SERK2v2: A new second‐order stabilized explicit Runge‐Kutta method for stiff problems

Traditionally, explicit numerical algorithms have not been used with stiff ordinary differential equations (ODEs) due to their stability. Implicit schemes are usually very expensive when used to solve systems of ODEs with very large dimension. Stabilized Runge‐Kutta methods (also called Runge–Kutta–Chebyshev methods) were proposed to try to avoid these difficulties. The Runge–Kutta methods are explicit methods with extended stability domains, usually along the negative real axis. They can easily be applied to large problem classes with low memory demand, they do not require algebra routines or the solution of large and complicated systems of nonlinear equations, and they are especially suited for discretizations using the method of lines of two and three dimensional parabolic partial differential equations. In Martín‐Vaquero and Janssen [Comput Phys Commun 180 (2009), 1802–1810], we showed that previous codes based on stabilized Runge–Kutta algorithms have some difficulties in solving problems with very large eigenvalues and we derived a new code, SERK2, based on sixth‐order polynomials. Here, we develop a new method based on second‐order polynomials with up to 250 stages and good stability properties. These methods are efficient numerical integrators of very stiff ODEs. Numerical experiments with both smooth and nonsmooth data support the efficiency and accuracy of the new algorithms when compared to other well‐known second‐order methods such as RKC and ROCK2. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

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...     

Weak second order S-ROCK methods for Stratonovich stochastic differential equations

It is well known that the numerical solution of stiff stochastic ordinary differential equations leads to a step size reduction when explicit methods are used. This has led to a plethora of implicit or semi-implicit methods with a wide variety of stability properties. However, for stiff stochastic problems in which the eigenvalues of a drift term lie near the negative real axis, such as those arising from stochastic partial differential equations, explicit methods with extended stability regions can be very effective. In the present paper our aim is to derive explicit Runge–Kutta schemes for non-commutative Stratonovich stochastic differential equations, which are of weak order two and which have large stability regions. This will be achieved by the use of a technique in Chebyshev methods for ordinary differential equations.     

General linear methods for y′′?=?f?(y?(t))

In this paper we consider the family of General Linear Methods (GLMs) for the numerical solution of special second order Ordinary Differential Equations (ODEs) of the type y????=?f(y(t)), with the aim to provide a unifying approach for the analysis of the properties of consistency, zero-stability and convergence. This class of methods properly includes all the classical methods already considered in the literature (e.g. linear multistep methods, Runge?CKutta?CNystr?m methods, two-step hybrid methods and two-step Runge?CKutta?CNystr?m methods) as special cases. We deal with formulation of GLMs and present some general results regarding consistency, zero-stability and convergence. The approach we use is the natural extension of the GLMs theory developed for first order ODEs.     

An improved class of generalized Runge–Kutta methods for stiff problems. Part I: The scalar case

A new family of p-stage methods for the numerical integration of some scalar equations and systems of ODEs is proposed. These methods can be seen as a generalization of the explicit p-stage Runge–Kutta ones, while providing better order and stability results. We will show in this first part that, at the cost of losing linearity in the formulas, it is possible to obtain explicit A-stable and L-stable methods for the numerical integration of scalar autonomous ODEs. Scalar autonomous ODEs are of very little interest in current applications. However, be begin studying this kind of problems because most of the work can be easily extended to a more general situation. In fact, we will show in a second part (entitled ‘The separated system case'), that it is possible to generalize our methods so that they can be applied to some non-autonomous scalar ODEs and systems. We will obtain linearly implicit L-stable methods which do not require Jacobian evaluations. In both parts, some numerical examples are discussed in order to show the good performance of the new schemes.     

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.     

Global error estimation and control in linearly-implicit parallel two-step peer W-methods

The class of linearly-implicit parallel two-step peer W-methods has been designed recently for efficient numerical solutions of stiff ordinary differential equations. Those schemes allow for parallelism across the method, that is an important feature for implementation on modern computational devices. Most importantly, all stage values of those methods possess the same properties in terms of stability and accuracy of numerical integration. This property results in the fact that no order reduction occurs when they are applied to very stiff problems. In this paper, we develop parallel local and global error estimation schemes that allow the numerical solution to be computed for a user-supplied accuracy requirement in automatic mode. An algorithm of such global error control and other technical particulars are also discussed here. Numerical examples confirm efficiency of the presented error estimation and stepsize control algorithm on a number of test problems with known exact solutions, including nonstiff, stiff, very stiff and large-scale differential equations. A comparison with the well-known stiff solver RODAS is also shown.     

An almost L‐stable BDF‐type method for the numerical solution of stiff ODEs arising from the method of lines

A new BDF‐type scheme is proposed for the numerical integration of the system of ordinary differential equations that arises in the Method of Lines solution of time‐dependent partial differential equations. This system is usually stiff, so it is desirable for the numerical method to solve it to have good properties concerning stability. The method proposed in this article is almost L‐stable and of algebraic order three. Numerical experiments illustrate the performance of the new method on different stiff systems of ODEs after discretizing in the space variable some PDE problems. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Global error estimation and control in linearly-implicit parallel two-step peer W-methods

The class of linearly-implicit parallel two-step peer W-methods has been designed recently for efficient numerical solutions of stiff ordinary differential equations. Those schemes allow for parallelism across the method, that is an important feature for implementation on modern computational devices. Most importantly, all stage values of those methods possess the same properties in terms of stability and accuracy of numerical integration. This property results in the fact that no order reduction occurs when they are applied to very stiff problems. In this paper, we develop parallel local and global error estimation schemes that allow the numerical solution to be computed for a user-supplied accuracy requirement in automatic mode. An algorithm of such global error control and other technical particulars are also discussed here. Numerical examples confirm efficiency of the presented error estimation and stepsize control algorithm on a number of test problems with known exact solutions, including nonstiff, stiff, very stiff and large-scale differential equations. A comparison with the well-known stiff solver RODAS is also shown.     

Efficient parallel multivalue hybrid methods for stiff differential equations

A class of efficient parallel multivalue hybrid methods for stiff differential equations are presented, which are all extremely stable at infinity,A-stable for orders 1–3 and A(α)-stable for orders 4–8. Each method of the class can be performed parallelly using two processors with each processor having almost the same computational amount per integration step as a backward differentiation formula (BDF) of the same order with the same stepsize performed in serial, whereas the former has not only much better stability properties but also a computational accuracy higher than the corresponding BDF. Theoretical analysis and numerical experiments show that the methods constructed in this paper are superior in many respects not only to BDFs but also to some other commonly used methods.     

Convergence of parallel multistep hybrid methods for singular perturbation problems

Parallel multistep hybrid methods (PHMs) can be implemented in parallel with two processors, accordingly have almost the same computational speed per integration step as BDF methods of the same order with the same stepsize. But PHMs have better stability properties than BDF methods of the same order for stiff differential equations. In the present paper, we give some results on error analysis of A(α)-stable PHMs for the initial value problems of ordinary differential equations in singular perturbation form. Our convergence results are similar to those of linear multistep methods (such as BDF methods), i.e. the convergence orders are equal to their classical convergence orders, and no order reduction occurs. Some numerical examples also confirm our results.     

一类具有L稳定和B稳定的Runge—Kutta方法

在求解刚性常微分方程的数值解法中,为了使获得的结果稳定,人们往往使用具有L稳定和B稳定的数值方法,本文利用W－变换构造了一类具有L稳定和B[稳定的Runge-Kutta(RK)方法。     

“Rescale and modify” implementation of IRKS methods

In this paper, we implement the “rescale and modify” approach in variable step size mode with both fixed and variable orders for stiff and nonstiff ODEs. A comparison of this approach and the “rescale” approach from the point of stability behavior and also step size and order selection provides very interesting results. To illustrate the efficiency of the method we have considered some standard test problems and report very useful tables and figures for step size and order changes, number of rejected or accepted steps, and also global error. As an optimal implementation, the numerical experiments suggest the application of this approach with both fixed and variable orders for nonstiff, and in variable order for stiff ODEs.     

Analysis of the Enright-Kamel Partitioning Method for Stiff Ordinary Differential Equations

The use of implicit formulae in the solution of stiff ODEs givesrise to systems of nonlinear equations which are usually solvediteratively by a modified Newton scheme. The linear algebracosts associated with such schemes may form a substantial partof the overall cost of the solution. The work of W. H. Enrightand M. S. Kamel attempts to reduce the cost of the iterationby automatically transforming and partitioning the system. Weprovide new theoretical justification for this method in thecase where the stiff eigenvalues of the Jacobian matrix usedin the modified Newton iteration are small in number and wellseparated from the other eigenvalues. The theory of Y. Saadis introduced and adapted to show that the method uses the projectionof the Jacobian onto a Krylov subspace which virtually containsthe dominant subspace. This is shown to have favourable consequences.Numerical evidence is provided to support the theory.     

Stability and convergence of boundary value methods for solving ODE †

we review some recent approaches to the numerical solution of ODE. They are based on the solution of IVP (Initial Value Problem) by means of suitable BVM (Boundary Value Methods). A discussion of their properties of stability and covergence is presented along with their practical construction. The advantages of such methods with respect to IVP methods consist in a higher accuracy in the case of stiff problems and in a more covenient implementation on parallel computers.     

On order‐reducible sinc discretizations and block‐diagonal preconditioning methods for linear third‐order ordinary differential equations

By introducing a variable substitution, we transform the two‐point boundary value problem of a third‐order ordinary differential equation into a system of two second‐order ordinary differential equations (ODEs). We discretize this order‐reduced system of ODEs by both sinc‐collocation and sinc‐Galerkin methods, and average these two discretized linear systems to obtain the target system of linear equations. We prove that the discrete solution resulting from the linear system converges exponentially to the true solution of the order‐reduced system of ODEs. The coefficient matrix of the linear system is of block two‐by‐two structure, and each of its blocks is a combination of Toeplitz and diagonal matrices. Because of its algebraic properties and matrix structures, the linear system can be effectively solved by Krylov subspace iteration methods such as GMRES preconditioned by block‐diagonal matrices. We demonstrate that the eigenvalues of certain approximation to the preconditioned matrix are uniformly bounded within a rectangle on the complex plane independent of the size of the discretized linear system, and we use numerical examples to illustrate the feasibility and effectiveness of this new approach. Copyright © 2013 John Wiley & Sons, Ltd.     

On the construction of high order <Emphasis Type="Italic">A</Emphasis>(<Emphasis Type="Italic">α</Emphasis>)-stable hybrid linear multistep methods for stiff IVPs in ODEs

In this paper, we present a class of A(α)-stable hybrid linear multistep methods for numerical solving stiff initial value problems (IVPs) in ordinary differential equations (ODEs). The method considered uses a second derivative like the Enright’s second derivative linear multistep methods for stiff IVPs in ODEs.     

Taylor series method for dynamical systems with control: Convergence and error estimates

To optimize a complicated function constructed from a solution of a system of ordinary differential equations (ODEs), it is very important to be able to approximate a solution of a system of ODEs very precisely. The precision delivered by the standard Runge-Kutta methods often is insufficient, resulting in a “noisy function” to optimize. We consider an initial-value problem for a system of ordinary differential equations having polynomial right-hand sides with respect to all dependent variables. First we show how to reduce a wide class of ODEs to such polynomial systems. Using the estimates for the Taylor series method, we construct a new “aggregative” Taylor series method and derive guaranteed a priori step-size and error estimates for Runge-Kutta methods of order r. Then we compare the 8,13-Prince-Dormand’s, Taylor series, and aggregative Taylor series methods using seven benchmark systems of equations, including van der Pol’s equations, the “brusselator,” equations of Jacobi’s elliptic functions, and linear and nonlinear stiff systems of equations. The numerical experiments show that the Taylor series method achieves the best precision, while the aggregative Taylor series method achieves the best computational time. The final section of this paper is devoted to a comparative study of the above numerical integration methods for systems of ODEs describing the optimal flight of a spacecraft from the Earth to the Moon. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 24, Dynamical Systems and Optimization, 2005.     

Mono-implicit Runge-Kutta schemes for the parallel solution of initial value ODEs

Among the numerical techniques commonly considered for the efficient solution of stiff initial value ordinary differential equations are the implicit Runge-Kutta (IRK) schemes. The calculation of the stages of the IRK method involves the solution of a nonlinear system of equations usually employing some variant of Newton's method. Since the costs of the linear algebra associated with the implementation of Newton's method generally dominate the overall cost of the computation, many subclasses of IRK schemes, such as diagonally implicit Runge-Kutta schemes, singly implicit Runge-Kutta schemes, and mono-implicit (MIRK) schemes, have been developed to attempt to reduce these costs. In this paper we are concerned with the design of MIRK schemes that are inherently parallel in that smaller systems of equations are apportioned to concurrent processors. This work builds on that of an earlier investigation in which a special subclass of the MIRK formulas were implemented in parallel. While suitable parallelism was achieved, the formulas were limited to some extent because they all had only stage order 1. This is of some concern since in the application of a Runge-Kutta method to a system of stiff ODEs the phenomenon of order reduction can arise; the IRK method can behave as if its order were only its stage order (or its stage order plus one), regardless of its classical order. The formulas derived in the current paper represent an improvement over the previous investigation in that the full class of MIRK formulas is considered and therefore it is possible to derive efficient, parallel formulas of orders 2, 3, and 4, having stage orders 2 or 3.     

The Stability Properties of Singly-implicit General Linear Methods

In this paper we examine the linear stability properties ofsingly-implicit general linear methods. We show numericallythat there exist A-stable methods of up to order 14. Based onvarious stability and implementation considerations we proposea family of methods of orders two to ten to be incorporatedinto a variable order, variable stepsize package suitable forsolving stiff ordinary differential equations.