二维输运方程离散格式的对称性

讨论了二维柱几何非定态中子输运方程离散格式的对称性问题，在几何空间和相空间连续的情况下，证明了时间离散方程的一维球对称性；而在时间和相空间离散的情况下，阐述了格式不具有一维球对称性；对时间和相空间离散情况下的几何空间间断有限元方程，得到了左右对称性。     

Lagrange柱坐标高阶中心型守恒格式

提出Lagrange柱坐标高阶中心型守恒格式.基于用对守恒律的单调迎风算法(MUSCL)构造的高阶子网格压力,引入了柱坐标高阶体权子网格力和柱坐标高阶面权子网格力,构造了柱坐标高阶体权中心型守恒格式和柱坐标高阶面权中心型格式.柱坐标高阶体权中心型守恒格式满足动量守恒、能量守恒,但不能确定保持一维球对称性.柱坐标高阶面权中心型格式满足能量守恒,保持一维球对称性.两种格式里,格点速度以与网格面的数值通量相容的方式计算.对Saltzman活塞问题等进行了数值模拟,数值结果显示Lagrange柱坐标高阶中心型守恒格式的有效性和精确性.     

粒子输运方程的子网格平衡格式的稳定性和收敛性

本文研究四边形网格上求解粒子输运方程的有限体积格式,其中角方向变量采用离散纵标(Sn)方法,空间离散采用子网格平衡(SCB)格式.利用能量估计方法,证明了在正交网格上该格式的稳定性和离散解的收敛性.数值实验结果验证了格式的稳定性和离散解的收敛性.     

保持球对称的相容流体算法的构造

重点研究如何在二维柱坐标系中保持流体运动的球对称问题.该问题需要对整个流体算法做细致的分析与设计,从而获得不同的方法.基于Caramana介绍的修正压强梯度算子构造了保持球对称的相容流体算法,该算法还包括了边人工粘性方法、子网格压力的计算方法,以及预估步与校正步相结合的策略.并且,考虑了总能量的守恒性,讨论了网格节点质量保持不变的必要性,以及基于相容格式证明了边人工粘性的耗散性.数值模拟结果与已知数据的比对验证了程序的正确性、稳定性.     

Robin型边界阻尼波动方程的半离散一致指数稳定逼近

对一维Robin型边界阻尼波动方程构造了一个新的等距网格上的半离散有限差分格式,该格式没有数值粘性项,且可以保持系统的一致指数稳定性.通过引进一个新的辅助函数,利用Lyapunov函数方法证明了半离散格式的一致指数稳定性.数值实验验证了理论结果.     

粘性机制下一类熵相容格式的对比研究

研究了一种人工和物理耗散机制下的离散熵相容格式,探讨数值粘性和物理粘性的大小以及它们所起的作用.所得结论是:在激波捕捉的过程中,粘性系数越大,则无需加入人工粘性项;粘性系数较小时,除了物理粘性项,还需要加入人工粘性项来得到熵相容格式.首先研究了一维粘性Burgers方程离散熵相容格式,再将其推广至Navier-Stokes方程.数值算例采用空间半离散格式,并结合显式三步三阶Runge-Kutta(RK3)方法进行时间推进.这两类方程的数值结果表明,最终选取的熵相容格式能够准确地捕捉到激波.     

一维Allen-Cahn方程紧差分格式的离散最大化原则和能量稳定性研究

本文主要研究相场模拟中的Allen-Cahn模型,考虑一维Allen-Cahn方程紧差分方法的数值逼近.建立具有O(∫τ2+h4)精度的全离散紧差分格式,证明在合理的步长比和时间步长的约束下,其数值解满足离散最大化原则,在此基础上,研究了全离散格式的能量稳定性.最后给出数值算例.     

极坐标系下非分裂PML及时域有限元实现

在弹性波传播的数值模拟中,吸收边界被广泛应用于截取有限空间进行无限空间问题的分析.完全匹配层（perfect matched layer, PML）吸收边界较其他吸收边界条件具有更优越的吸收性能,已被成功应用于直角坐标系下的弹性波方程正演模拟.考虑极坐标系下二阶弹性波动方程,通过采用辅助函数的方法,提出了一种非分裂格式的完全匹配层吸收边界条件.并且基于Galerkin近似技术,给出了非对称以及轴对称条件下的时域有限元计算格式.通过数值算例分析了该极坐标系下分裂格式的完全匹配层吸收边界的有效性.     

一类多维非线性Schrdinger方程及其方程组的数值计算问题

本文对一类多维非线性Schrdinger方程组提出了Galerkin有限元法,并由此证明了广义解的存在性.还对一类多维非线性Schrdinger方程采用有限差分法,证明了相应的四点隐式和六点隐式差分格式的收敛性和稳定性.我们对二维平面和一维球、柱对称非线性Schrdinger方程进行了数值计算,得到非线性Schrdinger方程多维孤立波坍塌(Collapse)的具体图象。     

保持球对称的相容流体算法的构造

重点研究如何在二维柱坐标系中保持流体运动的球对称问题．该问题需要对整个流体算法做细致的分析与设计,从而获得不同的方法．基于Caramana介绍的修正压强梯度算子构造了保持球对称的相容流体算法,该算法还包括了边人工粘性方法、子网格压力的计算方法,以及预估步与校正步相结合的策略．并且,考虑了总能量的守恒性,讨论了网格节点质量保持不变的必要性,以及基于相容格式证明了边人工粘性的耗散性．数值模拟结果与已知数据的比对验证了程序的正确性、稳定性.     

A cell-centered lagrangian scheme in two-dimensional cylindrical geometry

A new Lagrangian cell-centered scheme for two-dimensional compressible flows in planar geometry is proposed by Maire et al.The main new feature of the algorithm is that the vertex velocities and the numerical fluxes through the cell interfaces are all evaluated in a coherent manner contrary to standard approaches.In this paper the method introduced by Maire et al.is extended for the equations of Lagrangian gas dynamics in cylindrical symmetry.Two different schemes are proposed,whose difference is that one uses volume weighting and the other area weighting in the discretization of the momentum equation.In the both schemes the conservation of total energy is ensured,and the nodal solver is adopted which has the same formulation as that in Cartesian coordinates.The volume weighting scheme preserves the momentum conservation and the area-weighting scheme preserves spherical symmetry.The numerical examples demonstrate our theoretical considerations and the robustness of the new method.     

Node-pair finite volume/finite element schemes for the Euler equation in cylindrical and spherical coordinates

A numerical scheme is presented for the solution of the compressible Euler equations in both cylindrical and spherical coordinates. The unstructured grid solver is based on a mixed finite volume/finite element approach. Equivalence conditions linking the node-centered finite volume and the linear Lagrangian finite element scheme over unstructured grids are reported and used to devise a common framework for solving the discrete Euler equations in both the cylindrical and the spherical reference systems. Numerical simulations are presented for the explosion and implosion problems with spherical symmetry, which are solved in both the axial–radial cylindrical coordinates and the radial–azimuthal spherical coordinates. Numerical results are found to be in good agreement with one-dimensional simulations over a fine mesh.     

Radiant Heat Transfer Through an Aerosol Suspended in a Transparent Gas

The scattering of radiation by an aerosol in the cover-gas regionof a fast breeder nuclear reactor has been studied by meansof the equation of radiative transfer for a grey atmosphere.Some new results have been obtained in the form of an integralequation for the radiation intensity and heat flux. A variationalmethod is employed to obtain numerical results and the accuracyof an elementary theory based upon discrete ordinates is assessed.The variational approach leads to computationally useful andaccurate results for several quantities of practical interest.The interdisciplinary nature of the work is stressed by virtueof its close connection with neutron transport theory and rarefiedgas dynamics.     

中子输运方程误差估计及自适应计算

 本文研究了全离散方法求解二维中子输运方程的有限元自适应算法, 角度变量用离散纵坐标方法展开, 空间变量用间断元方法求解. 基于间断元方法给出了空间离散的残量型后验误差估计. 在后验误差估计的基础上, 我们设计了自适应有限元算法.由残量型后验估计可以给出局部加密网格的自适应算法. 最后, 我们给出了数值算例来验证我们的理论结果.     

Convergence of the modified discrete ordinates method for the anisotropic scattering transport equation

Criticality problem of nuclear tractors generally refers to an eigenvalue problem for the transport equations. In this paper, we deal with the eigenvalue of the anisotropic scattering transport equation in slab geometry. We propose a new discrete method which was called modified discrete ordinates method. It is constructed by redeveloping and improving discrete ordinates method in the space of L¹(X). Different from traditional methods, norm convergence of operator approximation is proved theoretically. Furthermore, convergence of eigenvalue approximation and the corresponding error estimation are obtained by analytical tools.     

Monte Carlo simulation of radiative transfer in a medium with varying refractive index specified at discrete points

A Monte Carlo method with discrete ray tracing is developed to simulate radiative transfer in a medium with a spatially varying refractive-index distribution available merely at a set of arbitrary discrete points. We solve the ray equation by a Runge−Kutta Dormand−Prince method to carry out the numerical ray tracing. To retrieve the refractive-index values and gradients needed in the discrete ray tracing, we apply cubic spline interpolation for one-dimensional simulation and a moving least square (MLS) method for two-dimensional simulation. The influence of the basis vectors and the numbers of sampled data used by the MLS method on ray tracing based on the retrieved refractive-index values and gradients has been examined. The results of radiative equilibrium in a planar medium and radiative transfer in two-dimensional media with different geometries and conditions obtained by the present methods are compared with those obtained by solving the integral equations of radiative transfer and the discrete ordinates method. The comparisons show that the present methods generate accurate results for radiative transfer with various geometries, parameters and refractive-index distributions specified at discrete points.     

RESEARCH ANNOUNCEMENTS High order Zumerical Solution of Integral Transport Equation in slab Geometry

There are some common numerical methods for solving neutron transport equation, which including the well-known discrete ordinates method, PN approximation and integral transport methods[1]. There exists certain singularities in the solution of transport equation near the boundary and interface[2]. It gives rise to the difficulty in the construction of high order accurate numerical methods. The numerical solution obtained by now can not attain the second order convergent accuracy[3,4].     

The IIM in polar coordinates and its application to electro capacitance tomography problems

In a numerical study of the electric field inside an Electro Capacitance Tomography (ECT) device and other applications, often a Poisson equation with a discontinuous coefficient needs to be solved in polar coordinates. This paper is devoted to the Immersed Interface Method (IIM) in polar coordinates and the application to the solution of the electric potential inside an ECT device. The numerical algorithm is based on a finite difference discretization on a uniform polar coordinates grid. The finite difference scheme is modified at grid points near and on the interface across which the coefficient is discontinuous so that the natural jump conditions are satisfied. The algorithm and analysis here is one step forward in applying the IIM for 3D problems in axisymmetric situations or in the spherical coordinates. Numerical examples against exact solutions and the application to ECT problems are also presented.     

An exact solution of a quasilinear Fisher equation in cylindrical coordinates

Lie symmetry method is applied to analyse Fisher equation in cylindrical coordinates. Symmetry algebra is found and symmetry invariance is used to reduce the equation to a first-order ODE. The first-order ODE is further analysed to obtain exact solution of Fisher equation in explicit form.