Marino Zennaro

Strong contractivity properties of numerical methods for ordinary and delay differential equations

Abstract In the last 20 years various stability test problems for DDE solvers have been considered. These are mainly generalizations of those used for ODEs. For the simplest autonomous case y ′( t )= λy ( t )+ μy ( t − τ ), the concepts of P- and GP-stability were introduced and a significant number of results have already been found for both classes of linear multistep methods and one-step Runge-Kutta methods. More difficult is the situation for the non-autonomous linear test equation y ′( t )= λ ( t ) y ( t )+ μ ( t ) y ( t − τ ) and for the general nonlinear case y ′( t )= f ( t , y ( t ), y ( t − τ )), which gave rise to the concepts of PN-, GPN-, RN- and GRN-stability. In particular, in order to study PN- and GPN-stability notion for ODE solvers based on the test equation with forcing term y′(t) = λ(t)y(t)+ƒ(t) , which is called AN ƒ -stability, has recently been found. This paper pursues this further and introduces the concepts of A ƒ -stability and BN ƒ -stability, which are based on the test equations y′(t)=λy(t)+ƒ(t) and y′(t)=ƒ(t,y(t),u(t)) , respectively. Some relationships are established among all these concepts of stability for ODEs and DDEs. The situation for the class of Runge–Kutta methods up to order 2 is thoroughly examined.

Contractivity of Runge-Kutta methods with respect to forcing terms

Abstract The present paper pursues the study of contractivity properties of Runge-Kutta methods for ODEs with respect to forcing terms which has been started in the papers by Torelli (1991) and Bellen and Zennaro (1992). Whereas these two papers addressed only the case of implicit methods for the discussion of optimal stability properties, here also explicit methods are considered and the regions of stability are introduced and investigated. Some of the problems opened in Bellen and Zennaro (1992) are partially settled and many examples of methods up to order p = 4 are discussed.

The use of Runge-Kutta formulae in waveform relaxation methods

Abstract We consider a very general class of waveform relaxation methods which are based on Runge-Kutta processes for the numerical solution of initial value problems for large systems of ordinary differential equations. We give general results about the convergence of the iterative schemes on arbitrarily long windows of integration, as well as about the order of accuracy of the limit methods. Finally, we briefly discuss a possible parallel implementation of some of these techniques.

Time-point relaxation Runge-Kutta methods for ordinary differential equations

Bellen, A., Z. Jackiewicz and M. Zennaro, Time-point relaxation Runge-Kutta methods for ordinary differential equations, Journal of Computational and Applied Mathematics 45 (1993) 121-137. We investigate convergence, order, and stability properties of time-point relaxation Runge-Kutta methods for systems of ordinary differential equations. These methods can be implemented in Gauss-Jacobi, or GaussSeidel modes denoted by TRGJRK(k) or TRGSRK(k), respectively, where k stands for the number of Picard-Lindelof iterations. As k --f ~0, these modes tend to the same one-step method for ordinary differential equations called diagonal split Runge- Kutta (DSRK) method. It is proved that these methods are convergent with order min{k, p), where p is the order of the underlying Runge-Kutta method. Recurrence relations resulting from application of TRGJRK(k), TRGSRK(k) and DSRK methods to the two-dimensional test system u’ = Au - pu, U’ = pu + Au, t 3 0, where A and p are real parameters, are derived and stability regions in the (h, p&plane are plotted for some methods using a variant of the boundary locus method. In most cases stability regions increase as the number of Picard-Lindeldf iterations k becomes larger.

Variable stepsize diagonally implicit multistage integration methods for ordinary differential equations

Abstract We study a class of variable stepsize general linear methods for the numerical solution of ordinary differential equations. These methods provide an alternative to the Nordsieck technique of changing the stepsize of integration. Order conditions are derived using a recent approach by Albrecht and examples of methods are given which are appropriate for stiff or nonstiff systems in sequential or parallel computing environments. A construction of variable stepsize continuous methods is also described which is facilitated by adding, in general, one extra external stage. Numerical experiments are presented which indicate that the implementation based on variable stepsize formulation is more accurate and more efficient than the implementation based on Nordsiecks technique for second-order DIMSIMs of type 1.

Multistep natural continuous extensions of Runge-Kutta methods: the potential for stable interpolation

Abstract The present paper develops a theory of multistep natural continuous extensions of Runge–Kutta methods, that is interpolants of multistep type that generalize the notion of natural continuous extension introduced by Zennaro [15]. The main motivation for the definition of such a type of interpolants is given by the need for interpolation procedures with strong stability properties and high order of accuracy, in view of interesting applications to the numerical solution of delay differential equations and to the waveform relaxation methods for large systems of ordinary differential equations.