一类显式的k阶线性k步法基本公式

本文给出了一类比Adams-Bashforth方法的局部截断误差主项系数小和绝对稳定区间大的显式k阶线性k步法基本公式.作者求出了公式的分数形式的精确系数,阶数和局部截断误差主项系数,给出了3-9步公式的绝对稳定区间,构造了由新公式的4阶显式公式和一个同阶隐式基本公式组合而成的特殊预估-校正方法,它的绝对稳定区间大于预估公式而且等于校正公式, 比著名的Adams-Bashforth-Moulton预估校正方法的绝对稳定区间大, 最后用数值试验对结果进行了验证,适合于求解常微分方程初值问题.     

一类A(α)稳定的k阶线性k步法公式

本文给出了一类与Gear方法类似的k阶线性k步法隐式公式.作者还求出了公式的分数形式的系数,阶数和局部截断误差主项系数,并验证了2-6步公式都具有A(α)稳定的,计算出了它们的幅角α.最后用对比数值实验验证了公式确实是稳定的,并且适合于求解刚性常微分方程.     

预估-校正方法的绝对稳定性讨论

预估-校正方法,即PECE方法,常被用于求解常微分方程的初值问题.而一般文献中常只讨论了单个线性多步法公式的稳定性问题,很少涉及由一个显式公式和一个隐式公式组合而成的PECE方法的稳定性.本文应用根轨迹法和对分法讨论了常用的PECE方法的稳定性,求出了一些常用PECE方法的组合公式的绝对稳定区间和绝对稳定区域,并用数值...     

一类A(α)稳定的k阶线性k步法公式

本文给出了一类与Gear方法类似的κ阶线性κ步法隐式公式.作者还求出了公式的分数形式的系数,阶数和局部截断误差主项系数,并验证了2-6步公式都具有A(α)稳定的,计算出了它们的幅角α.最后用对比数值实验验证了公式确实是稳定的,并且适合于求解刚性常微分方程.     

渐近A稳定的线性多步法

本文主要结果为: 1.构造了一类k步k+1阶隐式线性多步公式,它们是渐近A稳定的。 2.构造了一类k步k阶隐式线性多步公式,它们是stiff稳定且是渐近A稳定的。 3.构造了一类k步k—1阶显式线性多步公式,它们是渐近A稳定的。k为任意正整数。     

拓展的Adams-Moulton方法(英文)

本文以 k 步 Adams-Moulton(A—M)公式作为“预估式”,以带有拓展项 hβ_(k+1)f_(n+k+l)的 A—M 型公式作为校正式,形成一种具有隐式预估式的预估——校正法,即所谓拓展的A—M 方法。和 k 步 A—M 法相比,拓展的 A—M 法有较高的精度阶(p=k+2)和较大的稳定性区域,特别当 k=2时,方法是几乎 A 定稳的。计算量虽然大一些,但可设法节省许多。     

一类A(α)稳定的κ阶线性κ步法公式

本文给出了一类与Gear方法类似的κ阶线性κ步法隐式公式.作者还求出了公式的分数形式的系数,阶数和局部截断误差主项系数,并验证了2-6步公式都具有A(α)稳定的,计算出了它们的幅角α.最后用对比数值实验验证了公式确实是稳定的,并且适合于求解刚性常微分方程.     

求解常微分方程的预估——校正方法

本文提供了基于由Adams和Nystrom方法[1]的组合的一类预估——校正方法,它们是具有增大的绝对稳定域。对于K=3,4,5,6,7,给出这些公式的系数。     

关于求解常微分方程的一类预估——校正方法

在[1]中,我们研究了由显式的Adams一Bashforth(A—B)公式与Nystrom公式所组合的方法,以及由隐式的Adams-Moulton(A—M)公式与广义的Milne—Simpson(M—S)公式所组合的方法,获得了具有较大的绝大稳定域的一类方法。在此基础上,本文讨论用由上述组合的显式方法作为预估公式,而将隐式组合的方法作为校正公式,构造出含有双参数的P_KEC_KE方法。对于K=4,5,6,7,8,获得了具有增大的绝大稳定域的一类方法。     

基于切比雪夫多项式逼近的6级6阶隐式Runge-Kutta方法

以切比雪夫偏差点为插值点,利用切比雪夫多项式逼近理论和高斯-洛巴托-切比雪夫求积公式,构造了一个6级6阶的隐式Runge-Kutta方法.理论分析发现,该算法具有良好的稳定性——A_0稳定,较大α值的A(α)稳定,较小D值的刚性稳定和几乎L稳定.数值算例显示了该算法的有效性.     

两类线性隐式方法

众所周知,在Stiff常微分方程组初值问题的数值解法中,向后微分公式(即Gear方法)是目前最通用的方法之一(见[1]).但是,Gear方法是一类隐式方法,在数值解的过程中,一般说来,每向前积分一步,需要解一个非线性方程组,它的求解是采用Newton-Raphson迭代方法,因此需要给出适当精度的预估值和计算右函数f(t,y)的Jacobi阵以     

Methods for parallel integration of stiff systems of odes

This paper presents a class of parallel numerical integration methods for stiff systems of ordinary differential equations which can be partitioned into loosely coupled sub-systems. The formulas are called decoupled backward differentiation formulas, and they are derived from the classical formulas by restricting the implicit part to the diagnonal sub-system. With one or several sub-systems allocated to each processor, information only has to be exchanged after completion of a step but not during the solution of the nonlinear algebraic equations.The main emphasis is on the formula of order 1, the decoupled implicit Euler formula. It is proved that this formula even for a wide range of multirate formulations has an asymptotic global error expansion permitting extrapolation. Besides, sufficient conditions for absolute stability are presented.     

Backward differentiation formulas with extended regions of absolute stability

This paper presents a class of (p + 2)-step backward differentiation formulas of orderp. The two extra degrees of freedom obtained by limiting the order of a (p + 2)-step formula top are used to extend the region of absolute stability. A new formula of orderp has a region of absolute stability very similar to that of a classical backward differentiation formula of orderp - 1 forp being in the range 4–6. The backward differentiation formulas with extended regions of absolute stability are constructed by appending two exponential-trigonometric terms to the polynomial basis of the classical formulas. Besides the absolute stability, the paper discusses relative stability and contractivity. The principles of an experimental implementation of the new formulas are outlined, and a linear problem integrated with this computer program indicates that the extended regions of absolute stability can actually be exploited in practice.     

High‐order implicit staggered‐grid finite differences methods for the acoustic wave equation

Motivated by the idea that staggered‐grid methods give a greater stability and give energy conservation, this article presents a new family of high‐order implicit staggered‐grid finite difference methods with any order of accuracy to approximate partial differential equations involving second‐order derivatives. In particular, we numerically analyze our new methods for the solution of the one‐dimensional acoustic wave equation. The implicit formulation is based on the plane wave theory and the Taylor series expansion and only involves the solution of tridiagonal matrix equations resulting in an attractive method with higher order of accuracy but nearly the same computation cost as those of explicit formulation. The order of accuracy of the proposal staggered formulas are similar to the methods with conventional grids for a ‐point operator: the explicit formula is th‐order and the implicit formula is th‐order; however, the results demonstrate that new staggered methods are superior in terms of stability properties to the classical methods in the context of solving wave equations.     

一类含有稳定参数的Adams型隐式方法及其新算法

§1.引言 数值积分Stiff常微分方程初值问题,其积分过程的稳定性相当重要.用传统的数值方法,如Adams方法等,为保证计算稳定性,积分步长受到相当的限制.在stiff常微分方程初值问题的数值解法中,Gear方法是目前最通用的方法之一.但是,当阶p大     

Two-Stage Stochastic Runge-Kutta Methods for Stochastic Differential Equations

In this paper we discuss two-stage diagonally implicit stochastic Runge-Kutta methods with strong order 1.0 for strong solutions of Stratonovich stochastic differential equations. Five stochastic Runge-Kutta methods are presented in this paper. They are an explicit method with a large MS-stability region, a semi-implicit method with minimum principal error coefficients, a semi-implicit method with a large MS-stability region, an implicit method with minimum principal error coefficients and another implicit method. We also consider composite stochastic Runge-Kutta methods which are the combination of semi-implicit Runge-Kutta methods and implicit Runge-Kutta methods. Two composite methods are presented in this paper. Numerical results are reported to compare the convergence properties and stability properties of these stochastic Runge-Kutta methods.     

Construction of two-step Runge-Kutta methods of high order for ordinary differential equations

The construction of two-step Runge-Kutta methods of order p and stage order q=p with stability polynomial given in advance is described. This polynomial is chosen to have a large interval of absolute stability for explicit methods and to be A-stable and L-stable for implicit methods. After satisfying the order and stage order conditions the remaining free parameters are computed by minimizing the sum of squares of the difference between the stability function of the method and a given polynomial at a sufficiently large number of points in the complex plane. This revised version was published online in August 2006 with corrections to the Cover Date.