Partitioned adaptive Runge-Kutta methods and their stability

Summary This paper deals with the solution of partitioned systems of nonlinear stiff differential equations. Given a differential system, the user may specify some equations to be stiff and others to be nonstiff. For the numerical solution of such a system partitioned adaptive Runge-Kutta methods are studied. Nonstiff equations are integrated by an explicit Runge-Kutta method while an adaptive Runge-Kutta method is used for the stiff part of the system.The paper discusses numerical stability and contractivity as well as the implementation and usage of such compound methods. Test results for three partitioned stiff initial value problems for different tolerances are presented.     

Equilibria of Runge-Kutta methods

Summary It is known that certain Runge-Kutta methods share the property that, in a constant-step implementation, if a solution trajectory converges to a bounded limit then it must be a fixed point of the underlying differential system. Such methods are calledregular. In the present paper we provide a recursive test to check whether given method is regular. Moreover, by examining solution trajectories of linear equations, we prove that the order of ans-stage regular method may not exceed 2[(s+2)/2] and that the maximal order of regular Runge-Kutta method with an irreducible stability function is 4.     

Natural Runge-Kutta and projection methods

Summary Recently the author defined the class of natural Runge-Kutta methods and observed that it includes all the collocation methods. The present paper is devoted to a complete characterization of this class and it is shown that it coincides with the class of the projection methods in some polynomial spaces.This work was supported by the Italian Ministero della Pubblica Istruzione, funds 40%     

On error behaviour of partitioned linearly implicit runge-kutta methods for stiff and differential algebraic systems

This paper studies partitioned linearly implicit Runge-Kutta methods as applied to approximate the smooth solution of a perturbed problem with stepsizes larger than the stiffness parameter. Conditions are supplied for construction of methods of arbitrary order. The local and global error are analyzed and the limiting case 0 considered yielding a partitioned linearly implicit Runge-Kutta method for differential-algebraic equations of index one. Finally, some numerical experiments demonstrate our theoretical results.     

An algebraic characterization ofB-convergent Runge-Kutta methods

Summary In the analysis of discretization methods for stiff intial value problems, stability questions have received most part of the attention in the past.B-stability and the equivalent criterion algebraic stability are well known concepts for Runge-Kutta methods applied to dissipative problems. However, for the derivation ofB-convergence results — error bounds which are not affected by stiffness — it is not sufficient in many cases to requireB-stability alone. In this paper, necessary and sufficient conditions forB-convergence are determined.This paper was written while J. Schneid was visiting the Centre for Mathematics and Computer Science with an Erwin-Schrödinger stipend from the Fonds zur Förderung der wissenschaftlichen Forschung     

On the order conditions of Runge-Kutta methods with higher derivatives

Summary Runge-Kutta methods have been generalized to procedures with higher derivatives of the right side ofy=f(t,y) e.g. by Fehlberg 1964 and Kastlunger and Wanner 1972. In the present work some sufficient conditions for the order of consistence are derived for these methods using partially the degree of the corresponding numerical integration formulas. In particular, methods of Gauß, Radau, and Lobatto type are generalized to methods with higher derivatives and their maximum order property is proved. The applied technique was developed by Crouzeix 1975 for classical Runge-Kutta methods. Examples of simple explicit and semi-implicit methods are given up to order 7 and 6 respectively.     

On implicit Runge-Kutta methods with higher derivatives

Two families of implicit Runge-Kutta methods with higher derivatives are (re-)considered generalizing classical Runge-Kutta methods of Butcher type and f Ehle type. For generalized Butcher methods the characteristic functionG() is represented by means of the node polynomial directly, thereby showing that in methods of maximum order,G() is connected withs-orthogonal polynomials in exactly the same way as Padé approximations in the classical case.     

On the relation between algebraic stability andB-convergence for Runge-Kutta methods

Summary This paper is concerned with the numerical solution of stiff initial value problems for systems of ordinary differential equations using Runge-Kutta methods. For these and other methods Frank, Schneid and Ueberhuber [7] introduced the important concept ofB-convergence, i.e. convergence with error bounds only depending on the stepsizes, the smoothness of the exact solution and the so-called one-sided Lipschitz constant . Spijker [19] proved for the case <0 thatB-convergence follows from algebraic stability, the well-known criterion for contractivity (cf. [1, 2]). We show that the order ofB-convergence in this case is generally equal to the stage-order, improving by one half the order obtained in [19]. Further it is proved that algebraic stability is not only sufficient but also necessary forB-convergence.This study was completed while this author was visiting the Oxford University Computing Laboratory with a stipend from the Netherlands Organization for Scientific Research (N.W.O.)     

Stability andB-convergence of linearly implicit Runge-Kutta methods

Summary In this paper we study stability and convergence properties of linearly implicit Runge-Kutta methods applied to stiff semi-linear systems of differential equations. The stability analysis includes stability with respect to internal perturbations. All results presented in this paper are independent of the stiffness of the system.     

One-step and extrapolation methods for differential-algebraic systems

Summary The paper analyzes one-step methods for differential-algebraic equations (DAE) in terms of convergence order. In view of extrapolation methods, certain perturbed asymptotic expansions are shown to hold. For the special DAE extrapolation solver based on the semi-implicit Euler discretization, the perturbed order pattern of the extrapolation tableau is derived in detail. The theoretical results lead to modifications of the known code. The efficiency of the modifications is illustrated by numerical comparisons over critical examples mainly from chemical combustion.     

Monotonicity and boundedness in implicit Runge-Kutta methods

Summary This paper concerns the analysis of implicit Runge-Kutta methods for approximating the solutions to stiff initial value problems. The analysis includes the case of (nonlinear) systems of differential equations that are essentially more general than the classical test equationU=U (with a complex constant). The properties of monotonicity and boundedness of a method refer to specific moderate rates of growth of the approximations during the numerical calculations. This paper provides necessary conditions for these properties by using the important concept of algebraic stability (introduced by Burrage, Butcher and by Crouzeix). These properties will also be related to the concept of contractivity (B-stability) and to a weakened version of contractivity.     

Linearly implicit extrapolation methods for differential-algebraic systems

Summary We consider the numerical solution of implicit differential equations in which the solution derivative appears multiplied by a solution-dependent singular matrix. We study extrapolation methods based on two linearly implicit Euler discretizations. Their error behaviour is explained by perturbed asymptotic expansions.     

Numerical solution of delay differential equations by uniform corrections to an implicit Runge-Kutta method

Summary In this paper we develop a class of numerical methods to approximate the solutions of delay differential equations. They are essentially based on a modified version, in a predictor-corrector mode, of the one-step collocation method atn Gaussian points. These methods, applied to ODE's, provide a continuous approximate solution which is accurate of order 2n at the nodes and of ordern+1 uniformly in the whole interval. In order to extend the methods to delay differential equations, the uniform accuracy is raised to the order 2n by some a posteriori corrections. Numerical tests and comparisons with other methods are made on real-life problems.This work was supported by CNR within the Progetto Finalizzato Informatica-Sottopr. P1-SOFMAT     

Asymptotic error expansions for stiff equations: an analysis for the implicit midpoint and trapezoidal rules in the strongly stiff case

Summary The structure of the global discretization error is studied for the implicit midpoint and trapezoidal rules applied to nonlinearstiff initial value problems. The point is that, in general, the global error contains nonsmooth (oscillating) terms at the dominanth ²-level. However, it is shown in the present paper that for special classes of stiff problems these nonsmooth terms contain an additional factor (where-1/ is the magnitude of the stiff eigenvalues). In these cases a full asymptotic error expansion exists in thestrongly stiff case ( sufficiently small compared to the stepsizeh). The general case (where the oscillating error components areO(h ²) and notO(h ²)) and applications of our results (extrapolation and defect correction algorithims) will be studied in separate papers.     

Generalized Runge-Kutta methods of order four with stepsize control for stiff ordinary differential equations

Summary GeneralizedA()-stable Runge-Kutta methods of order four with stepsize control are studied. The equations of condition for this class of semiimplicit methods are solved taking the truncation error into consideration. For application anA-stable and anA(89.3°)-stable method with small truncation error are proposed and test results for 25 stiff initial value problems for different tolerances are discussed.     

General framework,stability and error analysis for numerical stiff boundary value methods

Summary This paper provides a general framework, called theoretical multiple shooting, within which various numerical methods for stiff boundary value ordinary differential problems can be analyzed. A global stability and error analysis is given, allowing (as much as possible) the specificities of an actual numerical method to come in only locally. We demonstrate the use of our results for both one-sided and symmetric difference schemes. The class of problems treated includes some with internal (e.g. turning point) layers.     

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.     

Rosenbrock methods for Differential Algebraic Equations

Summary This paper deals with the numerical solution of Differential/Algebraic Equations (DAE) of index one. It begins with the development of a general theory on the Taylor expansion for the exact solutions of these problems, which extends the well-known theory of Butcher for first order ordinary differential equations to DAE's of index one. As an application, we obtain Butcher-type results for Rosenbrock methods applied to DAE's of index one, we characterize numerical methods as applications of certain sets of trees. We derive convergent embedded methods of order 4(3) which require 4 or 5 evaluations of the functions, 1 evaluation of the Jacobian and 1 LU factorization per step.