On minimum configuration Rosenbrock methods

A new structure is proposed for Rosenbrock methods for solving stiff ordinary differential equations, which facilitates the development of minimum configuration processes (minimum computational work per step). A fourth order process is described.     

Combining Trust-Region Techniques and Rosenbrock Methods to Compute Stationary Points

Rosenbrock methods are popular for solving a stiff initial-value problem of ordinary differential equations. One advantage is that there is no need to solve a nonlinear equation at every iteration, as compared with other implicit methods such as backward difference formulas or implicit Runge–Kutta methods. In this article, we introduce a trust-region technique to select the time steps of a second-order Rosenbrock method for a special initial-value problem, namely, a gradient system obtained from an unconstrained optimization problem. The technique is different from the local error approach. Both local and global convergence properties of the new method for solving an equilibrium point of the gradient system are addressed. Finally, some promising numerical results are also presented. This research was supported in part by Grant 2007CB310604 from National Basic Research Program of China, and #DMS-0404537 from the United States National Science Foundation, and Grant #W911NF-05-1-0171 from the United States Army Research Office, and the Research Grant Council of Hong Kong.     

On improving the absolute stability of local extrapolation

Summary A widely used technique for improving the accuracy of solutions of initial value problems in ordinary differential equations is local extrapolation. It is well known, however, that when using methods appropriate for solving stiff systems of ODES, the stability of the method can be seriously degraded if local extrapolation is employed. This is due to the fact that performing local extrapolation on a low order method is equivalent to using a higher order formula and this high order formula may not be suitable for solving stiff systems. In the present paper a general approach is proposed whereby the correction term added on in the process of local extrapolation is in a sense a rational, rather than a polynomial, function. This approach allows high order formulae with bounded growth functions to be developed. As an example we derive anA-stable rational correction algorithm based on the trapezoidal rule. This new algorithm is found to be efficient when low accuracy is requested (say a relative accuracy of about 1%) and its performance is compared with that of the more familiar Richardson extrapolation method on a large set of stiff test problems.     

An effective modification of the homotopy perturbation method for stiff systems of ordinary differential equations

The present work deals with employing a new form of the homotopy perturbation method (NHPM) for solving stiff systems of linear and nonlinear ordinary differential equations (ODEs). In this scheme, the solution is considered as an infinite series that converges rapidly to the exact solution. Two problems are chosen as illustrative examples to show the effectiveness of the present method. In obtaining the exact solution for each case, the capability and the simplicity of the proposed technique is clarified.     

Rational Runge-Kutta methods are not suitable for stiff systems of ODEs

The present paper shows that rational RK-methods are not very appropriate to solve stiff differential equations. The CA₀-stability (i.e. componentwise contractivity) is defined and the non-existence of CA₀-stable rational RK-methods is demonstrated. Furthermore it is shown that the stepsizes which can be expected when solving a stiff differential system with a rational or with an explicit linear RK-method are of the same order of magnitude.     

一类自适应的单块法及其变步长实现方案

１．引言解 Ｓｔｉｆｆ ＯＤＥｓ初值问题的自开始型单块法已为 ［４, ５］所研究．这里, ｅ＝（１,１,……,１）Ｔ为单位矩阵,当 时见 ［４］,当 ０＜ ａ１＜ ａ２＜…＜ ａｒ＝ ｒ时见［５］。 众所周知,解（１．１）的有效方法通常是隐的．仅当有效地解决了其变步长计算问题并具有有效的迭代法求其解时,这样的方法才能有效地用于实际计算．后者是不言而喻的,前者是由于定步长计算或者往往带来精度的严重损失,或者会带来计算量的严重增加（当存在（ｔ０,Ｔ］的两个子区间,该两区间上的合理积分步长相差悬殊时,就会出现这种…     

Order and stability properties of explicit multivalue methods

Recent papers by Burrage and Moss [1] and Burrage [2] have studied in some detail the order properties of implicit multivalue (or general linear) methods and certain classes of these methods were proposed as being suitable for solving stiff differential equations. In this present paper we study the order and stability of explicit multivalue methods with a view to deriving new families of methods suitablefor solving non stiff problems.     

中立型随机比例延迟微分方程平衡半隐式Euler方法的均方收敛性

本文讨论求解刚性中立型随机比例延迟微分方程的平衡半隐式Euler方法。证明了中立型随机比例延迟微分方程的平衡半隐式Euler方法是1／2阶均方收敛的。     

Construction of a multirate RODAS method for stiff ODEs

Multirate time stepping is a numerical technique for efficiently solving large-scale ordinary differential equations (ODEs) with widely different time scales localized over the components. This technique enables one to use large time steps for slowly varying components, and small steps for rapidly varying ones. Multirate methods found in the literature are normally of low order, one or two. Focusing on stiff ODEs, in this paper we discuss the construction of a multirate method based on the fourth-order RODAS method. Special attention is paid to the treatment of the refinement interfaces with regard to the choice of the interpolant and the occurrence of order reduction. For stiff, linear systems containing a stiff source term, we propose modifications for the treatment of the source term which overcome order reduction originating from such terms and which we can implement in our multirate method.     

随机延迟微分方程平衡方法的均方收敛性与稳定性

本文讨论求解刚性随机延迟微分方程的平衡方法.证明了随机延迟微分方程平衡方法的均方收敛阶为1/2.给出了线性随机延迟微分方程平衡方法均方稳定的条件.     

一族多步二阶导数方法的收缩性

１．引言 １９７８年 Ｎｅｖａｎｌｉｎｎａ和 Ｌｉｎｉｇｅｒ［１,２］研究了常微分方程初值问题的单支方法和线性多步法的收缩性,就基于线性模型方程的收缩性建立了比较完整的理论．他们指出,收缩方法比绝对稳定方法能更好地给出间断问题的数值解,因而研究数值方法的收缩性具有重要理论和实践意义． １９７４年 Ｅｎｒｉｇｈｔ［３］构造了 ｋ步 ｋ ＋ ２阶二阶导数方法由于它是Ａｄｍａｓ型的且只含一个二阶导数项,因而方法在原点附近具有较理想的稳定性和稳定程度（参见［７］）,同时在 ∞处是极端稳定的．赵双锁和董国雄 ［４］…     

二维刚性边值问题的小波-吉尔解法

本文利用微分方程数值解的离散小波表示，讨论了此类方程在满足一定初始条件和边值条件下，在一个方向上利用小波伽辽金法，另一方向上利用吉尔方法进行求解，提出了一种解二维刚性初，边值问题的小波数值算法，计算结果表明，利用该方法所求得的数值解精度高，而且由小波特有的性质，它特别适用于求解带有奇异摄动的刚性问题。     

The adaptive operational Tau method for systems of ODEs

As the Tau method, like many other numerical methods, has the limitation of using a fixed step size with some high degree (order) of approximation for solving initial value problems over long intervals, we introduce here the adaptive operational Tau method. This limitation is very much problem dependent and in such case the fixed step size application of the Tau method loses the true track of the solution. But when we apply this new adaptive method the true solution is recovered with a reasonable number of steps. To illustrate the effectiveness of this method we apply it to some stiff systems of ordinary differential equations (ODEs). The numerical results confirm the efficiency of the method.     

非自治刚性随机微分方程正则EM分裂方法的收敛性和稳定性

本文研究了数值求解非自治随机微分方程的正则Euler-Maruyama分裂（CEMS）方法,该方程的漂移项系数带有刚性且允许超线性增长,扩散项系数满足全局Lipschitz条件.首先,证明了CEMS方法的强收敛性及收敛速度.其次,证明了在适当条件下CEMS方法是均方稳定的.进一步,利用离散半鞅收敛定理,研究了CEMS方法的几乎必然指数稳定性.结果表明,CEMS方法在漂移系数的刚性部分满足单边Lipschitz条件下可保持几乎必然指数稳定性.最后通过数值实验,检验了CEMS方法的有效性并证实了我们的理论结果.     

Variational methods of constructing monotone approximations for atmospheric chemistry models

A new method of constructing efficient monotone numerical schemes for solving direct, adjoint, and inverse atmospheric chemistry problems is presented. It is a synthesis of variational principles combined with splitting and decomposition methods and a constructive implementation of Euler integrating multipliers (EIM) bymeans of a local adjoint problem technique. To increase the efficiency of calculations, a method of decomposing the multicomponent substance transformation operators in terms of the mechanisms of reactions is also proposed. With analytical EIMs, the systems of stiff ODEs are decomposed and reduced to equivalent systems of integral equations solved by noniterative multistage algorithms of a given order of accuracy. An unconventional variational method of constructing mutually consistent algorithms for direct and adjoint problems and sensitivity studies for complex models with constraints is described.     

一类A－稳定或L－稳定的经济隐式单块法

一类Ａ－稳定或Ｌ－稳定的经济隐式单块法赵双锁，张国凤（兰州大学数学系）ＡＣＬＡＳＳＯＦＡ－ＳＴＡＢＬＥＯＲＬ－ＳＴＡＢＬＥＥＣＯＮＯＭＩＣＡＬＩＭＰＬＩＣＩＴＳＩＮＧＬＥ－ＢＬＯＣＫＭＥＴＨＯＤＳ￥ＺｈａｏＳｈｕａｎｇ－ｓｕｏ；ＺｈａｎｇＧｕｏ－ｆｅ...     

Second-order accurate projective integrators for multiscale problems

We introduce new projective versions of second-order accurate Runge–Kutta and Adams–Bashforth methods, and demonstrate their use as outer integrators in solving stiff differential systems. An important outcome is that the new outer integrators, when combined with an inner telescopic projective integrator, can result in fully explicit methods with adaptive outer step size selection and solution accuracy comparable to those obtained by implicit integrators. If the stiff differential equations are not directly available, our formulations and stability analysis are general enough to allow the combined outer–inner projective integrators to be applied to legacy codes or perform a coarse-grained time integration of microscopic systems to evolve macroscopic behavior, for example.     

Development and Application of an Exponential Method for Integrating Stiff Systems Based on the Classical Runge–Kutta Method

We study numerical methods for solving stiff systems of ordinary differential equations. We propose an exponential computational algorithm which is constructed by using an exponential change of variables based on the classical Runge–Kutta method of the fourth order. Nonlinear problems are used to prove and demonstrate the fourth order of convergence of the new method.     

Mesh selection for stiff two-point boundary value problems

This paper is concerned with the mesh selection algorithm of COLSYS, a well known collocation code for solving systems of boundary value problems. COLSYS was originally designed to solve non-stiff and mildly stiff problems only. In this paper we show that its performance for solving extremely stiff problems can be considerably improved by modifying its error estimation and mesh selection algorithms. Numerical examples indicate the superiority of the modified algorithm.Dedicated to John Butcher on the occasion of his sixtieth birthday     

Positivity-Preserving Local Discontinuous Galerkin Method for Pattern Formation Dynamical Model in Polymerizing Actin Flocks

In this paper, we apply local discontinuous Galerkin (LDG) methods for pattern formation dynamical model in polymerizing actin flocks. There are two main difficulties in designing effective numerical solvers. First of all, the density function is non-negative, and zero is an unstable equilibrium solution. Therefore, negative density values may yield blow-up solutions. To obtain positive numerical approximations, we apply the positivity-preserving (PP) techniques. Secondly, the model may contain stiff source. The most commonly used time integration for the PP technique is the strong-stability-preserving Runge-Kutta method. However, for problems with stiff source, such time discretizations may require strictly limited time step sizes, leading to large computational cost. Moreover, the stiff source any trigger spurious filament polarization, leading to wrong numerical approximations on coarse meshes. In this paper, we combine the PP LDG methods with the semi-implicit Runge-Kutta methods. Numerical experiments demonstrate that the proposed method can yield accurate numerical approximations with relatively large time steps.