共查询到20条相似文献,搜索用时 31 毫秒
1.
四阶R-K方法中一类新算法的分析 总被引:1,自引:0,他引:1
对常微分方程初值问题数值计算中的四阶R-K方法首次具体给出了一般格式中的参数所满足的方程,并提出了新的计算格式,这些新算法对某些初值问题其整体截断误差有明显的减少.这对常微分方程初值问题在社会、经济、生态等领域中的广泛应用将提供有益的新算法. 相似文献
2.
3.
Othmar Koch 《Numerische Mathematik》2005,101(1):143-164
We discuss an a posteriori error estimate for collocation methods applied to boundary value problems in ordinary differential equations with a singularity of the first kind. As an extension of previous results we show the asymptotical correctness of our error estimate for the most general class of singular problems where the coefficient matrix is allowed to have eigenvalues with positive real parts. This requires a new representation of the global error for the numerical solution obtained by piecewise polynomial collocation when applied to our problem class. 相似文献
4.
Ivar Lie 《BIT Numerical Mathematics》1990,30(1):126-144
Multistep collocation methods for initial value problems in ordinary differential equations are known to be a subclass of multistep Runge-Kutta methods and a generalisation of the well-known class of one-step collocation methods as well as of the one-leg methods of Dahlquist. In this paper we derive an error estimation method of embedded type for multistep collocation methods based on perturbed multistep collocation methods. This parallels and generalizes the results for one-step collocation methods by Nørsett and Wanner. Simple numerical experiments show that this error estimator agrees well with a theoretical error estimate which is a generalisation of an error estimate first derived by Dahlquist for one-leg methods. 相似文献
5.
L. F. Shampine 《Numerische Mathematik》1984,45(2):201-206
Summary Bulirsch and Stoer have shown how to construct asymptotic upper and lower bounds on the true (global) errors resulting from the solution by extrapolation of the initial value problem for a system of ordinary differential equations. It is shown here how to do this for any one-step method endowed with an asymptotically correct local error estimator. The one-step method can be changed at every step.This work performed at Sandia National Laboratories supported by the U.S. Department of Energy under contract number DE-AC04-76DP00789 相似文献
6.
Error estimates for shooting methods in two-point boundary value problems for second-order equations
This paper is concerned with a procedure for estimating the global discretization error arising when a boundary value problem for a system of second order differential equations is solved by the simple shooting method, without transforming the original problem in an equivalent first order problem. Expressions of the global discretization error are derived for both linear and nonlinear boundary value problems, which reduce the error estimation for a boundary value problem to that for an initial value problem of same dimension. The procedure extends to second order equations a technique for global error estimation given elsewhere for first order equations. As a practical result the accuracy of the estimates for a second order problem is increased compared with the estimates for the equivalent first order problem. 相似文献
7.
线性常微分方程初值问题求解在许多应用中起着重要作用.目前,已存在很多的数值方法和求解器用于计算离散网格点上的近似解,但很少有对全局误差(global error)进行估计和优化的方法.本文首先通过将离散数值解插值成为可微函数用来定义方程的残差;再给出残差与近似解的关系定理并推导出全局误差的上界;然后以最小化残差的二范数为目标将方程求解问题转化为优化求解问题;最后通过分析导出矩阵的结构,提出利用共轭梯度法对其进行求解.之后将该方法应用于滤波电路和汽车悬架系统等实际问题.实验分析表明,本文估计方法对线性常微分方程的初值问题的全局误差具有比较好的估计效果,优化求解方法能够在不增加网格点的情形下求解出线性常微分方程在插值解空间中的全局最优解. 相似文献
8.
Mingrong Cui 《Numerical Methods for Partial Differential Equations》2009,25(3):685-711
Finite difference scheme to the generalized one‐dimensional sine‐Gordon equation is considered in this paper. After approximating the second order derivative in the space variable by the compact finite difference, we transform the sine‐Gordon equation into an initial‐value problem of a second‐order ordinary differential equation. Then Padé approximant is used to approximate the time derivatives. The resulting fully discrete nonlinear finite‐difference equation is solved by a predictor‐corrector scheme. Both Dirichlet and Neumann boundary conditions are considered in our proposed algorithm. Stability analysis and error estimate are given for homogeneous Dirichlet boundary value problems using energy method. Numerical results are given to verify the condition for stability and convergence and to examine the accuracy and efficiency of the proposed algorithm. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 相似文献
9.
R.L. Brown 《Applied Mathematical Modelling》1978,2(2):115-118
The problem of modelling a dynamic system described by a system of ordinary differential equations which has unstable components for limited periods of time is discussed. It is shown that the global error in a multistep numerical method is the solution to a difference equation initial value problem, and the approximate solution is given for several popular multistep integration formulae. Inspection of the solution leads to the formulation of four criteria for integrators appropriate to unstable problems. A sample problem is solved numerically using three popular formulae and two different stepsizes to illustrate the appropriateness of the criteria. 相似文献
10.
In this paper a numerical method is presented to solve singularly perturbed two points boundary value problems for second order ordinary differential equations consisting a discontinuous source term. First, in this method, an asymptotic expansion approximation of the solution of the boundary value problem is constructed using the basic ideas of a well known perturbation method WKB. Then some initial value problems and terminal value problems are constructed such that their solutions are the terms of this asymptotic expansion. These initial value problems are happened to be singularly perturbed problems and therefore fitted mesh method (Shishkin mesh) are used to solve these problems. Necessary error estimates are derived and examples provided to illustrate the method. 相似文献
11.
Set-valued solutions to the Cauchy problem for hyperbolic systems of partial differential inclusions
Jean-Pierre Aubin Halina Frankowska 《NoDEA : Nonlinear Differential Equations and Applications》1997,4(2):149-168
We prove the existence of global set-valued solutions to the Cauchy problem for partial differential equations and inclusions,
with either single-valued or set-valued initial conditions. The method is based on the equivalence between this problem and
problem of finding viability tubes of the associated characteristic system of ordinary differential equations. As an application
we construct the value function of the Mayer problem arising in control theory.
Received August 25, 1995 相似文献
12.
D. V. Tunitsky 《Doklady Mathematics》2017,96(1):377-379
The communication concerns a theory of global solvability of initial value problem for nonlinear hyperbolic equations with two independent variables that is an immediate analog of a theory of global solvability of ordinary differential equations. 相似文献
13.
In this paper, we discuss the asymptotic properties and efficiency of several a posteriori estimates for the global error of collocation methods. Proofs of the asymptotic correctness are given for regular problems and for problems with a singularity of the first kind. We were also strongly interested in finding out which of our error estimates can be applied for the efficient solution of boundary value problems in ordinary differential equations with an essential singularity. Particularly, we compare estimates based on the defect correction principle with a strategy based on mesh halving.
AMS subject classification 65L05Supported in part by the Austrian Research Fund (FWF) grant P-15072-MAT and SFB Aurora. 相似文献
14.
G. H. Meyer 《Numerische Mathematik》1970,16(3):248-267
A numerical method for the solution of the one-phase Stefan problem is discussed. By discretizing the time variable the Stefan problem is reduced to a sequence of free boundary value problems for ordinary differential equations which are solved by conversion to initial value problems. The numerical solution is shown to converge to the solution of the Stefan problem with decreasing time increments. Sample calculations indicate that the method is stable provided the proper algorithm is chosen for integrating the initial value problems. 相似文献
15.
The paper consists of two parts. In the first part, we propose a procedure to estimate local errors of low order methods applied
to solve initial value problems in ordinary differential equations (ODEs) and index 1 differential-algebraic equations (DAEs).
Based on the idea of defect correction we develop local error estimates for the case when the problem data is only moderately
smooth. Numerical experiments illustrate the performance of the mesh adaptation based on the error estimation developed in
this paper. In the second part of the paper, we will consider the estimation of local errors in context of stochastic differential
equations with small noise.
AMS subject classification (2000) 65L06, 65L80, 65L50, 65L05 相似文献
16.
In this paper, the canonical dual function (Gao, 2004 [4]) is used to solve a global optimization. We find global minimizers by backward differential flows. The backward flow is created by the local solution to the initial value problem of an ordinary differential equation. Some examples and applications are presented. 相似文献
17.
In this paper, the existence and uniqueness of the local generalized solution and the local classical solution for the initial boundary value problem of the quasi-linear wave equation with viscous damping are proved. The nonexistence of the global solution for this problem is discussed by an ordinary differential inequality. Finally, an example is given. 相似文献
18.
A new formalism in the theory of linear boundary value problems involving causal functional differential equations is presented. The approach depends on the construction of a differentiable family of boundary problems into which the original boundary value problem is imbedded. The formalism then generates an initial value problem which is equivalent to the family of imbedded problems. An important aspect of the method is that the equations in the initial value algorithm are ordinary differential equations rather than functional differential equations, although nonlinear and of higher dimension. Applications of the theory to differential-delay and difference equations are given. 相似文献
19.
We consider issues related to the numerical solution of interval systems of ordinary differential equations. We suggest an algorithm that permits finding interval estimates of solutions with prescribed accuracy in reasonable time. The algorithm constructs an adaptive partition (a dynamic structured grid) based on a kd-tree over the space formed by interval initial conditions for the ordinary differential equations. In the operation of the algorithm, a piecewise polynomial function interpolating the dependence of the solution on the specific values of interval parameters is constructed at each step of solution of the original problem. We prove that the global error estimate linearly depends on the height of the kd-tree. The algorithm is tested on several examples; the test results show its efficiency when solving problems of the class under study. 相似文献
20.
Summary The extrapolated midpoint rule is a popular way to solve the initial value problem for a system of ordinary differential equations. As originally formulated by Gragg, the results are smoothed to remove the weak instability of the midpoint rule. It is shown that this smoothing is not necessary. A cheaper smoothing scheme is proposed. A way to exploit smoothing to increase the robustness of extrapolation codes is formulated.This work performed at Sandia National Laboratories supported by the US Department of Energy under contract number DE-AC04-76DP00789 相似文献