一维非线性对流占优扩散方程特征差分法的两重网格算法

针对一维非线性对流扩散方程，构造了特征差分的两重网络算法，并给出了误差估计和数值算例。此方法是先在粗网格上计算非线性问题，再在细网格上计算线性问题，数值算例表明，在计算精度保持不变的情况下，此算法可以极大提高非线性对流扩散问题的计算效率。     

结构动力响应数值分析的新的广义-α方法的频率域分析

介绍了一族用于结构动力响应数值计算的新的广义-α方法。用这族算法将结构动力学方程离散得到关于位移的差分方程，然后做Z-变换得到算法传递函数、算法频率响应函数。通过算法离散的频率响应和系统真实频率响应的比较，在频率域内分析了这族算法的性能，并与常用的Newmark平均加速度方法进行了比较。在分析中，以激励力频率为自变量分析了算法的Bode图，以系统固有频率和激励力频率与时间步长的乘积为自变量在三维空间内分析了算法频率响应幅值绝对误差，又以算法自由参数、阻尼比为自变量在三维空间内分析了算法频率响应幅值2-范数意义下的相对误差。结果表明，新算法的性能得到了更深刻的认识。     

抛物型方程移动界面的并行差分格式

本文构造了抛物型方程的带移动界面的一般并行差分格式,并证明了其稳定性.许多简单实用的差分格式都能从中导出.理论分析和数值试验检验了算法的稳定性,精度和并行性.     

抛物型方程移动界面的并行差分格式

本文构造了抛物型方程的带移动界面的一般并行差分格式,并证明了其稳定性。许多简单实用的差分格式都能从中导出。理论分析和数值试验检验了算法的稳定性,精度和并行性。     

求解带有不连续波数的二维变系数 Helmholtz方程的一种高精度紧致差分方法

很多实际物理问题都可以由带有不连续波数的变系数 Helmholtz 方程进行数值模拟。Helmholtz 方程的数值方法研究是热点问题之一,具有重要的理论和实际意义。由于波数的不连续性,使用传统的有限差分方法求解带有不连续波数的 Helmholtz 方程时通常无法达到原有差分格式的精度。结合浸入界面方法的思想,对带有不连续波数的二维变系数 Helmholtz 方程构造了一类新的四阶紧致有限差分格式,数值实验验证了新方法的可靠性和有效性。     

F—X域有限差分法叠后偏移方法研究

F—X域有限差分叠后偏移就是从地震单程波的波散关系出发,在频率一空间域利用有限差分求解波动方程的一种偏移处理方法。本文编程实现了对简单叠后剖面模型的数值模拟及偏移处理,并得到了较可靠的结果。通过利用不同适应角度的有限差分算法对模型进行试算,发现该方法可以适应地下速度场的纵横向变化,对于有倾角限制,低角度方程所引起的界面上移,其大角度方程可以很好的解决这一问题。     

一类复Ginzburg-Laudau方程的有限差分法

本文讨论的是一类复Ginzburg-Laudau方程的数值解法.通过数值解法中的差分方法,建立它的差分格式,并讨论它的二阶差分格式的收敛性和稳定性.     

三维定常/非定常Stokes方程的维数分裂算法

本文针对三维柱形区域提出了定常/非定常Stokes方程基于一致分裂格式的维数分裂算法(DSA).文章推导了三维定常/非定常Stokes方程维数分裂方法的数值迭代格式.新算法的优势在于一系列的二维问题能够并行执行,而且数值计算中避免了三维网格的生成.大量的数值结果表明新算法既能获得最优收敛阶,而且能获得比采用四面体元求解更精确的逼近解.最后,通过采用并行求解新算法能够得到比较好的加速比和并行效率.     

三维定常/非定常Stokes方程的维数分裂算法（英文）

本文针对三维柱形区域提出了定常/非定常Stokes方程基于一致分裂格式的维数分裂算法(DSA).文章推导了三维定常/非定常Stokes方程维数分裂方法的数值迭代格式.新算法的优势在于一系列的二维问题能够并行执行,而且数值计算中避免了三维网格的生成.大量的数值结果表明新算法既能获得最优收敛阶,而且能获得比采用四面体元求解更精确的逼近解.最后,通过采用并行求解新算法能够得到比较好的加速比和并行效率.     

北极冰的热传递模型和数值模拟

夏季北极冰的热传递主要在相变过程中进行,因此常规的温度场方程不适应这种状况。建立一个新模型是必要的。因此根据热力学中焓的概念,提出了焓度和焓扩散系数的概念,建立了焓的热传导方程并根据潜热把焓的热传导方程转化为温度场方程。提出了焓扩散系数的辨识模型,用半隐式差分格式和许瓦兹交替方向法来解热传导方程和灵敏度方程,用牛顿拉夫逊算法进行辨识。根据2003年8月第二次北极科学考察现场测试数据进行数值模拟,模拟结果说明本文的数学模型和算法是正确的和可行的。     

The Eulerian–Lagrangian method of fundamental solutions for two-dimensional unsteady Burgers’ equations

The Eulerian–Lagrangian method of fundamental solutions is proposed to solve the two-dimensional unsteady Burgers’ equations. Through the Eulerian–Lagrangian technique, the quasi-linear Burgers’ equations can be converted to the characteristic diffusion equations. The method of fundamental solutions is then adopted to solve the diffusion equation through the diffusion fundamental solution; in the meantime the convective term in the Burgers’ equations is retrieved by the back-tracking scheme along the characteristics. The proposed numerical scheme is free from mesh generation and numerical integration and is a truly meshless method. Two-dimensional Burgers’ equations of one and two unknown variables with and without considering the disturbance of noisy data are analyzed. The numerical results are compared very well with the analytical solutions as well as the results by other numerical schemes. By observing these comparisons, the proposed meshless numerical scheme is convinced to be an accurate, stable and simple method for the solutions of the Burgers’ equations with irregular domain even using very coarse collocating points.     

Approximate Riemann solutions of the two-dimensional shallow-water equations

A finite-difference scheme based on flux difference splitting is presented for the solution of the two-dimensional shallow-water equations of ideal fluid flow. A linearised problem, analogous to that of Riemann for gasdynamics, is defined and a scheme, based on numerical characteristic decomposition, is presented for obtaining approximate solutions to the linearised problem. The method of upwind differencing is used for the resulting scalar problems, together with a flux limiter for obtaining a second-order scheme which avoids non-physical, spurious oscillations. An extension to the two-dimensional equations with source terms, is included. The scheme is applied to a dam-break problem with cylindrical symmetry.     

无条件稳定的动态样条加权残数法

本文利用样条加权残数法建立了一种求解结构动力响应的直接积分法,所得的递推格式是无条件稳定的计算格式。因此,这种方法是无条件稳定的解法,对振型叠加法也适用。     

Numerical solution of the kinetic model equations for hypersonic flows

A numerical method for solving the model kinetic equations for hypersonic flows has been developed. The model equations for the distribution function are discretized in phase space using a second order upwind finite difference scheme for the spatial derivatives. The resulting system of ordinary differential equations in time is integrated by using a rational Runge-Kutta scheme. Calculations were carried out for hypersonic flow around a double ellipse under various free stream conditions. Calculated results are compared with the Navier-Stokes solutions and the Direct Simulation Monte Carlo (DSMC) method for the corresponding case. The agreement is quite excellent in general.     

Fast Computation of Stationary Inviscid Flow Around an Airfoil

The parallel performance of an implicit solver for the Euler equations on a structured grid is discussed. The flow studied is a two-dimensional transonic flow around an airfoil. The spatial discretization involves the MUSCL scheme, a higher-order Total Variation Diminishing scheme. The solver described in this paper is an implicit solver that is based on quasi Newton iteration and approximate factorization to solve the linear system of equations resulting from the Euler Backward scheme. It is shown that the implicit time-stepping method can be used as a smoother to obtain an efficient and stable multigrid process. Also, the solver has good properties for parallelization comparable with explicit time-stepping schemes. To preserve data locality domain decomposition is applied to obtain a parallelizable code. Although the domain decomposition slightly affects the efficiency of the approximate factorization method with respect to the number of time steps required to attain the stationary solution, the results show that this hardly affects the performance for practical purposes. The accuracy with which the linear system of equations is solved is found to be an important parameter. Because the method is equally applicable for the Navier-Stokes equations and in three-dimensions, the presented combination of efficient parallel execution and implicit time-integration provides an interesting perspective for time-dependent problems in computational fluid dynamics.     

A novel accurate and computationally efficient integration approach to viscoplastic constitutive model

A novel, accurate, and computationally efficient integration approach is developed to integrate small strain viscoplastic constitutive equations involving nonlinear coupled first-order ordinary differential equations. The developed integration scheme is achieved by a combination of the implicit backward Euler difference approximation and the implicit asymptotic integration. For the uniaxial loading case, the developed integration scheme produces accurate results irrespective of time steps. For the multiaxial loading case, the accuracy and computational efficiency of the developed integration scheme are better than those of either the implicit backward Euler difference approximation or the implicit asymptotic integration. The simplicity of the developed integration scheme is equivalent to that of the implicit backward Euler difference approximation since it also reduces the solution of integrated constitutive equations to the solution of a single nonlinear equation. The algorithm tangent constitutive matrix derived for the developed integration scheme is consistent with the integration algorithm and preserves the quadratic convergence of the Newton–Raphson method for global iterations.     

Description of the orthogonal collocation method for modelling flames in furnaces

The orthogonal collocation method has been used to solve some of the governing equations encountered in modelling flames with the boundary layer approximation. A split trial function is used to permit greater resolution of the region close to the reactor inlet. The governing equations constituting a set of coupled parabolic partial differential equations are transformed to a set of ordinary differential equations and integrated numerically by an explicit marching method. The details of the numerical scheme are elaborated.     

Boltzmann schemes for continuum gas dynamics

Many problems arising in the aerodynamic design of aerospace vehicles require the numerical solution of the Euler equations of gas dynamics. These are nonlinear partial differential equations admitting weak solutions such as shock waves and constructing robust numerical schemes for these equations is a challenging task. A new line of research called Boltzmann or kinetic schemes discussed in the present paper exploits the connection between the Boltzmann equation of the kinetic theory of gases and the Euler equations for inviscid compressible flows. Because of this connection, a suitable moment of a numerical scheme for the Boltzmann equation yields a numerical scheme for the Euler equations. This idea called the “moment method strategy” turns out to be an extremely rich methodology for developing robust numerical schemes for the Euler equations. The richness is demonstrated by developing a variety of kinetic schemes such as kinetic numerical method, kinetic flux vector splitting method, thermal velocity based splitting, multidirectional upwind method and least squares weak upwind scheme. A 3-D time-marching Euler code calledbheema based on the kinetic flux vector splitting method and its variants involving equilibrium chemistry have been developed for computing hypersonic reentry flows. The results obtained from the codebheema demonstrate the robustness and the utility of the kinetic flux vector splitting method as a design tool in aerodynamics. The work presented in this paper is based on the research work done by several graduate students at our laboratory and collaborators from research and development organizations within the country.     

A FULLY SPECTRAL METHOD FOR HYPERBOLIC EQUATIONS

This investigation presents a fully spectral method for solving coupled hyperbolic partial differential equations. The spectral method is based on the Galerkin–collocation technique. Two different preconditioners, the Preissmann and upwind schemes, are evaluated for their performance in solving the discretized equations. It has been found, for the cases considered, that the upwind scheme is a viable preconditioner for the fully spectral discretization of hyperbolic PDEs. Its performance as a preconditioner is in every way superior to that of the Preissmann scheme. It is established that the relative accuracy of different numerical solutions is reliably indicated by the root-mean-square average of their residuals obtained by the discretization. It is also established that the scheme gives much better accuracy than the finite-difference Preissmann scheme, for the same amount of computational effort, for both linear and non-linear problems.     

An Iterative Scheme of Arbitrary Odd Order and Its Basins of Attraction for Nonlinear Systems

In this paper, we propose a fifth-order scheme for solving systems of nonlinear equations. The convergence analysis of the proposed technique is discussed. The proposed method is generalized and extended to be of any odd order of the form 2n − 1. The scheme is composed of three steps, of which the first two steps are based on the two-step Homeier’s method with cubic convergence, and the last is a Newton step with an appropriate approximation for the derivative. Every iteration of the presented method requires the evaluation of two functions, two Fréchet derivatives, and three matrix inversions. A comparison between the efficiency index and the computational efficiency index of the presented scheme with existing methods is performed. The basins of attraction of the proposed scheme illustrated and compared to other schemes of the same order. Different test problems including large systems of equations are considered to compare the performance of the proposed method according to other methods of the same order. As an application, we apply the new scheme to some real-life problems, including the mixed Hammerstein integral equation and Burgers’ equation. Comparisons and examples show that the presented method is efficient and comparable to the existing techniques of the same order.