椭圆-抛物偏微分方程奇异摄动问题的差分解法

本文讨论了含有小参数在高阶导数项的椭圆型方程奇异摄动问题的差分解法.当ε=0时椭圆型方程退化为抛物型方程.作者根据此问题解的边界层性质,构造了特殊的差分格式:研究了它的收敛性和解的渐近性态.最后给出一个数值例题.     

Balance-characteristic scheme as applied to the shallow water equations over a rough bottom

The CABARET scheme is used for the numerical solution of the one-dimensional shallow water equations over a rough bottom. The scheme involves conservative and flux variables, whose values at a new time level are calculated by applying the characteristic properties of the shallow water equations. The scheme is verified using a series of test and model problems.     

Finite-dimensional linear approximations of solutions to general irregular nonlinear operator equations and equations with quadratic operators

A general scheme for improving approximate solutions to irregular nonlinear operator equations in Hilbert spaces is proposed and analyzed in the presence of errors. A modification of this scheme designed for equations with quadratic operators is also examined. The technique of universal linear approximations of irregular equations is combined with the projection onto finite-dimensional subspaces of a special form. It is shown that, for finite-dimensional quadratic problems, the proposed scheme provides information about the global geometric properties of the intersections of quadrics.     

一类变系数微分方程差分格式的收敛性(英文)

本文首先对一类变系数微分方程建立有限差分格式.然后利用矩阵的特征值和范数理论,讨论该格式解的收敛性和唯一性.通过数值算例,说明该格式既有效又便于模拟.并且文中所用方法还能用于高阶微分方程和某些非线性微分方程问题的研究.     

A New Exponential Compact Scheme for the Two-Dimensional Unsteady Nonlinear Burgers' and Navier-Stokes Equations in Polar Cylindrical Coordinates

In this article, a new compact difference scheme is proposed in exponential form to solve two-dimensional unsteady nonlinear Burgers' and Navier-Stokes equations of motion in polar cylindrical coordinates by using half-step discretization. At each time level by using only nine grid points in space, the proposed scheme gives accuracy of order four in space and two in time. The method is directly applicable to the equations having singularities at boundary points. Stability analysis is explained in detail and many benchmark problems like Burgers', Navier-Stokes and Taylor-vortex problems in polar cylindrical coordinates are solved to verify the accuracy and efficiency of the scheme.     

Approximation schemes for nonlinear neutral optimal control systems

We discuss methods of approximating stable neutral functional differential equations and associated optimal control problems by sequences of optimal control problems for ordinary differential equations. By introducing a class of “mollified” neutral functional differential equations, convergence of the linear interpolating spline and the averaging approximation scheme is proved. A number of numerical examples are included.     

非定常不可压Navier-Stokes方程的高效和稳健的差分格式Ⅱ

A second order accurate implicit finite difference scheme CNMT2 is proposed in this paper for the unsteady incompressible Navier-Stokes equations. It is proved that the scheme is unconditionally nonlinearly stable on smoothly nonuniform halfstaggered meshes; this stability also holds for this scheme with the pressure correction projection method. However, it is found that the pressure correction projection method may lead to deviation problems in practical simulation of high Re flow;the reason and the cure is given in this paper in terms of differential-algebraic equations.     

Difference spline scheme for modeling of convective flows using full Navier — Stokes equations

A difference scheme is constructed, in which enhanced stability is achieved by simultaneous solutions of the equations of motion, energy, and continuity. Spline approximations of spatial derivatives (with the original equations written in divergence form) substantially improve the accuracy of the scheme compared with the standard difference scheme using symmetric differences. The efficiency of the scheme is demonstrated for some problems of convective flow of compressible gas with lateral and bottom heating.Translated from Matematicheskoe Modelirovanie i Reshenie Obratnykh Zadach. Matematicheskoi Fiziki, pp. 38–45, 1993.     

Upwind compact finite difference scheme for time-accurate solution of the incompressible Navier-Stokes equations

This article presents a time-accurate numerical method using high-order accurate compact finite difference scheme for the incompressible Navier-Stokes equations. The method relies on the artificial compressibility formulation, which endows the governing equations a hyperbolic-parabolic nature. The convective terms are discretized with a third-order upwind compact scheme based on flux-difference splitting, and the viscous terms are approximated with a fourth-order central compact scheme. Dual-time stepping is implemented for time-accurate calculation in conjunction with Beam-Warming approximate factorization scheme. The present compact scheme is compared with an established non-compact scheme via analysis in a model equation and numerical tests in four benchmark flow problems. Comparisons demonstrate that the present third-order upwind compact scheme is more accurate than the non-compact scheme while having the same computational cost as the latter.     

A second-order ADI scheme for three-dimensional parabolic differential equations

A second-order unconditionally stable ADI scheme has been developed for solving three-dimensional parabolic equations. This scheme reduces three-dimensional problems to a succession of one-dimensional problems. Further, the scheme is suitable for simulating fast transient phenomena. Numerical examples show that the scheme gives an accurate solution for the parabolic equation and converges rapidly to the steady state solution. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14:159–168, 1998     

On the linear two-point boundary-value problem

In this note, a noniterative scheme for solving two-point boundary-value problems for single and multi-input, linear constant systems is developed. The scheme requires the solution of 2n differential equations.     

小波Galerkin法在非线性分岔问题求解中的应用

通过一个典型的Bratu问题,研究了小波Galerkin法(WGM)在非线性分岔问题求解方面的应用.首先,利用基于Coiflet的小波Galerkin法,对一维和二维Bratu方程进行离散;然后针对单参数问题,推导了追踪解曲线的伪弧长格式和直接计算极值型分岔点的扩展方程;针对双参数问题,推导了追踪稳定边界的伪弧长格式和...     

A two-grid discretization scheme for eigenvalue problems

A two-grid discretization scheme is proposed for solving eigenvalue problems, including both partial differential equations and integral equations. With this new scheme, the solution of an eigenvalue problem on a fine grid is reduced to the solution of an eigenvalue problem on a much coarser grid, and the solution of a linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy.

     

A new third order exponentially fitted discretization for the solution of non-linear two point boundary value problems on a graded mesh

This paper puts forward a novel graded mesh implicit scheme resting upon full step discretization of order three for computation of non-linear two point boundary value problems. The suggested method is compact and employs three nodal points for the unknown function $u(x)$ in spatial axis. We have also performed error analysis of the cited method. The given method was tried (implemented) upon multiple problems in Cartesian and Polar coordinates with extremely favorable outcomes. This method, though meant for scalar equations, was further extended to compute the vector equations of two point nonlinear boundary value problems. To check the validity of the proposed scheme, we applied it to multiple problems and obtained supporting numerical computations.     

Modulation equations and Reynolds averaging for finite-amplitude non-linear waves in an incompressible fluid

A formal perturbation scheme is developed to determine originalmodulation equations for laminar finite-amplitude non-linearwaves in an incompressible fluid. Three idealized problems areanalysed. The modulation equations comprise conservation ofwaves, averaged conditions for conservation of mass, momentum,kinetic energy and angular momentum and the averaged projectionof the Navier–Stokes equations onto the vorticity vector.The last of these modulation equations, which is related tovortex stretching, only appears in 3D problems. The techniqueof Reynolds averaging is also employed to obtain equations forthe mean velocities and pressure. The Reynolds-averaged Navier–Stokesequations correspond to the modulation equations for conservationof mass and momentum. However, the Reynolds stress transportequations are shown to be inconsistent with the other necessarymodulation equations. In two further idealized problems, exactsolutions of the Navier–Stokes equations are obtainedby employing the modulation equations.     

Viscosity-splitting scheme for the Navier-Stokes equations

A viscosity-splitting scheme for the initial boundary value problems of the Navier-Stokes equations is considered. In the scheme, the Stokes equation is solved in conjunction with a nonhomogeneous boundary condition which connects the tangent flow with a no-slip flow. Convergence is proved.     

双相滞热传导方程的有限元分析

本文考虑一类具有广泛应用背景的双相滞热传导方程混合边界问题．建立了其有限元和交替方向有限元的两种数值逼近格式．利用微分方程的先验估计理论与技巧,作出了数值解的L^2—范数估计结果．基于一系列的误差估计,也研究了两种逼近格式数值的稳定性和收敛性。     

A new extension of boundary elements method for classical elliptic partial differential equations with constant coefficients

This article is devoted to an extension of boundary elements method (BEM) for solving elliptic partial differential equations of general type with constant coefficients. As the fundamental solution of these equations was not available in the literature, BEM was not able to handle them, directly. So the dual reciprocity method (DRM) has been applied to tackle these problems. In this work, a fundamental solution for these equations is obtained and a new formulation is derived to solve them. Besides, we show that the rate of convergence of the new scheme is quadratic when singular (boundary and domain) integrals are calculated, accurately. The new scheme is applicable on complex domains, without needing internal nodes, just same as conventional BEM. So the CPU time of the new scheme is much less than that of the DRM. Numerical examples presented in the article show ability and efficiency of the new scheme in solving two‐dimensional nonhomogenous elliptic boundary value problems, clearly. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 2027–2042, 2015     

Fast Solvers of Fredholm Optimal Control Problems

<正>The formulation of optimal control problems governed by Fredholm integral equations of second kind and an efficient computational framework for solving these control problems is presented.Existence and uniqueness of optimal solutions is proved. A collective Gauss-Seidel scheme and a multigrid scheme are discussed.Optimal computational performance of these iterative schemes is proved by local Fourier analysis and demonstrated by results of numerical experiments.