非线性中子输运问题的蒙特卡罗模拟

针对考虑中子之间相互碰撞的非线性中子输运方程,提出一种线性化近似处理方法,导出相应的积分输运方程,利用蒙特卡罗方法求解此方程,数值实验表明算法的有效性,为研究超高能中子产生与输运问题提供了必要的模拟工具.     

中子中子碰撞产生超高能中子问题模拟

ICF的聚变靶丸中,高能中子相互之间发生碰撞将导致超高能中子的产生,推导中子中子碰撞与超高能中子产生的关系,研究利用蒙特卡罗方法确定碰撞后中子速度的方法,通过求解非线性中子输运方程及对DT聚变小球中的中子输运问题进行数值模拟,研究中子与中子碰撞产生超高能中子及其与聚变中子源强的关系问题.     

球几何中子输运保正线性间断有限元格式

针对球几何中子输运方程线性间断有限元方法计算的负中子通量问题,构造了保正线性间断有限元格式,该格式保持中子角通量0阶矩和1阶矩。现有方法计算中子角通量非负时,采用传统的线性间断有限元方法,求解线性方程组;原方法计算出现负通量,则采用构造的保正格式,求解非线性方程组。编制了球几何中子输运问题保正格式程序模块,并集成到应用程序。数值算例表明构造的保正格式计算的中子通量非负,有效降低数值误差,提高数值计算的精度。     

一维球几何菱形差分中子输运方程的扩散综合加速解法

本文导出了一维球几何定态中子输运方程菱形格式的扩散综合加速方程,并给出了差分公式。所给出的加速方法可以加速菱形格式的输运方程的迭代求解。并给出了部分模型的数值计算结果。     

有限元方法在中子测井数值模拟中的应用

讨论了有限元方法在中子测量井数值模拟中的应用。用有限元方法求解多群Ｐ１近似中子输运方程，编制了二维有限元程序ＦＥＭＬＯＧ，并对中子测井问题进行了数值计算，所得结果同国际通用二维ＳＮ程序的计算结果进行了比较，二者符合良好，但其计算速度比ＳＮ方法快得多。     

中子测井问题中输运方程的谱有限元方法

讨论了谱有限元方法在中子测井数值模拟中的应用,用球谐函数谱展开和有限元耦合方法求解Boltzmann中子输运方程,得到了这种耦合方法的收敛性。研制了三维有限元程序,实现了中子测井问题数值模拟的正演计算,实际算例表明此方法是有效的。     

提高超高能中子计算效率的蒙特卡罗抽样技巧

利用蒙特卡罗方法模拟中子与中子碰撞及超高能中子的产生与输运,原有抽样方法需要模拟大量样本方能使与超高能中子有关统计量的计算误差达到要求.针对ICF聚变靶中子输运问题的特点,发展一种"加权赌分裂抽样方法",以增加重要区域的抽样数、减少非重要区域的抽样数,同时通过权修正保证计算结果无偏.典型模型数值模拟计算结果显示,该方法能够有效增加聚变区的中子抽样数、增加中子相互碰撞数,使超高能中子通量计算误差显著降低,达到提高超高能中子计算效率的目的.     

一维辐射输运方程的近似求解

给出强激光与靶相互作用时带有反射边界条件的辐射输运方程的严格解,介绍了对辐射输运方程的一种近似简化以及数值求解辐射输运方程的一般方法,并给出在一种简单情况下进行数值计算的例子。     

中子输运方程基于界面修正的源迭代并行计算方法

中子输运方程的计算量非常大。在现有的计算机条件下，进行精密物理的数值模拟所需要的中子计算仍是非常的费时间和费内存的，不采用并行计算是难以承受的。并且，由于中子输运隐式离散纵标方法引起的数据强相关，以及计算过程必须严格沿中子运动方向进行(否则会出现计算不稳定)，因此会出现相当严重的算法同步的问题，使得隐式格式在大型并行计算机上实施时所能得到的并行度十分有限，严格限制了其实现具有高并行度的迭代计算的可能性。因此，对中子输运方程隐式差分格式进行并行改造是十分必要的。     

解二维输运问题的三角网间断有限元法

本文研究了求解二维柱对称几何中子输运问题的三角网间断有限元法,证明了数值方法的稳定性,给出了一些数值计算结果,并与精确解和DSN等方法的计算结果进行了比较,取得了满意的结果。     

Numerical method satisfying the first two conservation laws for the Korteweg–de Vries equation

In this paper, we develop a finite-volume scheme for the KdV equation which conserves both the momentum and energy. The main ingredient of the method is a numerical device we developed in recent years that enables us to construct numerical method for a PDE that also simulates its related equations. In the method, numerical approximations to both the momentum and energy are conservatively computed. The operator splitting approach is adopted in constructing the method in which the conservation and dispersion parts of the equation are alternatively solved; our numerical device is applied in solving the conservation part of the equation. The feasibility and stability of the method is discussed, which involves an important property of the method, the so-called Jensen condition. The truncation error of the method is analyzed, which shows that the method is second-order accurate. Finally, several numerical examples, including the Zabusky–Kruskal’s example, are presented to show the good stability property of the method for long-time numerical integration.     

含强间断导热系数的三维扩散方程数值计算

针对惯性约束聚变领域需要求解的含强间断导热系数的扩散方程,借鉴各向异性导热的思想,给出了一种导热系数定义在网格节点的支撑算子改进格式。改进后的方法保留了支撑算子格式的算子相容性优点。数值算例表明,新方法可以很好地处理含强间断和强非线性的热扩散问题,计算结果保持较高的精度,适合于辐射流体力学问题的模拟。     

Two-Dimensional Legendre Wavelets for Solving Time-Fractional Telegraph Equation

In this paper, we develop an accurate and efficient Legendre wavelets method for numerical solution of the well known time-fractional telegraph equation. In the proposed method we have employed both of the operational matrices of fractional integration and differentiation to get numerical solution of the time-telegraph equation. The power of this manageable method is confirmed. Moreover, the use of Legendre wavelet is found to be accurate, simple and fast.     

Numerical method for two dimensional fractional reaction subdiffusion equation

In this paper, a new numerical algorithm for solving two dimensional fractional reaction subdiffusion equation is proposed. The stability and convergency of this method are investigated by the Fourier analysis. Theoretical analysis and numerical experiment demonstrate that the proposed method is effective for solving the two dimensional fractional reaction subdiffusion equation.     

分析完全电离等离子体扩散问题的边界单元法

一、扩散方程 完全电离等离子体的扩散问题,可归结为下面的微分方程定解问题。 在内; 在Γ_1上; 在Γ_2上; 式中,A=ηκB~Z/B~Z,n=n(r,t)是等离子体密度,κ是玻尔兹曼常数,T是绝对温度,B是磁感应强度,η是电导率,Ω是由Γ=Γ_1 Γ_2界定的区域,ω是边界的外法向方向,和是边界上的已知函数。     

Numerical simulation of the soliton solutions for a complex modified Korteweg—de Vries equation by a finite difference method

In this paper,a Crank-Nicolson-type finite difference method is proposed for computing the soliton solutions of a complex modifed Korteweg de Vries(MKdV)equation(which is equivalent to the Sasa-Satsuma equation)with the vanishing boundary condition.It is proved that such a numerical scheme has the second order accuracy both in space and time,and conserves the mass in the discrete level.Meanwhile,the resuling scheme is shown to be unconditionally stable via the von Nuemann analysis.In addition,an iterative method and the Thomas algorithm are used together to enhance the computational efficiency.In numerical experiments,this method is used to simulate the single-soliton propagation and two-soliton collisions in the complex MKdV equation.The numerical accuracy,mass conservation and linear stability are tested to assess the scheme's performance.     

Numerical solution of fractional differential equations via a Volterra integral equation approach

The main focus of this paper is to present a numerical method for the solution of fractional differential equations. In this method, the properties of the Caputo derivative are used to reduce the given fractional differential equation into a Volterra integral equation. The entire domain is divided into several small domains, and by collocating the integral equation at two adjacent points a system of two algebraic equations in two unknowns is obtained. The method is applied to solve linear and nonlinear fractional differential equations. Also the error analysis is presented. Some examples are given and the numerical simulations are also provided to illustrate the effectiveness of the new method.     

Numerical Method for the Time Fractional Fokker-Planck Equation

In this paper, a new numerical algorithm for solving the time fractional Fokker-Planck equation is proposed. The analysis of local truncation error and the stability of this method are investigated. Theoretical analysis and numerical experiments show that the proposed method has higher order of accuracy for solving the time fractional Fokker-Planck equation.     

Direct discontinuous Galerkin method for the generalized Burgers—Fisher equation

In this study, we use the direct discontinuous Galerkin method to solve the generalized Burgers-Fisher equation. The method is based on the direct weak formulation of the Burgers-Fisher equation. The two adjacent cells are jointed by a numerical flux that includes the convection numerical flux and the diffusion numerical flux. We solve the ordinary differential equations arising in the direct Galerkin method by using the strong stability preserving Runge-Kutta method. Numerical results are compared with the exact solution and the other results to show the accuracy and reliability of the method.     

Modified Burgers equation by local discontinuous Galerkin method

In this paper, we present the local discontinuous Galerkin method for solving Burgers’ equation and the modified Burgers’ equation. We describe the algorithm formulation and practical implementation of the local discontinuous Galerkin method in detail. The method is applied to the solution of the one-dimensional viscous Burgers’ equation and two forms of the modified Burgers’ equation. The numerical results indicate that the method is very accurate and efficient.