弱凝胶调驱数值模拟

以组分模型为基础,结合黑油模型的推导方式,建立了三维三相八组分的弱凝胶调驱的数学模型,模型中综合考虑了重力、毛管力、流体和岩石压缩性等各种影响因素.在模型的求解上采用了改进的IMPES方法即隐式求解压力,高阶显式求取饱和度的方法进行求解,以此提高饱和度求解的精度,并编制了相应的数值模拟软件,对弱凝胶体系的调驱效果进行了模拟研究.     

黏弹性聚合物驱驱替动态与不同注入参数分析

基于质量守恒定律,建立三相五组分考虑聚合物弹性增黏、降低残余油和稠油黏弹性效应的聚合物驱数学模型,并采用IMPES方法进行求解.计算分析表明,黏弹性聚合物驱可降低残余油饱和度,增加驱油体系黏度,提高原油采收率;采收率随注入浓度增大先迅速增加后趋于稳定,最优值为2 000 mg·L^-1;随注入速度先增加后降低,存在最优值;随聚合物段塞增大先迅速增加后缓慢增加并趋于稳定,最佳注入量约为0.6 PV;随原油松弛时间增加呈线性增加.     

非结构网格上的B-B单方程模型湍流计算

研究如何在非结构网格上进行Navier Stokes(N-S)方程湍流计算.采用格心有限体积方法离散N-S方程.为了适应非结构网格,计算所用的湍流模型特别选用Baldwin Barth(B-B)单方程模型.此模型由一个单一的具有源项的对流扩散方程组成.为了能在非结构网格上求解B B单方程模型,提出一显式有限体积格式,并直接对带源项的格式进行稳定性分析,得到了相应的时间步长限制条件.最后以平板、RAE 2822翼型、多段翼型绕流等数值算例验证了计算方法的有效性.     

模拟短脉冲激光在介质中传输的扩散综合有限元法

建立有限元模型,通过求解瞬态辐射传输方程模拟短脉冲激光在半透明介质中的传输.针对散射占优性半透明介质内辐射传输求解效率较差的问题,采用扩散综合加速迭代算法,提高计算效率,缩短计算时间.结果表明:采用精确解析式描述脉冲激光散射源项的求解策略可以获得准确的计算结果,精确地模拟快速变化的波前,不会产生数值扩散和数值振荡.此外,扩散综合迭代算法的计算时间仅为源项迭代的50%~60%.     

JFNK在高温堆扩散计算中的应用

研究了JFNK框架下高温堆中子扩散问题的求解方法。研究结果表明,JFNK方法在求解与源迭代相同形式中子扩散方程时,相对残差下降趋势为逐渐加快并趋于稳定,有利于更高求解精度的实现。使用通量归一化附加方程可以获得更好的JFNK非线性迭代特性,但在算例中其部分牛顿修正方程求解时间偏多,总计算时间高于显式有效增殖系数附加方程法,需要研究更高效的JFNK预处理方法对线性求解环节进行改善。     

三维非结构网格Euler方程的LU-SGS算法及其改进

将LU-SGS隐式时间推进格式运用到非结构网格Euler方程的求解中,并对传统LU-SGS格式进行改进,结合网格重排序,发展了一套效率更高的三维Euler方程求解器.以M6机翼及超临界LANN机翼的跨音速无粘流场为算例,将改进LU-SGS格式与四步龙格-库塔显式格式及传统LU-SGS格式进行了比较.计算结果表明:所有格式的计算结果与实验结果都符合很好;传统LU-SGS格式计算效率为显式格式的3倍多,而改进LU-SGS格式计算效率为显式格式的7倍多.     

JFNK在高温堆扩散计算中的应用

研究了JFNK框架下高温堆中子扩散问题的求解方法。研究结果表明,JFNK方法在求解与源迭代相同形式中子扩散方程时,相对残差下降趋势为逐渐加快并趋于稳定,有利于更高求解精度的实现。使用通量归一化附加方程可以获得更好的JFNK非线性迭代特性,但在算例中其部分牛顿修正方程求解时间偏多,总计算时间高于显式有效增殖系数附加方程法,需要研究更高效的JFNK预处理方法对线性求解环节进行改善。     

非线性薛定谔方程的高阶分裂改进光滑粒子动力学算法

为提高传统光滑粒子动力学(SPH)方法求解高维非线性薛定谔(nonlinear Schr?dinger/Gross-Pitaevskii equation, NLS/GP)方程的数值精度和计算效率,本文首先基于高阶时间分裂思想将非线性薛定谔方程分解成线性导数项和非线性项,其次拓展一阶对称SPH方法对复数域上线性导数部分进行显式求解,最后引入MPI并行技术,结合边界施加虚粒子方法给出一种能够准确、高效地求解高维NLS/GP方程的高阶分裂修正并行SPH方法.数值模拟中,首先对带有周期性和Dirichlet边界条件的NLS方程进行求解,并与解析解做对比,准确地得到了周期边界下孤立波的奇异性,且对提出方法的数值精度、收敛速度和计算效率进行了分析;随后,运用给出的高阶分裂粒子方法对复杂二维和三维NLS/GP问题进行了数值预测,并与其他数值结果进行比较,准确地展现了非线性孤立波传播中的奇异现象和玻色-爱因斯坦凝聚态中带外旋转项的量子涡旋变化过程.     

甲烷平面射流扩散火焰的大涡模拟

本文对甲烷-空气平面自由射流扩散火焰进行了大涡模拟,采用分步投影法求解动量方程,湍流亚格子项采用动态模式模拟,化学反应速率亚格子项采用动态相似模式模拟,压力泊松方程采用修正的循环消去法快速求解,空间方向采用二阶精度的差分格式,在时间方向上采用二阶精度的显式差分格式。模拟结果给出了湍流扩散火焰的瞬态发展变化过程,表明射流扩散火焰的发展过程存在着“湍流控制”和“化学反应控制”两个不同阶段。 “湍流控制”阶段仅存在于火焰发展初期的极短时间内。     

Richtmyer-Meshkov不稳定性强化混合参变机理

在不同参数条件下, 计算分析了H₂O和N₂等混合物界面上激波诱导Richtmyer-Meshkov(R-M)不稳定性过程.采用有限差分方法数值求解了二维可压缩Navier-Stokes方程, 对流项以5阶特征紧致-WENO混合格式离散, 输运项以6阶对称紧致格式离散, 时间方向以3阶显式Runge-Kutta方法推进.研究表明, 界面振幅和激波强度增大, 均可增强界面附近涡量场, 强化混合.      

求解波动方程的准粒子离散修正辛算法

针对波动方程求解,在Hamilton体系下建立了对空间离散的准粒子体系,该准粒子体系实现简单,物理意义明确;在时间离散方面,构造了一种适合高效声波模拟的修正辛格式,该格式是在常规的二阶Partitioned Runge-Kutta(PRK)基础之上构造而成,其具有三阶时间精度,从理论上分析了修正辛格式的数值稳定性和频散性能.数值结果表明,本文提出的方法在计算时间,计算精度和计算存储量等各方面性能都有相应改善。     

Explicit multi-symplectic extended leap-frog methods for Hamiltonian wave equations

In this paper, we study the integration of Hamiltonian wave equations whose solutions have oscillatory behaviors in time and/or space. We are mainly concerned with the research for multi-symplectic extended Runge–Kutta–Nyström (ERKN) discretizations and the corresponding discrete conservation laws. We first show that the discretizations to the Hamiltonian wave equations using two symplectic ERKN methods in space and time respectively lead to an explicit multi-symplectic integrator (Eleap-frogI). Then we derive another multi-symplectic discretization using a symplectic ERKN method in time and a symplectic partitioned Runge–Kutta method, which is equivalent to the well-known Störmer–Verlet method in space (Eleap-frogII). These two new multi-symplectic schemes are extensions of the leap-frog method. The numerical stability and dispersive properties of the new schemes are analyzed. Numerical experiments with comparisons are presented, where the two new explicit multi-symplectic methods and the leap-frog method are applied to the linear wave equation and the Sine–Gordon equation. The numerical results confirm the superior performance and some significant advantages of our new integrators in the sense of structure preservation.     

Rogue waves in Lugiato-Lefever equation with variable coefficients

In this paper, we theoretically investigate the generation of optical rogue waves from a Lugiato-Lefever equation with variable coefficients by using the nonlinear Schrödinger equation-based constructive method. Exact explicit rogue-wave solutions of the Lugiato-Lefever equation with constant dispersion, detuning and dissipation are derived and presented. The bright rogue wave, intermediate rogue wave and the dark rogue wave are obtained by changing the value of one parameter in the exact explicit solutions corresponding to the external pump power of a continuous-wave laser.     

A discontinuous Galerkin method for inviscid low Mach number flows

In this work we extend the high-order discontinuous Galerkin (DG) finite element method to inviscid low Mach number flows. The method here presented is designed to improve the accuracy and efficiency of the solution at low Mach numbers using both explicit and implicit schemes for the temporal discretization of the compressible Euler equations. The algorithm is based on a classical preconditioning technique that in general entails modifying both the instationary term of the governing equations and the dissipative term of the numerical flux function (full preconditioning approach). In the paper we show that full preconditioning is beneficial for explicit time integration while the implicit scheme turns out to be efficient and accurate using just the modified numerical flux function. Thus the implicit scheme could also be used for time accurate computations. The performance of the method is demonstrated by solving an inviscid flow past a NACA0012 airfoil at different low Mach numbers using various degrees of polynomial approximations. Computations with and without preconditioning are performed on different grid topologies to analyze the influence of the spatial discretization on the accuracy of the DG solutions at low Mach numbers.     

Lie symmetry analysis,conservation laws and analytical solutions for a generalized time-fractional modified KdV equation

In this paper, a generalized time fractional modified KdV equation is investigated, which is used for representing physical models in various physical phenomena. By Lie group analysis method, the invariance properties and the vector fields of the equation are presented. Then the symmetry reductions are provided. Moreover, we construct the explicit solutions of the equation by using sub-equation method. Based on the power series theory, the approximate analytical solution for the equation are also constructed. Finally, the new conservation theorem is applied to constructed conservation laws for the equation.     

Localized light waves: Paraxial and exact solutions of the wave equation (a review)

Simple explicit localized solutions are systematized over the whole space of a linear wave equation, which models the propagation of optical radiation in a linear approximation. Much attention has been paid to exact solutions (which date back to the Bateman findings) that describe wave beams (including Bessel-Gauss beams) and wave packets with a Gaussian localization with respect to the spatial variables and time. Their asymptotics with respect to free parameters and at large distances are presented. A similarity between these exact solutions and harmonic in time fields obtained in the paraxial approximation based on the Leontovich-Fock parabolic equation has been studied. Higher-order modes are considered systematically using the separation of variables method. The application of the Bateman solutions of the wave equation to the construction of solutions to equations with dispersion and nonlinearity and their use in wavelet analysis, as well as the summation of Gaussian beams, are discussed. In addition, solutions localized at infinity known as the Moses-Prosser “acoustic bullets”, as well as their harmonic in time counterparts, “X waves”, waves from complex sources, etc., have been considered. Everywhere possible, the most elementary mathematical formalism is used.     

等离子体平衡逆问题的一种数值解法

本文借助离散化算子,把滤波方法推广到非圆截面,对具有非圆截面的等离子体平衡逆问题建立了一个稳定的显式逆推滤波算法;给出了部分非圆截面的计算结果;对圆截面情形的数值解与解析解作了比较。     

CLOSED FORM SOLUTIONS TO THE INTEGRABLE DISCRETE NONLINEAR SCHRÖDINGER EQUATION

In this article we derive explicit solutions of the matrix integrable discrete nonlinear Schrödinger equation by using the inverse scattering transform and the Marchenko method. The Marchenko equation is solved by separation of variables, where the Marchenko kernel is represented in separated form, using a matrix triplet (A, B, C). Here A has only eigenvalues of modulus larger than one. The class of solutions obtained contains the N-soliton and breather solutions as special cases. We also prove that these solutions reduce to known continuous matrix NLS solutions as the discretization step vanishes.     

Fast finite difference solvers for singular solutions of the elliptic Monge–Ampère equation

The elliptic Monge–Ampère equation is a fully nonlinear Partial Differential Equation which originated in geometric surface theory, and has been applied in dynamic meteorology, elasticity, geometric optics, image processing and image registration. Solutions can be singular, in which case standard numerical approaches fail.In this article we build a finite difference solver for the Monge–Ampère equation, which converges even for singular solutions. Regularity results are used to select a priori between a stable, provably convergent monotone discretization and an accurate finite difference discretization in different regions of the computational domain. This allows singular solutions to be computed using a stable method, and regular solutions to be computed more accurately. The resulting nonlinear equations are then solved by Newton’s method.Computational results in two and three-dimensions validate the claims of accuracy and solution speed. A computational example is presented which demonstrates the necessity of the use of the monotone scheme near singularities.     

An Adaptive Wavelet–Vaguelette Algorithm for the Solution of PDEs

The paper first describes a fast algorithm for the discrete orthonormal wavelet transform and its inverse without using the scaling function. This approach permits to compute the decomposition of a function into a lacunary wavelet basis, i.e., a basis constituted of a subset of all basis functions up to a certain scale, without modification. The construction is then extended to operator-adapted biorthogonal wavelets. This is relevant for the solution of certain nonlinear evolutionary PDEs where a priori information about the significant coefficients is available. We pursue the approach described in (J. Fröhlich and K. Schneider,Europ. J. Mech. B/Fluids13,439, 1994) which is based on the explicit computation of the scalewise contributions of the approximated function to the values at points of hierarchical grids. Here, we present an improved construction employing the cardinal function of the multiresolution. The new method is applied to the Helmholtz equation and illustrated by comparative numerical results. It is then extended for the solution of a nonlinear parabolic PDE with semi-implicit discretization in time and self-adaptive wavelet discretization in space. Results with full adaptivity of the spatial wavelet discretization are presented for a one-dimensional flame front as well as for a two-dimensional problem.