This paper studies Rosenbrock methods when they are applied to stiff differential equations containing a small stiffness parametere. The basic ideas and techniques are the same as those developed for Runge-Kutta methods in an earlier paper of the authors. The results obtained here are essentially those obtained for Diagonally Implicit Runge-Kutta methods.

Error of Runge-Kutta methods for stiff problems studied via differential algebraic equations

The ordinary differential equations occurring in linear boundary value problems characteristically have both stable and unstable solution modes. Therefore a stable numerical algorithm should avoid both forward and backward integration of solutions on large intervals. It is shown that most methods (like multiple shooting, collocation, invariant imbedding and difference methods) derive their stability from the fact that they all decouple the continuous or the discrete problem sooner or later (for instance when solving a linear system). This decoupling is related to the dichotomy of the ordinary differential equations. In fact it turns out that the inherent initial value instability is an important prerequisite for a stable utilization of the decoupled representations from which the solutions are computed. How this stability is related to the use of the boundary conditions is also investigated.

/pdf/decoupling-and-stability-of-algorithms-for-boundary-value-r238fvtu9p.pdf

Decoupling and Stability of Algorithms for Boundary Value Problems

http://www.math.leidenuniv.nl/~spijker/Spijker(1996)-JCAM.pdf

Stiffness in numerical initial-value problems

Petri Nets (PN) are a graphical formalism which is gaining popularity in recent years as a tool for the representation of complex logical interactions (like synchronization, sequentiality, concurrency and conflict) among physical components or activities in a system. This notes are devoted to introduce the formalism of Petri nets with particular emphasis on the application of the methodology in the area of the performance and reliability modelling and analysis of systems. The quantitative analysis of the behaviour of systems in time requires the superposition of a stochastic timing mechanism to the classical representation of PN. Timed Petri nets and, in particular, Stochastic Petri nets (SPN) are the object of the second part of the notes. Finally, some fully developed examples enlighten peculiar aspects which differentiate PNs from other modelling techniques usual in reliability analysis. In few words, the goal of these notes is to show that the proposed methodology based on the PN formalism can be conveniently used as a user-friendly language to represent and evaluate complex stochastic systems.

/pdf/system-modelling-with-petri-nets-1s7gb3a0h9.pdf

System Modelling with Petri Nets

Finding asymptotic solutions to two-point boundary value problems for linear singularly perturbed second-order ordinary differential equations is now quite well understood. Troublesome exceptions involving turning points, known as problems of boundary layer resonance, have been studied by many experts since the initial work of Ackerberg and O'Malley [Stud. Appl. Math., 49 (1970), pp. 277-295]; this significant work is surveyed here. Closely related literature involves the canard phenomena. New, generally more geometric, approaches, including Gevrey asymptotics and “blowup," promise improved understanding of such sensitive asymptotics and their generalizations.

Singularly Perturbed Linear Two-Point Boundary Value Problems

Numerical Methods for Stiff Equations

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Numerical Methods for Stiff Equations