基于SIMPLE算法求解Navier—Stokes方程

介绍求解Navier-Stokes65数值解法。针对不可压缩流体的的数值解法有涡量一流函数方法@SIMPLE57法．对基于同位网格的65SIMPLE法作详细讨论,给出该算法的推导过程,最终得出求解SIMPLE算法的求解步骤。应用该求解步骤对具体实例求解,得出结论。     

用余弦微分求积法数值求解KdV—Burgers方程

采用余弦微分求积法（CDQM）对（1＋1）维非线性KdV—Burgers方程进行了数值求解．结果表明,所得数值解与方程的精确解相比具有明显的高精度且稳定性高,相对于其他常用方法,且公式简单,使用方便;计算量小,时间复杂性好．     

无约束非线性lp问题的区间极大熵方法

针对信号处理、系统识别等领域中涉及到的无约束非线性lp问题,为减小由于二进制编码的舍入误差对该问题计算结果的影响,对求解该问题的极大熵方法进行了区间扩张.证明了区间扩张后的极大熵函数至少具有二阶收敛性,并设计了具有多项式时间复杂度的区间算法进行求解,举例进行了数值计算.数值计算结果显示,该区间算法可靠,计算结果与区间扩张前相比,结果更加精确.     

求解强非线性动力系统响应的一种新方法

将同伦理论和参数变换技术相结合提出了一种可适用于求解强非线性动力系统响应的新方法．即PE-HAM方法(基于参数展开的同伦分析技术)．其主要思想是通过构造合适的同伦映射,将一非线性动力系统的求解问题,转化为一线性微分方程组的求解问题,然后借助于参数展开技术消除长期项,进而得到系统的解析近似解．为了检验所提方法的有效性,研究了具有精确周期的保守Duffing系统的响应,求出了其解析的近似解表达式．在与精确周期的比较中,可以得出：在非线性强度。很大,甚至在α→∞时,近似解的周期与原系统精确周期的误差也只有2．17％．数值模拟结果说明了新方法的有效性．     

基于SIMPLE算法求解Navier-Stokes方程

介绍求解Navier-Stokes的数值解法,针对不可压缩流体的的数值解法有涡量-流函数方法和SIMPLE方法,对基于同位网格的SIMPLE算法作详细讨论,给出该算法的推导过程,最终得出求解SIMPLE算法的求解步骤,应用该求解步骤对具体实例求解,得出结论。     

基于二阶锥规划的宽带波束形成器设计

宽带波束形成器有两种典型的实现方式,分别基于FIR滤波器和基于长方形阵列,将这两种波束形成器的波束响应表达为统一的形式,并根据参考波束的选择情况,为该类波束优化问题的求解建立了数学模型,该数学模型属于二阶锥约束问题,可以用二阶锥规划方法进行求解。仿真结果表明,提出的求解宽带恒定束宽响应波束形成器权系数的数学模型具有通用性,并且把二阶锥规划方法运用到宽带波束形成器优化设计中,其约束控制灵活,问题求解方便,设计结果精确。     

SPH方法核近似形式改进及算法研究

光滑粒子流体动力学(Smoothed Particle Hydrodynamics,SPH)方法是一种无网格拉格朗日粒子法,目前在流体力学领域以及大变形和冲击载荷等问题的模拟方面具有广泛的应用,众多学者在SPH算法方面开展了大量的研究,以提高SPH算法的计算速度和精度.针对现有SPH方法在边界附近粒子近似精度下降的问题,本文在CSPH方法和MSPH方法基础上提出了一种改进的核近似形式,在求解场函数、一阶导数近似值以及二阶导数近似值过程中,对含二阶导数项的方程进行优化,减少了二阶导数项近似值的求解个数,相比MSPH方法减少了计算量.此外,本文基于改进的SPH算法,建立了二维数值波浪水槽模拟推板造波,通过数值模拟造波将SPH算法生成的波浪参数与理论值进行对比,验证了改进的SPH方法在波浪生成和传播上具有较好的模拟效果,为后续研究内波、畸形波以及非线性波相互作用提供了算法研究基础.     

参数线性规划问题的新型光滑精确罚函数神经网络

针对不等式约束条件下,目标函数和约束条件中含有参数的线性规划问题,提出一种基于新型光滑精确罚函数的神经网络计算方法．引入误差函数构造单位阶跃函数的近似函数,给出一种更加精确地逼近于Ll精确罚函数的光滑罚函数,讨论了其基本性质;利用所提光滑精确罚函数建立了求解参数线性规划问题的神经网络模型,证明了该网络模型的稳定性和收敛性,并给出了详细的算法步骤．数值仿真验证了所提方法具有罚因子取值小、结构简单、计算精度高等优点．     

求解非线性方程组的改进不精确雅可比牛顿法

在分析不精确雅可比牛顿法的基础上,进一步研究了不精确雅可比矩阵在精确解附近奇异的求解方法。利用雅可比矩阵与函数自身,在不增加新的计算量前提下,得到改进的求解非线性方程组的不精确雅可比牛顿算法。数值结果表明,改进后算法与原不精确雅可比牛顿法具有相同的计算效率,而且在使用上更为方便,有效。     

求解相对论流体力学方程的低耗散中心迎风格式

针对一维相对论流体力学方程,给出一种数值求解方法.该方法以低耗散中心迎风数值通量为基础,通过分片线性重构来获得空间上的二阶精度,最后采用强稳定龙格库塔方法在时间方向上推进.数值算例验证了该方法的有效性和基本无振荡性.     

Incompressible Navier–Stokes solutions by a triangular spectral/p element projection method

The application of the fractional step projection method recently proposed by Guermond and Quartapelle to the numerical approximation of unsteady Navier–Stokes solutions by means of a spectral/p element method is considered. In particular we illustrate the second-order pressure correction technique and evaluate its accuracy properties in some test cases. Stability with respect to the compatibility condition between the approximation spaces for velocity and pressure is also addressed. The high (spectral) accuracy in space and the second-order accuracy in time are verified by two simple test cases with analytical solution. A more interesting problem is solved showing the ability of the method to produce very accurate results also for problems in complex geometries.     

Iterative methods for stationary solutions to the steady-state compressible Navier-Stokes equations

Numerical experiments are presented for the solution of the steady-state compressible Navier-Stokes equations. One test problem is fixed supersonic flow past a double ellipse, and the various solution methods studied. The problem is discretized using Osher's scheme, first- and second-order accurate. The fastest convergence to steady state is obtained using Newton's method. Simplifications of Newton's method based on domain decomposition are shown to perform well, whereas line relaxation methods meet with difficulties.     

Accurate shock-capturing finite volume method for advection-dominated flow and pollution transport

This paper describes an accurate shock-capturing finite volume numerical method to solve a two-dimensional flow and solute transport problem in shallow water. Hydrodynamic and advection-diffusion equations are simultaneously solved by means of a Strang operator-splitting approach. The advective part is solved in time by a third-order TDV Runge-Kutta method and in space by a second-order WAF method coupled with a fifth-order WENO reconstruction. The diffusion part is solved in time and space by a second-order accurate method. Thus the overall accuracy is second-order both in time and space. Nevertheless the Strang splitting approach allows the advective part of the equations to be solved with a reconstruction of high order, where at lower orders it shows excessive numerical diffusion and damping, especially for very long time simulations. Very good results have been obtained applying the model to standard long time numerical tests.     

Numerical solution of nonlinear system of second-order boundary value problems using cubic B-spline scaling functions

A numerical technique is presented for the solution of nonlinear system of second-order boundary value problems. This method uses the cubic B-spline scaling functions. The method consists of expanding the required approximate solution as the elements of cubic B-spline scaling function. Using the operational matrix of derivative, we reduce the problem to a set of algebraic equations. Numerical examples are included to demonstrate the validity and applicability of the technique. The method is easy to implement and produces very accurate results.     

Second-Order Accurate Godunov Scheme for Multicomponent Flows on Moving Triangular Meshes

This paper presents a second-order accurate adaptive Godunov method for two-dimensional (2D) compressible multicomponent flows, which is an extension of the previous adaptive moving mesh method of Tang et al. (SIAM J. Numer. Anal. 41:487–515, 2003) to unstructured triangular meshes in place of the structured quadrangular meshes. The current algorithm solves the governing equations of 2D multicomponent flows and the finite-volume approximations of the mesh equations by a fully conservative, second-order accurate Godunov scheme and a relaxed Jacobi-type iteration, respectively. The geometry-based conservative interpolation is employed to remap the solutions from the old mesh to the newly resulting mesh, and a simple slope limiter and a new monitor function are chosen to obtain oscillation-free solutions, and track and resolve both small, local, and large solution gradients automatically. Several numerical experiments are conducted to demonstrate robustness and efficiency of the proposed method. They are a quasi-2D Riemann problem, the double-Mach reflection problem, the forward facing step problem, and two shock wave and bubble interaction problems.     

A Spectral Element Method to Price European Options. I. Single Asset with and without Jump Diffusion

We develop a spectral element method to price single factor European options with and without jump diffusion. The method uses piecewise high order Legendre polynomial expansions to approximate the option price represented pointwise on a Gauss-Lobatto mesh within each element, which allows an exact representation of the non-smooth payoff function. The convolution integral is approximated by high order Gauss-Lobatto quadratures. A second order implicit/explicit (IMEX) approximation is used to integrate in time, with the convolution integral integrated explicitly. The method is spectrally accurate (exponentially convergent) in space for the solution and Greeks, and second-order accurate in time. The spectral element solution to the Black-Scholes equation is ten to one hundred times faster than commonly used second order finite difference methods.     

求解带有间断系数泊松方程的修正有限体积

利用修正的有限体积方法求解带有间断系数的泊松方程,改进是对基于笛卡尔坐标系下的调和平均系数进行的。数值实验表明新格式二阶逐点收敛并且在界面处具有二阶精度,新方法较已有的求解不连续扩散系数的算术平均法和调和平均法,特别是在系数跳跃较大的情况下更具优势。     

A parameter uniform B-spline collocation method for solving singularly perturbed turning point problem having twin boundary layers

A numerical scheme is proposed to solve singularly perturbed two-point boundary value problems with a turning point exhibiting twin boundary layers. The scheme comprises B-spline collocation method on a non-uniform mesh of Shishkin type. Asymptotic bounds are established for the derivative of the analytical solution of a turning point problem. The present method is boundary layer resolving as well as second-order accurate in the maximum norm. A brief analysis has been carried out to prove the uniform convergence with respect to the singular perturbation parameter ? by decomposing the solution into smooth and singular components. Some relevant numerical examples are also illustrated to verify computationally the theoretical aspects.     

Higher-order split-step Fourier schemes for the generalized nonlinear Schrödinger equation

The generalized nonlinear Schrödinger (GNLS) equation is solved numerically by a split-step Fourier method. The first, second and fourth-order versions of the method are presented. A classical problem concerning the motion of a single solitary wave is used to compare the first, second and fourth-order schemes in terms of the accuracy and the computational cost. This numerical experiment shows that the split-step Fourier method provides highly accurate solutions for the GNLS equation and that the fourth-order scheme is computationally more efficient than the first-order and second-order schemes. Furthermore, two test problems concerning the interaction of two solitary waves and an exact solution that blows up in finite time, respectively, are investigated by using the fourth-order split-step scheme and particular attention is paid to the conserved quantities as an indicator of the accuracy. The question how the present numerical results are related to those obtained in the literature is discussed.     

A minisum location problem with regional demand considering farthest Euclidean distances

We consider a continuous multi-facility location-allocation problem that aims to minimize the sum of weighted farthest Euclidean distances between (closed convex) polygonal and/or circular demand regions, and facilities they are assigned to. We show that the single facility version of the problem has a straightforward second-order cone programming formulation and can therefore be efficiently solved to optimality. To solve large size instances, we adapt a multi-dimensional direct search descent algorithm to our problem which is not guaranteed to find the optimal solution. In a special case with circular and rectangular demand regions, this algorithm, if converges, finds the optimal solution. We also apply a simple subgradient method to the problem. Furthermore, we review the algorithms proposed for the problem in the literature and compare all these algorithms in terms of both solution quality and time. Finally, we consider the multi-facility version of the problem and model it as a mixed integer second-order cone programming problem. As this formulation is weak, we use the alternate location-allocation heuristic to solve large size instances.