基于移动网格的熵稳定格式

提出一种基于移动网格的熵稳定格式求解双曲型守恒律方程.该方法利用等分布原理得到新的网格分布,基于守恒型插值公式计算新的网格上的物理量,使用熵稳定数值通量和三阶强稳定Runge-Kutta时间推进方法得到下一时刻的数值解.数值算例表明该格式不仅能有效提高解在间断处的分辨率,而且能消除可能产生的伪振荡.     

单个守恒型方程熵耗散格式中熵耗散函数的构造

对于一维单个守恒律方程,文[8]设计了一种非线性守恒型差分格式.此格式为二阶Godunov型的,用的是分片线性重构(reconstruction),重构函数的斜率是根据熵耗散得到的.格式满足熵条件.与传统的守恒格式不同的是此格式在计算过程中不仅用到了数值解还用到了数值熵.在此格式中一个所谓的熵耗散函数起到了很重要的作用,它在每一个网格的计算中耗散熵,以保证格式满足熵条件.文[8]中设计的熵耗散函数比较复杂,并且不是很完善.故数值地分析了在格式的构造中为何应给熵以一定的耗散,及应耗散多少.并且给出了一个新的以数值解的二阶差分作为基本模块的熵耗散函数.最后给出了相应的数值算例.     

二维交错网格的GAUSS型格式

利用Gauss型求积公式在交错网格的情况下构造了一类不需解Riemann问题的求解二维双曲守恒律的二阶显式Gauss型差分格式,该格式在CFL条件限制下为MmB格式.并将格式推广到二维方程组,进行了数值试验.     

解流体力学方程组的一种隐式完全守恒差分格式

对Lagrange非守恒流体力学方程组给出了一种隐式完全守恒差分格式,既保证了质量、动量和总能量守恒的差分近似,又能满足内能与动能的平衡特性,提高了数值解的精度。并用该格式对两个可压缩理想流体模型进行了数值计算,并与其它差分格式作了比较。     

基于非结构网格CE/SE算法圆柱绕流气动声场模拟

本文采用时空守恒(CE/SE)格式求解Navier-Stoks方程与Euler方程,数值模拟了绕圆柱流动引发的气动噪声源及传播问题.对近声场与远声场采用了不同的控制方程和边界条件,分别得到了流场与声场解.将其与采用大涡模拟和FW-H方程得到的结果进行了对比,表明本文的数值方法能够很好地反映流场与声场的形态,能较好地模拟声场指向性及流场中的拟声现象,相比LES/FW-H方法能更精确地反映流场与声场的相互关系.     

双曲型守恒律的一种三阶半离散中心迎风格式

对一维双曲型守恒律,给出了一种具有较小数值耗散的三阶半离散中心迎风格式.该格式以Liu和Tadmor提出的三阶无振荡重构为基础,同时考虑了波传播的单侧局部速度.时间离散用保持强稳定性的三阶Runge-Kutta方法.由于不需用Riemann解算器,避免了特征分解过程,保持了中心格式简单的优点.数值算例验证本方法可进一步减小数值耗散,提高分辨率.     

多介质流体力学计算的守恒型高分辨率格式

应用Lagrange坐标系下的守恒型格式计算多介质流体力学问题,在物质交界面附近采用一阶格式的数值通量,而在其余部分采用高分辨率格式的数值通量,不仅保持了高分辨率的良好性质,而且消除了一般的守恒型格式在界面附近所产生的震荡.     

解Hamilton-Jacobi方程的不连续有限元方法

将两类具有不同基函数的有限元应用于Hamilton Jacobi方程,得到了求解Hamilton Jacobi方程的不连续有限元数值格式,并证明了这两类格式数值解在一定条件下收敛于Hamilton Jacobi方程的弱解.数值实例比较了两类格式的精度和分辨间断的能力.     

一维双曲守恒律的龙格-库塔控制体积间断有限元方法

给出数值求解一维双曲守恒律方程的新方法——龙格-库塔控制体积间断有限元方法(RKCVDFEM),其中空间离散基于控制体积有限元方法,时间离散基于二阶TVB Runge-Kutta技术,有限元空间选取为分段线性函数空间.理论分析表明,格式具有总变差有界(TVB)的性质,而且空间和时间离散形式上具有二阶精度.数值算例表明,数值解收敛到熵解并且对光滑解的收敛阶是最优的,优于龙格-库塔间断Galerkin方法(RKDGM)的计算结果.     

用NND格式模拟Kelvin-Helmholtz不稳定性

使用局部Steger-Warming通量分裂方法,利用NND有限差分格式求解守恒型流体力学方程组,实现对Kelvin-Helmholtz不稳定性的数值模拟.数值模拟给出的线性增长率与线性稳定性分析给出的结果相符合,得到锐利的界面变形图像.     

Uniqueness and nonuniqueness in the Einstein constraints

The conformal thin-sandwich (CTS) equations are a set of four of the Einstein equations, which generalize the Laplace-Poisson equation of Newton's theory. We examine numerically solutions of the CTS equations describing perturbed Minkowski space, and find only one solution. However, we find two distinct solutions, one even containing a black hole, when the lapse is determined by a fifth elliptic equation through specification of the mean curvature. While the relationship of the two systems and their solutions is a fundamental property of general relativity, this fairly simple example of an elliptic system with nonunique solutions is also of broader interest.     

Example of indeterminacy in classical dynamics

The case of a particle moving along a nonsmooth constraint under the action of uniform gravity is presented as an example of indeterminancy in a classical situation. The indeterminacy arises from certain initial conditions having nonunique solutions and is due to the failure of the Lipschitz condition at the corresponding points in the phase space of the equation of motion.     

A Laplace Decomposition Method for Nonlinear Partial Differential Equations with Nonlinear Term of Any Order

A Laplace decomposition algorithm is adopted to investigate numerical solutions of a class of nonlinear partial differential equations with nonlinear term of any order, utt ＋ auxx ＋ bu ＋ cup ＋ du^2p-1 = 0, which contains some important equations of mathematical physics. Three distinct initial conditions are constructed and generalized numerical solutions are thereby obtained, including numerical hyperbolic function solutions and doubly periodic ones. Illustrative figures and comparisons between the numerical and exact solutions with different values of p are used to test the efficiency of the proposed method, which shows good results are azhieved.     

Exact Solutions of the Two-Dimensional Cubic-Quintic Nonlinear Schrdinger Equation with Spatially Modulated Nonlinearities

Applying the similarity transformation,we construct the exact vortex solutions for topological charge S ≥ 1 and the approximate fundamental soliton solutions for S = 0 of the two-dimensional cubic-quintic nonlinear Schrdinger equation with spatially modulated nonlinearities and harmonic potential.The linear stability analysis and numerical simulation are used to exam the stability of these solutions.In different profiles of cubic-quintic nonlinearities,some stable solutions for S ≥ 0 and the lowest radial quantum number n = 1 are found.However,the solutions for n ≥ 2 are all unstable.     

Nonuniqueness of a massive spin-2 theory

The nonunique nature of massive spin-2 fields is explicitly shown in this paper through the construction of all possible field equations, using Dirac formalism for spin-1/2 fields. Out of these four possible theories, we point out two that do not show up scalar representations.     

Transient motions of bi-lattices

We develop formal solutions for the propagation of transient pulses on a variety of bi-lattice models. The lattices are composed of a finite homogeneous chain connected in series with a different semi-infinite homogeneous chain at a common location occupied by a single mass which is different from the masses of both chains. Exact analytic solutions of this general case are not possible. Some analytic solutions are, however, possible for a variety of special cases. The general solutions are illustrated by numerically inverting the Laplace transform functions. The exact solutions are found to correlate very well with the numerical inversion scheme. Such correlations give confidence in the numerical scheme's predictions of the solutions of the more complicated chains.     

Numerical study of paraxial beam formation

A method for numerical study of different integral problems with sufficiently strongly localized solutions is considered. Approximations of the solution with different degrees of accuracy were constructed and, using the existing approximation, an approach to an increase of the solution accuracy is proposed. Numerical solutions are obtained for the problem of free oscillations of laser cavities the optical surface of which was approximated by polynomials of the second and fourth degree inclusive. The obtained numerical solutions were stable.     

Solutions of time-fractional Kudryashov–Sinelshchikov equation arising in the pressure waves in the liquid with gas bubbles

In this paper, analytical approximate solutions for time-fractional Kudryashov–Sinelshchikov equation have been obtained. Two different techniques have been implemented to calculate the solutions, namely, homotopy analysis method and residual power series method. The approximate solutions are represented numerically and graphically for different values of fractional order of derivative. The numerical results are expressed in Tables 1, 2, 3 and 4 which show that the approximate solutions are in good agreement with the exact solution. The comparative study of the numerical results reveal that both methods are reliable and effective tools for the solution of time-fractional Kudryashov–Sinelshchikov equation.