Partitioned adaptive Runge-Kutta methods for the solution of nonstiff and stiff systems

Summary For the numerical solution of initial value problems of ordinary differential equations partitioned adaptive Runge-Kutta methods are studied. These methods consist of an adaptive Runge-Kutta methods for the treatment of a stiff system and a corresponding explicit Runge-Kutta method for a nonstiff system. First we modify the theory of Butcher series for partitioned adaptive Runge-Kutta methods. We show that for any explicit Runge-Kutta method there exists a translation invariant partitoned adaptive Runge-Kutta method of the same order. Secondly we derive a special translaton invariant partitioned adaptive Runge-Kutta method of order 3. An automatic stiffness detection and a stepsize control basing on Richardson-extrapolation are performed. Extensive tests and comparisons with the partitioned RKF4RW-algorithm from Rentrop [16] and the partitioned algorithm LSODA from Hindmarsh [9] and Petzold [15] show that the partitoned adaptive Runge-Kutta algorithm works reliable and gives good numericals results. Furthermore these tests show that the automatic stiffness detection in this algorithm is effective.     

Runge-Kutta theory for Volterra integrodifferential equations

Summary The present paper develops the theory of general Runge-Kutta methods for Volterra integrodifferential equations. The local order is characterized in terms of the coefficients of the method. We investigate the global convergence of mixed and extended Runge-Kutta methods and give results on asymptotic error expansions. In a further section we construct examples of methods up to order 4.     

A note onB-stability of Runge-Kutta methods

Summary Burrage and Butcher [1, 2] and Crouzeix [4] introduced for Runge-Kutta methods the concepts ofB-stability,BN-stability and algebraic stability. In this paper we prove that for any irreducible Runge-Kutta method these three stability concepts are equivalent.Chapters 1–3 of this article have been written by the second author, whereas chapter 4 has been written by the first author     

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.     

Perturbed collocation and Runge-Kutta methods

Summary It is well known thatsome implicit Runge-Kutta methods are equivalent to collocation methods. This fact permits very short and natural proofs of order andA, B, AN, BN-stability properties for this subclass of methods (see [9] and [10]). The present paper answers the natural question, ifall RK methods can be considered as a somewhat perturbed collocation. After having introduced this notion we give a proof on the order of the method and discuss their stability properties. Much of known theory becomes simple and beautiful.     

The reliability/cost trade-off for a class of ODE solvers

In the numerical solution of ODEs, it is now possible to develop efficient techniques that will deliver approximate solutions that are piecewise polynomials. The resulting methods can be designed so that the piecewise polynomial will satisfy a perturbed ODE with an associated defect (or residual) that is directly controlled in a consistent fashion. We will investigate the reliability/cost trade off that one faces when implementing and using such methods, when the methods are based on an underlying discrete Runge-Kutta formula. In particular we will identify a new class of continuous Runge-Kutta methods with a very reliable defect estimator and a validity check that reflects the credibility of the estimate. We will introduce different measures of the “reliability” of an approximate solution that are based on the accuracy of the approximate solution; the maximum magnitude of the defect of the approximate solution; and how well the method is able to estimate the maximum magnitude of the defect of the approximate solution. We will also consider how methods can be implemented to detect and cope with special difficulties such as the effect of round-off error (on a single step) or the ability of a method to estimate the magnitude of the defect when the stepsize is large (as might happen when using a high-order method at relaxed accuracy requests). Numerical results on a wide selection of problems will be summarized for methods of orders five, six and eight. It will be shown that a modest increase in the cost per step can lead to a significant improvement in the quality of the approximate solutions and the reliability of the method. For example, the numerical results demonstrate that, if one is willing to increase the cost per step by 50%, then a method can deliver approximate solutions where the reported estimated maximum defect is within 1% of its true value on 95% of the steps.     

Combination method for parallel computation in ODE

In this paper, a 4th order parallel computation method with four processes for solving ODEs is discussed. This method is the Runge-Kutta method combined with a linear multistep method, which overcomes the difficulties of the 4th order parallel Runge-Kutta method discussed in [1]. The concept of critical speedup for parallel methods is also defined, and speedups of some methods are analyzed by using this concept.     

Order conditions for Rosenbrock type methods

Summary Based on the theory of Butcher series this paper developes the order conditions for Rosenbrock methods and its extensions to Runge-Kutta methods with exact Jacobian dependent coefficients. As an application a third order modified Rosenbrock method with local error estimate is constructed and tested on some examples.     

Conditions for the coefficients of Runge-Kutta methods for systems of n-th order differential equations

To derive order conditions for Runge-Kutta methods of Nyström or Fehlberg type, applicable to arbitrary order differential equations, a theory similar to that about Runge-Kutta methods for first order systems, due to Butcher [1], is developed. By a new definition of elementary differentials, which is independent of the order of the given system, each condition to be satisfied by the coefficients of the method directly follows from the representation of the corresponding elementary differential.     

Order conditions for numerical methods for partitioned ordinary differential equations

Summary Motivated by the consideration of Runge-Kutta formulas for partitioned systems, the theory of P-series is studied. This theory yields the general structure of the order conditions for numerical methods for partitioned systems, and in addition for Nyström methods fory=f(y,y), for Rosenbrock-type methods with inexact Jacobian (W-methods). It is a direct generalization of the theory of Butcher series [7, 8]. In a later publication, the theory ofP-series will be used for the derivation of order conditions for Runge-Kutta-type methods for Volterra integral equations [1].     

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.     

A special family of Runge-Kutta methods for solving stiff differential equations

An efficient way of implementing Implicit Runge-Kutta Methods was proposed by Butcher [3]. He showed that the most efficient methods when using this implementation are those whose characteristic polynomial of the Runge-Kutta matrix has a single reals-fold zero. In this paper we will construct such a family of methods and give some results concerning their maximum attainable order and stability properties. Some consideration is also given to showing how these methods can be efficiently implemented and, in particular, how local error estimates can be obtained by the use of embedding techniques.     

Local error estimation for multistep collocation methods

Multistep collocation methods for initial value problems in ordinary differential equations are known to be a subclass of multistep Runge-Kutta methods and a generalisation of the well-known class of one-step collocation methods as well as of the one-leg methods of Dahlquist. In this paper we derive an error estimation method of embedded type for multistep collocation methods based on perturbed multistep collocation methods. This parallels and generalizes the results for one-step collocation methods by Nørsett and Wanner. Simple numerical experiments show that this error estimator agrees well with a theoretical error estimate which is a generalisation of an error estimate first derived by Dahlquist for one-leg methods.     

The orders of embedded continuous explicit Runge-Kutta methods

Previously, the authors [9] classified various types of continuous explicit Runge-Kutta methods of order 5. Here, new lower bounds on the numbers of stages required for a sequence of continuous methods of increasing orders which are embedded in a continuouss-stage method of orderp are obtained. Carnicer [2] showed for each continuous explicit Runge-Kutta method of orderp in a mildly restricted family that at least 2p – 2 stages are required. Here, the same bound is established for all such methods of orderp.This research was supported by the Natural Sciences and Engineering Research Council of Canada, and the Information Technology Research Centre of Ontario. In addition, the second author was supported by the Ministero dell'Università e della Ricerca Scientifica e Tecnologica of Italy.     

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.     

High order algebraically stable Runge-Kutta methods

In Burrage and Butcher [3] the concept of Algebraic Stability was introduced in the study of Runge-Kutta methods. In this paper an analysis is made of the family ofs-stage Runge-Kutta methods of order 2s—2 or more which possesses this property.     

Stability analysis and a priori error estimate of explicit Runge-Kutta discontinuous Galerkin methods for correlated random walk with density-dependent turning rates

In this paper,we analyze the explicit Runge-Kutta discontinuous Galerkin(RKDG)methods for the semilinear hyperbolic system of a correlated random walk model describing movement of animals and cells in biology.The RKDG methods use a third order explicit total-variation-diminishing Runge-Kutta(TVDRK3)time discretization and upwinding numerical fluxes.By using the energy method,under a standard CourantFriedrichs-Lewy(CFL)condition,we obtain L2stability for general solutions and a priori error estimates when the solutions are smooth enough.The theoretical results are proved for piecewise polynomials with any degree k 1.Finally,since the solutions to this system are non-negative,we discuss a positivity-preserving limiter to preserve positivity without compromising accuracy.Numerical results are provided to demonstrate these RKDG methods.     

A stability property ofA-stable natural Runge-Kutta methods for systems of delay differential equations

A natural Runge-Kutta method is a special type of Runge-Kutta method for delay differential equations (DDEs); it is known that any one-step collocation method is equivalent to one of such methods. In this paper, we consider a linear constant-coefficient system of DDEs with a constant delay, and discuss the application of natural Runge-Kutta methods to the system. We show that anA-stable method preserves the asymptotic stability property of the analytical solutions of the system.     

Stability of Runge-Kutta methods for linear delay differential equations

Summary. This paper investigates the stability of Runge-Kutta methods when they are applied to the complex linear scalar delay differential equation . This kind of stability is called stability. We give a characterization of stable Runge-Kutta methods and then we prove that implicit Euler method is stable. Received November 3, 1998 / Revised version received March 23, 1999 / Published online July 12, 2000     

Natural Continuous Extensions of Runge-Kutta methods for Volterra integrodifferential equations

Summary In this paper we deal with a very general class of Runge-Kutta methods for the numerical solution of Volterra integrodifferential equations. Our main contribution is the development of the theory of Natural Continuous Extensions (NCEs), i.e. piecewice polynomial functions which interpolate the values given by the RK-method at the mesh points. The particular features of these NCEs allow us to construct tail approximations which are quite efficient since they require a minimal number of kernel evaluations.