一个半隐式指数型差分格式

为了用数值方法解对流-扩散方程,Allen-Southwell于1955年提出一种特殊形式的差分格式.这种格式与通常用差商代替微商所得到的差分方程不同,其系数带有指数函数,通常称此类差分格式为指数型格式.此后一直到1969年,苏联学者bH才首先证明了它对小参数的一致收敛性,使这类格式得到广泛的研究和应用.近几年来,许多人将隐式指数型格式用于解时间相关的对流-扩散方程,其最大缺点是:解多维     

The Milstein Scheme for Stochastic Delay Differential Equations Without Using Anticipative Calculus

The Milstein scheme is the simplest nontrivial numerical scheme for stochastic differential equations with a strong order of convergence one. The scheme has been extended to the stochastic delay differential equations but the analysis of the convergence is technically complicated due to anticipative integrals in the remainder terms. This article employs an elementary method to derive the Milstein scheme and its first order strong rate of convergence for stochastic delay differential equations.     

对流占优扩散方程的一种特征差分算法

A new kind of characteristic-difference scheme for convection-diffusion equations is constructed by characteristic method and bilinear interpolation method. The convergence of the scheme is proved. The advantages of this scheme are to obtain the solutions of the convection diffusion equations with variable coefficient expediently and to reduce the numerical oscillations of the convectiondominanted diffusion equations effectively.     

An iterative solver for a coupled system of Helmholtz equations

An iterative solver for a pair of coupled partial differential equations that are related to the Maxwell equations is discussed. The convergence of the scheme depends on the choice of two parameters. When the first parameter is fixed, the scheme is seen to be a successive under-relaxation scheme in the other parameter. A theory for convergence of the scheme is discussed for a special case of the equations, and several numerical examples are presented. © 1996 John Wiley & Sons, Inc.     

Convergence of a finite-volume scheme for the drift-diffusion equations in 1D

We construct an approximate solution to one-dimensional nonlineardrift–diffusion equations by a finite-volume scheme. Theconvergence of the scheme is proved, which yields the globalexistence of solutions to the equations. This result is valideven in the presence of vacuum state: the densities of freecarriers of charge vanish for some times and positions, whichimplies that the continuity equations are degenerate parabolicequations. Finally, numerical simulations are performed by thefinite-volume scheme.     

Integrable Quasilinear Equations

We develop a classification scheme for integrable third-order scalar evolution equations using the symmetry approach to integrability. We use this scheme to study quasilinear equations of a particular type and prove that several equations that were suspected to be integrable can be reduced to the well-known Korteweg–de Vries and Krichever–Novikov equations via a Miura-type differential substitution.     

无波动,无自由参数,耗散的隐式差分格式

本文建立了求解NS方程和Euler方程无波动、无自由参数、耗散的隐式差分格式.该格式是TVD的和无条件稳定的.其隐式部分在1,2,3维情况下仅分别依赖于3,5,9个点,且系数矩阵是主对角占优的.计算例题表明,该方法可获得和显式方法相同的精度,能很好地捕捉激波和剪切层,且计算时间比显式有较多的节省.     

随机微分方程欧拉格式算法分析

首先给出了线性随机微分方程的欧拉格式算法,然后给出了非线性随机微分方程变步长的欧拉格式算法,接着讨论了其对初值的连续依赖性和收敛性.     

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.     

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.     

THE ASYMPTOTIC PRESERVING UNIFIED GAS KINETIC SCHEME FOR GRAY RADIATIVE TRANSFER EQUATIONS ON DISTORTED QUADRILATERAL MESHES

In this paper, we consider the multi-dimensional asymptotic preserving unified gas kinetic scheme for gray radiative transfer equations on distorted quadrilateral meshes. Different from the former scheme [J. Comput. Phys. 285(2015), 265-279] on uniform meshes, in this paper, in order to obtain the boundary fluxes based on the framework of unified gas kinetic scheme (UGKS), we use the real multi-dimensional reconstruction for the initial data and the macro-terms in the equation of the gray transfer equations. We can prove that the scheme is asymptotic preserving, and especially for the distorted quadrilateral meshes, a nine-point scheme [SIAM J. SCI. COMPUT. 30(2008), 1341-1361] for the diffusion limit equations is obtained, which is naturally reduced to standard five-point scheme for the orthogonal meshes. The numerical examples on distorted meshes are included to validate the current approach.     

Stability and convergence of staggered Runge-Kutta schemes for semilinear wave equations

A staggered Runge-Kutta (staggered RK) scheme is a Runge-Kutta type scheme using a time staggered grid, as proposed by Ghrist et al. in 2000 [6]. Afterwards, Verwer in two papers investigated the efficiency of a scheme proposed by Ghrist et al. [6] for linear wave equations. We study stability and convergence properties of this scheme for semilinear wave equations. In particular, we prove convergence of a fully discrete scheme obtained by applying the staggered RK scheme to the MOL approximation of the equation.     

Numeric solution of advection–diffusion equations by a discrete time random walk scheme

Explicit numerical finite difference schemes for partial differential equations are well known to be easy to implement but they are particularly problematic for solving equations whose solutions admit shocks, blowups, and discontinuities. Here we present an explicit numerical scheme for solving nonlinear advection–diffusion equations admitting shock solutions that is both easy to implement and stable. The numerical scheme is obtained by considering the continuum limit of a discrete time and space stochastic process for nonlinear advection–diffusion. The stochastic process is well posed and this guarantees the stability of the scheme. Several examples are provided to highlight the importance of the formulation of the stochastic process in obtaining a stable and accurate numerical scheme.     

On nonlinear Fredholm integral equations with non-differentiable Nemystkii operator

From decomposition method for operators, we consider a Newton-Steffensen iterative scheme for approximating a solution of nonlinear Fredholm integral equations with non-differentiable Nemystkii operator. By means of a convergence study of the iterative scheme applied to this type of nonlinear Fredholm integral equations, we obtain domains of existence and uniqueness of solution for these equations. In addition, we illustrate this study with a numerical experiment.     

The Minimal Radius of Galerkin Information for the Fredholm Problem of the First Kind

We propose a new scheme of discretization for solving Fredholm integral equations of the first kind and show that for some classes of equations this scheme is order-optimal in the sense of amount of used Galerkin information.     

解偏微分方程的多步-小波-Galerkin方法

推广Lax-Wendroff多步方法,建立一类新的显式和隐式相结合的多步格式,并以此为基础提出了一类显隐多步-小波-Galerkin方法,可以用来求解依赖时间的偏微分方程.不同于Taylor-Galerkin方法,文中的方案在提高时间离散精度时不包含任何新的高阶导数.由于引入了隐式部分,与传统的多步方法相比该方案有更好的稳定性,适合于求解非线性偏微分方程,理论分析和数值例子都说明了方法的有效性.     

Two difference schemes for the numerical solution of Maxwell’s equations as applied to extremely and super low frequency signal propagation in the Earth-ionosphere waveguide

Two explicit two-time-level difference schemes for the numerical solution of Maxwell’s equations are proposed to simulate propagation of small-amplitude extremely and super low frequency electromagnetic signals (200 Hz and lower) in the Earth-ionosphere waveguide with allowance for the tensor conductivity of the ionosphere. Both schemes rely on a new approach to time approximation, specifically, on Maxwell’s equations represented in integral form with respect to time. The spatial derivatives in both schemes are approximated to fourth-order accuracy. The first scheme uses field equations and is second-order accurate in time. The second scheme uses potential equations and is fourth-order accurate in time. Comparative test computations show that the schemes have a number of important advantages over those based on finite-difference approximations of time derivatives. Additionally, the potential scheme is shown to possess better properties than the field scheme.     

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.     

An implicit finite-difference algorithm for the numerical simulation of supersonic flow over blunted bodies

In order to compute axisymmetric laminar supersonic flow we use an unsteady implicit finite-difference scheme. This scheme solves numerically either the inviscid Euler equations or the ‘thin-layer’ Navier-Stokes equations. In both cases the scheme leads to large sparse non-linear systems, which can be solved by a genuine iteration process. The convergence of this process is shown and numerical results are given.     

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.