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     

Modified defect correction algorithms for ODEs. Part II: Stiff initial value problems

As shown in part I of this paper and references therein, the classical method of Iterated Defect Correction (IDeC) can be modified in several nontrivial ways, extending the flexibility and range of applications of this approach. The essential point is an adequate definition of the defect, resulting in a significantly more robust convergence behavior of the IDeC iteration, in particular, for nonequidistant grids. The present part II is devoted to the efficient high-order integration of stiff initial value problems. By means of model problem investigation and systematic numerical experiments with a set of stiff test problems, our new versions of defect correction are systematically evaluated, and further algorithmic measures are proposed for the stiff case. The performance of the different variants under consideration is compared, and it is shown how strong coupling between non-stiff and stiff components can be successfully handled. AMS subject classification 65L05 Supported by the Austrian Research Fund (FWF) grant P-15030.     

Parallel iterated method based on multistep Runge-Kutta of Radau type for stiff problems

Research on parallel iterated methods based on Runge-Kutta formulas both for stiff and non-stiff problems has been pioneered by van der Houwen et al., for example see [8-11]. Burrage and Suhartanto have adopted their ideas and generalized their work to methods based on Multistep Runge-Kutta of Radau type [2] for non-stiff problems. In this paper we discuss our methods for stiff problems and study their performance.     

Extrapolation-based implicit-explicit general linear methods

For many systems of differential equations modeling problems in science and engineering, there are natural splittings of the right hand side into two parts, one non-stiff or mildly stiff, and the other one stiff. For such systems implicit-explicit (IMEX) integration combines an explicit scheme for the non-stiff part with an implicit scheme for the stiff part. In a recent series of papers two of the authors (Sandu and Zhang) have developed IMEX GLMs, a family of implicit-explicit schemes based on general linear methods. It has been shown that, due to their high stage order, IMEX GLMs require no additional coupling order conditions, and are not marred by order reduction. This work develops a new extrapolation-based approach to construct practical IMEX GLM pairs of high order. We look for methods with large absolute stability region, assuming that the implicit part of the method is A- or L-stable. We provide examples of IMEX GLMs with optimal stability properties. Their application to a two dimensional test problem confirms the theoretical findings.     

Jacobian Matrix Properties and Their Impact on Choice of Software for Stiff ODE Systems

Information is presented about the spectral and other propertiesof Jacobian matrices occurring in the numerical solution ofa number of large, very stiff ODE problems, arising from massaction kinetics. These properties demonstrate that the conceptof a few "stiff" eigenvalues, the rest being "non-stiff", isnot valid for such problems; consequently, it is argued thatpartitioning and exponential-fitting methods are inappropriatefor use in general-purpose software for stiff systems. Moreover,second-derivative methods and all but a very few formulationsof implicit Runge-Kutta methods would be at a grave disadvantagewhen applied to large, very stiff problems.     

High-order splitting schemes for semilinear evolution equations

We first derive necessary and sufficient stiff order conditions, up to order four, for exponential splitting schemes applied to semilinear evolution equations. The main idea is to identify the local splitting error as a sum of quadrature errors. The order conditions of the quadrature rules then yield the stiff order conditions in an explicit fashion, similarly to that of Runge–Kutta schemes. Furthermore, the derived stiff conditions coincide with the classical non-stiff conditions. Secondly, we propose an abstract convergence analysis, where the linear part of the vector field is assumed to generate a group or a semigroup and the nonlinear part is assumed to be smooth and to satisfy a set of compatibility requirements. Concrete applications include nonlinear wave equations and diffusion-reaction processes. The convergence analysis also extends to the case where the nonlinear flows in the exponential splitting scheme are approximated by a sufficiently accurate one-step method.     

Dynamic adaptive selection of integration algorithms when solving ODE:S

The specific area of stiffness detection has traditionally required user interaction. This paper presents an attempt to switch dynamically between stiff and non-stiff algorithms.     

High order methods for the numerical integration of ordinary differential equations

Summary High order implicit integration formulae with a large region of absolute stability are developed for the approximate numerical integration of both stiff and non-stiff systems of ordinary differential equations. The algorithms derived behave essentially like one step methods and are demonstrated by direct application to certain particular examples.     

On the choice of correctors for semi-implicit Picard deferred correction methods

The goal of this study is to assess the implications of the choice of correctors for semi-implicit Picard integral deferred correction (SIPIDC) methods. The SIPIDC methods previously developed compute a high-order approximation by first computing a low-order provisional solution using a semi-implicit method and then using a first-order semi-implicit method to solve a series of correction equations, each of which raises the order of accuracy of the solution by one. In this study, we examine the efficiency of SIPIDC methods that instead use standard second-order semi-implicit methods to solve the correction equations. The accuracy, efficiency, and stability of the resulting methods are compared to previously developed methods, in the context of both nonstiff and stiff problems.     

Asymptotic behavior of averaged and firmly nonexpansive mappings in geodesic spaces

We further study averaged and firmly nonexpansive mappings in the setting of geodesic spaces with a main focus on the asymptotic behavior of their Picard iterates. We use methods of proof mining to obtain an explicit quantitative version of a generalization to geodesic spaces of a result on the asymptotic behavior of Picard iterates for firmly nonexpansive mappings proved by Reich and Shafrir. From this result we obtain effective uniform bounds on the asymptotic regularity for firmly nonexpansive mappings. Besides this, we derive effective rates of asymptotic regularity for sequences generated by two algorithms used in the study of the convex feasibility problem in a nonlinear setting.     

On non-standard finite difference models of reaction–diffusion equations

Reaction–diffusion equations arise in many fields of science and engineering. Often, their solutions enjoy a number of physical properties. We design, in a systematic way, new non-standard finite difference schemes, which replicate three of these properties. The first property is the stability/instability of the fixed points of the associated space independent equation. This property is preserved by non-standard one- and two-stage theta methods, presented in the general setting of stiff or non-stiff systems of differential equations. Schemes, which preserve the principle of conservation of energy for the corresponding stationary equation (second property) are constructed by non-local approximation of nonlinear reactions. Assembling of theta-methods in the time variable with energy-preserving schemes in the space variable yields non-standard schemes which, under suitable functional relation between step sizes, display the boundedness and positivity of the solution (third property). A spectral method in the space variable coupled with a suitable non-standard scheme in the time variable is also presented. Numerical experiments are provided.     

Extrapolation at stiff differential equations

Summary Asymptotic expansions of the global error of numerical methods are well-understood, if the differential equation is non-stiff. This paper is concerned with such expansions for the implicit Euler method, the linearly implicit Euler method and the linearly implicit mid-point rule, when they are applied tostiff differential equations. In this case perturbation terms are present, whose dominant one is given explicitly. This permits us to better understand the behaviour ofextrapolation methods at stiff differential equations. Numerical examples, supporting the theoretical results, are included.     

两平行圆板间径向层流进口段效应分析

本文首先将B.B.方法[1]推广至二平行圆板间的径向扩散流动,由边界层运动方程式同时推导出动量积分方程式和能量积分方程式,而后再用Picard逐次逼近法[2]解能量积分方程式,求得进口段通道长随边界层厚度而改变的二级近似显函数表达式.从而为进口段效应诸系数的直接解析分析提供了可能.特别是当圆板外径小于进口段长度时,更加突出地表现了本方法的优越性.由于采用了能量积分方程式,则压力损失系数的各项才得以从理论上独立地推导出来.本文所提供的压力损失系数计算值,在进口修正雷诺数Re<100时,和文献[3]比较与实验值更为接近.因此在该范围内本文的结果既可靠又简便.     

Error bounds of finite difference schemes for multi-dimensional scalar conservation laws with source terms

This paper studies explicit and semi=implicit finite differenceschemes approximating non-homogeneous multi-dimensional scalarconservation laws with both stiff and non-stiff source terms.Error bounds of order O(t) are presented for both cases.     

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.     

On explicit two-derivative Runge-Kutta methods

The theory of Runge-Kutta methods for problems of the form y′?=?f(y) is extended to include the second derivative y′′?=?g(y):?=?f′(y)f(y). We present an approach to the order conditions based on Butcher’s algebraic theory of trees (Butcher, Math Comp 26:79–106, 1972), and derive methods that take advantage of cheap computations of the second derivatives. Only explicit methods are considered here where attention is given to the construction of methods that involve one evaluation of f and many evaluations of g per step. Methods with stages up to five and of order up to seven including some embedded pairs are presented. The first part of the paper discusses a theoretical formulation used for the derivation of these methods which are also of wider applicability. The second part presents experimental results for non-stiff and mildly stiff problems. The methods include those with the computation of one second derivative (plus many first derivatives) per step, and embedded methods for changing stepsize as well as those involving one first derivative (plus many second derivatives) per step. The experiments have been performed on standard problems and comparisons made with some standard explicit Runge-Kutta methods.     

How to avoid accuracy and order reduction in Runge–Kutta methods as applied to stiff problems

The solution of stiff problems is frequently accompanied by a phenomenon known as order reduction. The reduction in the actual order can be avoided by applying methods with a fairly high stage order, ideally coinciding with the classical order. However, the stage order sometimes fails to be increased; moreover, this is not possible for explicit and diagonally implicit Runge–Kutta methods. An alternative approach is proposed that yields an effect similar to an increase in the stage order. New implicit and stabilized explicit Runge–Kutta methods are constructed that preserve their order when applied to stiff problems.     

Towards explicit methods for differential algebraic equations

Explicit methods have previously been proposed for parabolic PDEs and for stiff ODEs with widely separated time constants. We discuss ways in which Differential Algebraic Equations (DAEs) might be regularized so that they can be efficiently integrated by explicit methods. The effectiveness of this approach is illustrated for some simple index three problems. In memory of Germund Dahlquist (1925–2005).AMS subject classification (2000) 65-L80, 34-04     

Singly Diagonally Implicit Runge–Kutta Methods with an Explicit First Stage

The purpose of this paper is to construct methods for solving stiff ODEs, in particular singular perturbation problems. We consider embedded pairs of singly diagonally implicit Runge–Kutta methods with an explicit first stage (ESDIRKs). Stiffly accurate pairs of order 3/2, 4/3 and 5/4 are constructed.