Voherra方程数值解中带自适应步长控制的Runge—Kutta方法

在本文中我们构造了解第二类Volterra方程的一般Runge—Kutta方法,并且研究了第二类Voherra方程数值解法的自适应步长控制。     

Volterra积分方程的蒙特卡罗数值求解方法

本文讨论了第一类、第二类以及具有奇异核的Volterra积分方程的数值解问题.利用重要抽样蒙特卡罗随机模拟方法获得积分方程解的近似计算结果.通过对文献中算例的实现表明文中所提方法扩展了Volterra型积分方程的数值求解方法,     

压电弹性动力学中第二类Volterra积分方程的数值解法

对于径向变形的压电空心圆柱和空心球弹性动力学问题,丁皓江等最近的研究表明,可以将它转变为关于一个时间函数的第二类Volterra积分方程,使求解工作得到极大的简化,又使探索第二类Volterra积分方程的快速而又高精度的数值解法成为一个关键．采用插值逼近方法,成功地导出了两个新型的递推公式,不仅计算速度快,且在较大时间步长时仍具有足够的精度,有着广泛的应用价值．     

利用随机模拟方法求解第二类积分方程

给出一种求解第二类Fredholm和Volterra积分方程的数值算法,算法在数值积分技术的基础上使用Monte Carlo随机模拟方法求积分方程的近似解.通过数值例子证明了该算法是有效的.     

一维热传导方程移动边界识别的计算方法

构造了一种正则化的积分方程方法来由Cauchy数据确定一维热传导方程的移动边界．在将区域延拓至规则区域后,通过Fourier方法将问题转化为一个第一类Volterra积分方程．然后分别用Lavrentiev正则化方法以及Tikhonov正则化方法将不稳定的第一类Volterra积分方程转化为适定的第二类积分方程,并分别将积分方程转化为常微分方程组,并用Runge—Kutta方法数值求解,以及直接离散来求解．最后通过自由边界上的条件得到数值的移动边界．通过一些数值试验表明此方法是有效可行的,并且给出的方法无需迭代,数值计算较简单．     

随机Volterra积分方程相容解的稳定性

本文考虑了随机Volterra积分方程相容解的稳定性.应用Lyapunov第二方法,并以推广的Ito公式为工具,给出了随机Volterra积分方程相容解的几乎确定指数稳定和矩指数稳定的充分性原则.     

一种改进的Adomian分解方法

给出了一种新的改进Adomian分解方法,新方法能有效地解决传统Adomian分解方法及其改进方法的不足.将新改进方法应用于第二类Volterra积分方程、积分-微分方程求解,并与传统Adomian分解方法及其改进方法作比较分析,结果表明提出的新改进方法能返回方程精确解析解.     

关于求解第二类Volterra积分方程的Picard迭代收敛性的一点注记

考虑第二类Volterra积分方程: φ(x)+integral from n=0 to x(K(x,y)φ(y)dy)=f(x),x∈[0,L],(1)其中f(x)∈C([0,L]),核函数 K(x,y)对y可积,且     

具可微卷积核的第二类Volterra积分方程的行列式级数解法

§1.引言 对具可微卷积核的第二类Volterra积分方程 y(x)=f(x)+λ integral from n=a to x(K(x-t)y(t)dt),(1)通常的解法有迭代法与Laplace变换法以及化为微分方程求解等.毫无疑义,这些方法对于方程(1)的求解是重要的.但这些方法也有其本质的缺点,即在求解过程中,往往涉     

具无限时滞和无界时滞的卷积Volterra方程稳定价的等价

本文证明了具无界时滞的Volterra方程 的一致渐近稳定性与具无限时滞的Volterra方程 的一致渐近稳定性是等价的     

A distributed combustion solver for engine simulations on grids

Multi-dimensional models for predictive simulations of modern engines are an example of multi-physics and multi-scale mathematical models, since lots of thermofluiddynamic processes in complex geometrical configurations have to be considered. Typical models involve different submodels, including turbulence, spray and combustion models, with different characteristic time scales. The predictive capability of the complete models depends on the accuracy of the submodels as well as on the reliability of the numerical solution algorithms. In this work we propose a multi-solver approach for reliable and efficient solution of the stiff Ordinary Differential Equation (ODE) systems arising from detailed chemical reaction mechanisms for combustion modeling. Main aim was to obtain high-performance parallel solution of combustion submodels in the overall procedure for simulation of engines on distributed heterogeneous computing platforms. To this aim we interfaced our solver with the CHEMKIN-II package and the KIVA3V-II code and carried out multi-computer simulations of realistic engines. Numerical experiments devoted to test reliability of the simulation results and efficiency of the distributed combustion solver are presented and discussed.     

Multiresolution algorithms for the numerical solution of hyperbolic conservation laws

Given any scheme in conservation form and an appropriate uniform grid for the numerical solution of the initial value problem for one-dimensional hyperbolic conservation laws we describe a multiresolution algorithm that approximates this numerical solution to a prescribed tolerance in an efficient manner. To do so we consider the grid-averages of the numerical solution for a hierarchy of nested diadic grids in which the given grid is the finest, and introduce an equivalent multiresolution representation. The multiresolution representation of the numerical solution consists of its grid-averages for the coarsest grid and the set of errors in predicting the grid-averages of each level of resolution in this hierarchy from those of the next coarser one. Once the numerical solution is resolved to our satisfaction in a certain locality of some grid, then the prediction errors there are small for this particular grid and all finer ones; this enables us to compress data by setting to zero small components of the representation which fall below a prescribed tolerance. Therefore instead of computing the time-evolution of the numerical solution on the given grid we compute the time-evolution of its compressed multiresolution representation. Algorithmically this amounts to computing the numerical fluxes of the given scheme at the points of the given grid by a hierarchical algorithm which starts with the computation of these numerical fluxes at the points of the coarsest grid and then proceeds through diadic refinements to the given grid. At each step of refinement we add the values of the numerical flux at the center of the coarser cells. The information in the multiresolution representation of the numerical solution is used to determine whether the solution is locally well-resolved. When this is the case we replace the costly exact value of the numerical flux with an accurate enough approximate value which is obtained by an inexpensive interpolation from the coarser grid. The computational efficiency of this multiresolution algorithm is proportional to the rate of data compression (for a prescribed level of tolerance) that can be achieved for the numerical solution of the given scheme.     

带Markov状态转换的跳扩散方程的数值解

研究了一类带Markov状态转换的跳扩散方程的数值解的问题,为讨论这类方程精确解的数值计算问题,我们给出了一种基于Euler格式的方程解的跳适应算法,并在一定的条件下,证明了基于这种新的跳适应算法所得到的方程的数值解是收敛于它的精确解,同时还给出了数值解收敛到其精确解的收敛阶数.最后,本文通过两个例子说明了这种跳适应算法的计算有效性.     

A comparison of the String Gradient Weighted Moving Finite Element method and a Parabolic Moving Mesh Partial Differential Equation method for solutions of partial differential equations

We compare numerical experiments from the String Gradient Weighted Moving Finite Element method and a Parabolic Moving Mesh Partial Differential Equation method, applied to three benchmark problems based on two different partial differential equations. Both methods are described in detail and we highlight some strengths and weaknesses of each method via the numerical comparisons. The two equations used in the benchmark problems are the viscous Burgers’ equation and the porous medium equation, both in one dimension. Simulations are made for the two methods for: a) a travelling wave solution for the viscous Burgers’ equation, b) the Barenblatt selfsimilar analytical solution of the porous medium equation, and c) a waiting-time solution for the porous medium equation. Simulations are carried out for varying mesh sizes, and the numerical solutions are compared by computing errors in two ways. In the case of an analytic solution being available, the errors in the numerical solutions are computed directly from the analytic solution. In the case of no availability of an analytic solution, an approximation to the error is computed using a very fine mesh numerical solution as the reference solution.     

PROBABILISTIC NUMERICAL APPROACH FOR PDE AND ITS APPLICATION IN THE VALUATION OF EUROPEAN OPTIONS

1. IlltroductionThis paPer is aimed to give a probabilistic numerical aPproach for PDE. Probabilistic numerical method can get the solution one by one, whiCh differs from other nu-merical methods,such aJs the Anite e1ement and drite dffeence method, and rea1ize total parallel computing easily Another advallage of this method is that it suits for problems of highdimension becauseit is dimension indep endent.Consider the fOl1owing Cauchy problem of convectiondiffusion equations. FOr simplic…     

On convergence of numerical methods for variational–hemivariational inequalities under minimal solution regularity

Hemivariational inequalities have been successfully employed for mathematical and numerical studies of application problems involving nonsmooth, nonmonotone and multivalued relations. In recent years, error estimates have been derived for numerical solutions of hemivariational inequalities under additional solution regularity assumptions. Since the solution regularity properties have not been rigorously proved for hemivariational inequalities, it is important to explore the convergence of numerical solutions of hemivariational inequalities without assuming additional solution regularity. In this paper, we present a general convergence result enhancing existing results in the literature.     

Zu einigen mathematischen problemen in der theorie der transonischen strömung

The work deals with a definition of a weak solution of steady plane transonic flows past a thin profile, with the properties of the solution across a shock wave, and with a derivation of a conservative difference scheme suitable for numerical solution of the above mentioned problem by a finite difference method. The work presents several examples of numerical solution of transonic flows past a profile, through a plane cascade and some three-dimensional results. The numerical results presented are compared with experimental results or with numerical results by other authors.     

Exact Solutions of Extended Boussinesq Equations

The problem addressed in this paper is the verification of numerical solutions of nonlinear dispersive wave equations such as Boussinesq-like system of equations. A practical verification tool for numerical results is to compare the numerical solution to an exact solution if available. In this work, we derive some exact solitary wave solutions and several invariants of motion for a wide range of Boussinesq-like equations using Maple software. The exact solitary wave solutions can be used to specify initial data for the incident waves in the Boussinesq numerical model and for the verification of the associated computed solution. The invariants of motions can be used as verification tools for the conservation properties of the numerical model.     

On the numerical solution of Cauchy type singular integral equations by the collocation method

The collocation method for the numerical solution of Fredholm integral equations of the second kind is applied, properly modified, to the numerical solution of Cauchy type singular integral equations of the first or the second kind but with constant coefficients. This direct method of numerical solution of Cauchy type singular integral equations is compared afterwards with the corresponding method resulting from applying the collocation method to the Fredholm integral equation of the second kind equivalent to the Cauchy type singular integral equation, as well as with another method, based also on the regularization procedure, for the numerical solution of the same class of equations. Finally, the convergence of the method is discussed.     

An efficient approach for the numerical solution of the Monge-Ampère equation

In this paper, we present a new method to compute the numerical solution of the elliptic Monge-Ampère equation. This method is based on solving a parabolic Monge-Ampère equation for the steady state solution. We study the problem of global existence, uniqueness, and convergence of the solution of the fully nonlinear parabolic PDE to the unique solution of the elliptic Monge-Ampère equation. Some numerical experiments are presented to show the convergence and the regularity of the numerical solution.