该方法是一种高精度、高效率的 方法，在求解混沌系统上比传统方法有很大的优势. 关键词： 非线性系统 任意项精细积分多步法 混沌系统     

任意阶球谐函数法求解辐射传输问题的误差分析

针对各向异性散射介质内的红外辐射传输开展数值方法研究.应用球谐函数展开推导了一维吸收、发射、散射性灰介质的辐射传输近似方程组,采用差分方法结合三对角矩阵解法建立了任意阶辐射传输近似方程数值解法.并通过计算算例验证了高阶PN方法的数值无关性问题.计算结果表明,高阶球谐函数数值解与理论解吻合得很好,同时计算验证了不同展开阶...     

局部时间步长间断有限元方法求解三维欧拉方程

使用间断有限元方法求解三维流体力学方程.空间剖分采用非结构四面体网格,为了克服显格式在单元网格尺寸差别较大时计算效率低下的问题,在格式中采用局部时间步长技术(LTS),即控制方程在空间、时间上积分得到一种单步格式,既可以局部计算每个单元又避免了Runge-Kutta高精度格式处理三维问题时存储量过大的问题.为了提高流体力学方程计算精度,在计算单元边界的数值流通量时使用任意高阶精度方法(ADER).数值算例表明格式稳定有效.     

基于通量重构高阶算法的无矩阵预处理求解

基于通量重构形式的高阶算法,在保持间断Galerkin算法局部重构特性和非结构网格中任意高阶精度优点的同时,其计算量大大减小,且具有形式简单、灵活性高等特点。使用显式Runge-Kutta法,隐式非线性LU-SGS法,以及使用无矩阵预处理的广义极小残值法(generalized minimal residual,GMRES)进行求解,并使用p型多重网格在低阶次上光顺低频误差以加快求解。一至四阶精度结果显示使用p型多重网格对显式Runge-Kutta求解以及LU-SGS均具有明显的加速效果,而基于无矩阵预处理的GMRES解法具有更好的稳定性和更快的求解速度。本文提出的基于Gauss-Seidel迭代的无矩阵预处理方法,具有高效和稳定的特征,存储量大大小于ILU预处理。     

随机电磁高阶Bessel-Gaussian光束在海洋湍流中的传输特性

利用广义惠更斯-菲涅耳衍射积分公式得到了随机电磁高阶Bessel-Gaussian光束在海洋湍流中传输的交叉谱密度矩阵的一般表达式,通过数值计算主要研究了随机电磁高阶Bessel-Gaussian光束在海洋湍流中传输时其在远场输出面的统计特性的变化,包括归一化光谱强度、光谱偏振度、两点的光谱相干度等.数值模拟结果显示海洋湍流能够对随机电磁高阶Bessel-Gaussian光束的归一化光谱强度分布产生影响,随着传输距离的增加,零阶Bessel-Gaussian光束中心出现凹陷,高阶Bessel-Gaussian光束中心会变平坦继而又凹陷下去,不管零阶还是高阶,当传输距离增加到足够远,光强分布都会演变成最终的类高斯分布.x轴上各点的偏振度改变与相干长度δ_(xx),δ_(yy)以及海洋湍流参数有关.x轴上任意一点和原点这两点的光谱相干度也随x的增加而呈振荡变化,并且海洋的均方温度耗散率χT对光谱相干度有影响.     

水平层状各向异性介质中电磁场并矢Green函数的一种高效算法

利用高阶窗函数结合连分式展开等技术研究并建立一种水平层状各向异性介质中电磁场并矢Green函数的快速有效算法. 首先借助于高阶窗函数将构成并矢Green函数的Sommerfeld积分转化成广义快速下降路径上积分,并给出高阶窗函数Hankel变换的一种新的更高阶幂级数展开式以及严格的Lommel函数表达式,以满足在全空间上高精度计算并矢Green函数的要求. 在此基础上,用Bessel函数的零点将积分路径划分成一系列小区间并通过改进的自适应Gauss求积公式确定各个小区间上的积分值,然后引入连分式展开法对各个区间上的积分值求和,从而使整个积分的收敛效率得到大大提高. 最后通过数值结果验证本方法的有效性. 关键词： 高阶窗函数 连分式展开 并矢Green函数 层状各向异性介质     

各向异性介质三维电磁响应模拟的Ho-GEBA算法

本文基于积分方程法研究并建立了一种模拟横向同性介质中任意各向异性异常 体三维电磁响应的高阶广义扩展Born近似(Ho-GEBA)算法. 首先利用逐次迭代技术给出积分方程的广义级数展开解, 为保证其收敛性, 引入一种各向异性条件下满足压缩映射的迭代算子. 然后利用异常体区域分解技术, 并结合扩展Born近似原理, 得到各向异性介质三维电磁响应的Ho-GEBA解. 为提高效率, 计算过程中采用并矢Green函数的解析表达式. 最后通过数值计算实例对比验证了本文算法的有效性. 关键词： 高阶广义扩展Born近似 积分方程 电磁模拟 解析Green函数     

高阶Schrdinger方程的高精度辛格式

提出由第三类生成函数法构造高阶Schr dinger方程ut=i(-1)m2mux2m的高精度辛格式.首先,给出它的典则Hamilton方程组;然后,成功地克服了本质上是困难的高阶变分导数的计算,并利用第三类生成函数法得到在时间方向具有任意阶精度的半离散方程,进而得到原始方程相关的修正方程的离散形式,最后得到各种精度的辛格式.数值结果表明该格式是有效的,具有高精度及良好的长时间数值行为等特性.     

空间分数阶导数“反常”扩散方程数值算法的比较

分别采用显式差分格式、隐式差分格式以及Crank-Nicholson差分格式数值求解空间分数阶导数,并从局部截断误差、稳定性、计算量三个方面进行比较分析;通过数值算例验证分析结果.     

Transient heat conduction analysis using the MLPG method and modified precise time step integration method

The meshless local Petrov–Galerkin (MLPG) method in conjunction with the modified precise time step integration method in the time domain is proposed for transient heat conduction analysis in this paper. The MLPG method is often referred to as a truly meshless method because it requires no elements or background cells for either field interpolation or background integration. Local weak forms are developed using weighted residual method locally from the partial differential equation of transient heat conduction. In order to simplify the treatment of essential boundary conditions, the natural neighbour interpolation (NNI) is employed for the construction of trial functions. Moreover, the three-node triangular FEM shape functions are taken as test functions to reduce the order of integrands involved in domain integrals. The semi-discrete heat conduction equation is solved numerically with modified precise time step integration method in the time domain. The availability and accuracy of the present method for transient heat conduction analysis are tested through numerical examples.     

A fully second order implicit/explicit time integration technique for hydrodynamics plus nonlinear heat conduction problems

We present a fully second order implicit/explicit time integration technique for solving hydrodynamics coupled with nonlinear heat conduction problems. The idea is to hybridize an implicit and an explicit discretization in such a way to achieve second order time convergent calculations. In this scope, the hydrodynamics equations are discretized explicitly making use of the capability of well-understood explicit schemes. On the other hand, the nonlinear heat conduction is solved implicitly. Such methods are often referred to as IMEX methods [2], [1], [3]. The Jacobian-Free Newton Krylov (JFNK) method (e.g. [10], [9]) is applied to the problem in such a way as to render a nonlinearly iterated IMEX method. We solve three test problems in order to validate the numerical order of the scheme. For each test, we established second order time convergence. We support these numerical results with a modified equation analysis (MEA) [21], [20]. The set of equations studied here constitute a base model for radiation hydrodynamics.     

A second order self-consistent IMEX method for radiation hydrodynamics

We present a second order self-consistent implicit/explicit (methods that use the combination of implicit and explicit discretizations are often referred to as IMEX (implicit/explicit) methods ,  and ) time integration technique for solving radiation hydrodynamics problems. The operators of the radiation hydrodynamics are splitted as such that the hydrodynamics equations are solved explicitly making use of the capability of well-understood explicit schemes. On the other hand, the radiation diffusion part is solved implicitly. The idea of the self-consistent IMEX method is to hybridize the implicit and explicit time discretizations in a nonlinearly consistent way to achieve second order time convergent calculations. In our self-consistent IMEX method, we solve the hydrodynamics equations inside the implicit block as part of the nonlinear function evaluation making use of the Jacobian-free Newton Krylov (JFNK) method ,  and . This is done to avoid order reductions in time convergence due to the operator splitting. We present results from several test calculations in order to validate the numerical order of our scheme. For each test, we have established second order time convergence.     

New embedded pairs of explicit Runge-Kutta methods with FSAL properties adapted to the numerical integration of oscillatory problems

A higher order Runge-Kutta (pair) method specially adapted to the numerical integration of IVPs with oscillatory solutions is presented. This method is based on the adapted methods proposed by Franco (see Ref. [J.M. Franco, Appl. Numer. Math. 50 (2004) 427]). We give explicit method (up to order 5) as well as pairs of embedded Runge-Kutta methods of order 5 and 4 designed using the FSAL properties. The stability of the new methods is analyzed. The numerical experiments are carried out to show the efficiency and robustness of our methods in comparison with some efficient methods.     

光子晶体传输特性的时域精细积分法分析

将精细积分法应用于时域有限差分法中,提出了一种求解光子晶体传输特性的时域精细积分法,并对其计算精度及稳定性进行了分析.从一阶麦克斯韦方程出发,在空间上采用Yee元胞进行差分离散,结合吸收边界条件及激励源表达式将方程整理为标准的一阶常微分方程组形式.通过时间步长的精细划分和指数矩阵的加法定理,在时间上利用精细积分法对齐次微分方程进行积分求解,并结合激励向量的特解得到空间离散的场分量,最终通过傅里叶变换求得方程的解.利用时域精细积分法对光子晶体进行了实例计算,并将其结果分别与时域有限差分法和四阶龙格库塔法在精度、稳定性等方面进行了比较,结果表明时域精细积分法具有更高的计算精度,并且克服了时域有限差分法以及四阶龙格库塔法在计算稳定性上对时间步长的限制.提出的方法具有精确、稳定的特点,为光子晶体传输特性的研究提供了一种新的有效的分析方法.     

Fast simulation code for heating, phase changes and dopant diffusion in silicon laser processing using the alternating direction explicit (ADE) method

Enhancements to our existing finite-differences code for the simulation of laser heating, melting and evaporation of silicon are presented. The Knudsen evaporation model has been added to the previously used enthalpy-based model in order to simulate laser pulses with pulse lengths down to a few nanoseconds. Fick’s diffusion law has also been incorporated allowing laser doping by dopant diffusion in silicon melt to be described. Finally, the basic equations for the alternating direction explicit method (ADE) have been adapted to consider nonlinear temperature-enthalpy relations, thus including affects due to phase changes. This improved the simulation speed by up to factor of 100 compared to standard explicit and implicit time integration methods. Details of the ADE algorithm and numerical stability issues are presented in this paper. Validation of the code is presented by comparing to different integration methods and to experimental results. The final code successfully simulates melting, evaporation and dopant diffusion by multiple laser pulses in three dimensions in an acceptable computing time.     

Efficient numerical evaluation of Feynman integrals

Feynman loop integrals are a key ingredient for the calculation of higher order radiation effects, and are responsible for reliable and accurate theoretical prediction. We improve the efficiency of numerical integration in sector decomposition by implementing a quasi-Monte Carlo method associated with the CUDA/GPU technique. For demonstration we present the results of several Feynman integrals up to two loops in both Euclidean and physical kinematic regions in comparison with those obtained from FIESTA3. It is shown that both planar and non-planar two-loop master integrals in the physical kinematic region can be evaluated in less than half a minute with O(10~(-3))accuracy, which makes the direct numerical approach viable for precise investigation of higher order effects in multiloop processes, e.g. the next-to-leading order QCD effect in Higgs pair production via gluon fusion with a finite top quark mass.     

Nonuniform time-step Runge–Kutta discontinuous Galerkin method for Computational Aeroacoustics

With many superior features, Runge–Kutta discontinuous Galerkin method (RKDG), which adopts Discontinuous Galerkin method (DG) for space discretization and Runge–Kutta method (RK) for time integration, has been an attractive alternative to the finite difference based high-order Computational Aeroacoustics (CAA) approaches. However, when it comes to complex physical problems, especially the ones involving irregular geometries, the time step size of an explicit RK scheme is limited by the smallest grid size in the computational domain, demanding a high computational cost for obtaining time accurate numerical solutions in CAA. For computational efficiency, high-order RK method with nonuniform time step sizes on nonuniform meshes is developed in this paper. In order to ensure correct communication of solutions on the interfaces of grids with different time step sizes, the values at intermediate-stages of the Runge–Kutta time integration on the elements neighboring such interfaces are coupled with minimal dissipation and dispersion errors. Based upon the general form of an explicit p-stage RK scheme, a linear coupling procedure is proposed, with details on the coefficient matrices and execution steps at common time-levels and intermediate time-levels. Applications of the coupling procedures to Runge–Kutta schemes frequently used in simulation of fluid flow and acoustics are given, including the third-order TVD scheme, and low-storage low dissipation and low dispersion (LDDRK) schemes. In addition, an analysis on the stability of coupling procedures on a nonuniform grid is carried out. For validation, numerical experiments on one-dimensional and two-dimensional problems are presented to illustrate the stability and accuracy of proposed nonuniform time-step RKDG scheme, as well as the computational benefits it brings. Application to a one-dimensional nonlinear problem is also investigated.     

干斜压大气拉格朗日原始方程组的半解析解法和非线性密度流数值试验

以差分方程代替微分方程给大气原始方程组求解带来了诸多难以解决的问题, 对于(半)拉格朗日模式来说质点轨迹的计算与Helmholtz方程的求解是两大难题. 本文通过对气压变量代换, 并在积分时间步长内将原始方程组线性化, 近似为常微分方程组, 求出方程组的半解析解, 再采用精细积分法求解半解析解. 半解析方法可同时计算风、气压和位移, 无需求解Helmholtz方程, 质点的位移采用积分风的半解析解得到, 相比采用风速外推的计算方法, 半解析方法更科学合理. 非线性密度流试验检验表明: 半解析模式能够清晰地模拟Kelvin-Helmholtz 切变不稳定涡旋的发生和发展过程; 模拟的气压场和风场环流结构与标准解非常相似, 且数值解是收敛的, 同时, 总质量和总能量具有较好的守恒性. 试验初步证明了采用半解析方法求解大气原始方程组是可行的, 为大气数值模式的构建提供了一个新的思路.     

Locally implicit discontinuous Galerkin method for time domain electromagnetics

In the recent years, there has been an increasing interest in discontinuous Galerkin time domain (DGTD) methods for the solution of the unsteady Maxwell equations modeling electromagnetic wave propagation. One of the main features of DGTD methods is their ability to deal with unstructured meshes which are particularly well suited to the discretization of the geometrical details and heterogeneous media that characterize realistic propagation problems. Such DGTD methods most often rely on explicit time integration schemes and lead to block diagonal mass matrices. However, explicit DGTD methods are also constrained by a stability condition that can be very restrictive on highly refined meshes and when the local approximation relies on high order polynomial interpolation. An implicit time integration scheme is a natural way to obtain a time domain method which is unconditionally stable but at the expense of the inversion of a global linear system at each time step. A more viable approach consists of applying an implicit time integration scheme locally in the refined regions of the mesh while preserving an explicit time scheme in the complementary part, resulting in an hybrid explicit–implicit (or locally implicit) time integration strategy. In this paper, we report on our recent efforts towards the development of such a hybrid explicit–implicit DGTD method for solving the time domain Maxwell equations on unstructured simplicial meshes. Numerical experiments for 3D propagation problems in homogeneous and heterogeneous media illustrate the possibilities of the method for simulations involving locally refined meshes.