The Essential Stability of Local Error Control for Dynamical Systems

Although most adaptive software for initial value problems is designed with an accuracy requirement—control of the local error—it is frequently observed that stability is imparted by the adaptation. This relationship between local error control and numerical stability is given a firm theoretical underpinning. The dynamics of numerical methods with local error control are studied for three classes of ordinary differential equations: dissipative, contractive, and gradient systems. Dissipative dynamical systems are characterised by having a bounded absorbing set B which all trajectories eventually enter and remain inside. The exponentially contractive problems studied have a unique, globally exponentially attracting equilibrium point and thus they are also dissipative since the absorbing set B may be chosen to be a ball of arbitrarily small radius around the equilibrium point. The gradient systems studied are those for which the set of equilibria comprises isolated points and all trajectories are bounded so that each trajectory converges to an equilibrium point as t → ∞. If the set of equilibria is bounded then the gradient systems are also dissipative. Conditions under which numerical methods with local error control replicate these large-time dynamical features are described. The results are proved without recourse to asymptotic expansions for the truncation error. Standard embedded Runge–Kutta pairs are analysed together with several nonstandard error control strategies. Both error per step and error per unit step strategies are considered. Certain embedded pairs are identified for which the sequence generated can be viewed as coming from a small perturbation of an algebraically stable scheme, with the size of the perturbation proportional to the tolerance τ. Such embedded pairs are defined to be essentially algebraically stable and explicit essentially stable pairs are identified. Conditions on the tolerance τ are identified under which appropriate discrete analogues of the properties of the underlying differential equation may be proved for certain essentially stable embedded pairs. In particular, it is shown that for dissipative problems the discrete dynamical system has an absorbing set B_τ and is hence dissipative. For exponentially contractive problems the radius of B_τ is proved to be proportional to τ. For gradient systems the numerical solution enters and remains in a small ball about one of the equilibria and the radius of the ball is proportional to τ. Thus the local error control mechanisms confer desirable global properties on the numerical solution. It is shown that for error per unit step strategies the conditions on the tolerance τ are independent of initial data while for error per step strategies the conditions are initial-data dependent. Thus error per unit step strategies are considerably more robust

صف من الطرائق الشرائحية بثلاث نقاط تجميعية لحل معادلات تفاضلية متأخرة

In this paper a study of the existence, uniqueness, stability and convergence of a class of C2-spline collocation methods for solving delay differential equations (DDEs) is introduced. The presented methods are based on C2-Spline with three collocation points , , and in each subinterval  It turns out that the proposed methods for DDEs are stable iff , and they possess convergence rate of order 6 if , in the remaining cases the order is 5. Moreover, the methods are P-stable for . Numerical results illustrating the behavior of the methods when faced with some difficult problems are presented and the numerical results are compared to those obtained by other methods. نقدم في هذا البحث صفاً من الطرائق الشرائحية التجميعية لإيجاد الحل العددي للمعادلات التفاضلية المتأخرة. تعتمد الطرائق المذكورة على إنشاء تقريبات شرائحية في الفضاء C2 باستخدام ثلاث نقاط تجميعية  في كل مجال جزئي  حيث   ,و،. تم إثبات وجود حل تقريبي شرائحي وحيد لمثل هذه المعادلات، وجرت دراسة استقرار وتقارب ومعدل التقارب لهذه الطرائق. تبين الدراسة أن الطرائق لأجل المعادلات المذكورة تكون مستقرة إذا كان ، علاوة على ذلك ، الطرائق تكون متقاربة, وهذا التقارب من المرتبة السادسة لأجل بارامترات تحقق المعادلة ، وفي حالات أخرى يكون التقارب من المرتبة الخامسة. بالإضافة إلى ذلك، يظهر تحليل الاستقرار أن الطرائق تكون في حالة P-استقرار لأجل . كما تم اختبار الطرائق المقدمة بحل بعض المسائل ذات السلوك القاسي ولأجل دالة بدء إما أن تكون غير ملساء أو تملك تذبذبات عالية، حيث تشير النَتائِج العددية إلى فعالية وكفاءة طرائقنا مقارنة مع بعض الطرائقِ الأخرى

صف من الطرائق الشرائحية بثلاث نقاط تجميعية لحل معادلات تفاضلية متأخرة

In this paper a study of the existence, uniqueness, stability and convergence of a class of C2-spline collocation methods for solving delay differential equations (DDEs) is introduced. The presented methods are based on C2-Spline with three collocation points , , and in each subinterval  It turns out that the proposed methods for DDEs are stable iff , and they possess convergence rate of order 6 if , in the remaining cases the order is 5. Moreover, the methods are P-stable for . Numerical results illustrating the behavior of the methods when faced with some difficult problems are presented and the numerical results are compared to those obtained by other methods. نقدم في هذا البحث صفاً من الطرائق الشرائحية التجميعية لإيجاد الحل العددي للمعادلات التفاضلية المتأخرة. تعتمد الطرائق المذكورة على إنشاء تقريبات شرائحية في الفضاء C2 باستخدام ثلاث نقاط تجميعية  في كل مجال جزئي  حيث   ,و،. تم إثبات وجود حل تقريبي شرائحي وحيد لمثل هذه المعادلات، وجرت دراسة استقرار وتقارب ومعدل التقارب لهذه الطرائق. تبين الدراسة أن الطرائق لأجل المعادلات المذكورة تكون مستقرة إذا كان ، علاوة على ذلك ، الطرائق تكون متقاربة, وهذا التقارب من المرتبة السادسة لأجل بارامترات تحقق المعادلة ، وفي حالات أخرى يكون التقارب من المرتبة الخامسة. بالإضافة إلى ذلك، يظهر تحليل الاستقرار أن الطرائق تكون في حالة P-استقرار لأجل . كما تم اختبار الطرائق المقدمة بحل بعض المسائل ذات السلوك القاسي ولأجل دالة بدء إما أن تكون غير ملساء أو تملك تذبذبات عالية، حيث تشير النَتائِج العددية إلى فعالية وكفاءة طرائقنا مقارنة مع بعض الطرائقِ الأخرى

Schémas d'intégration dédiés à l'étude, l'analyse et la synthèse dans le formalisme Hamiltonien à ports

This thesis work dealing with finite dimensional approximation of infinite dimension system. The class considered is that of Hamiltonian systems in ports. We study initially ordinary differential equations systems. Based on an energy integrator, we define a class of discrete passive dynamics is invariant interconnection. We obtain the stability conditions (LMI) for dynamic network in the presence of delays and uncertainties, and propose a method of stabilizing energy synthesis. These developments were experimentally validated by the implementation of an energy control a power converter (Buck). We then study the Hamiltonian formalism in infinite dimensions. We offer an approximation that combines a semi-discretization and an energy integrator. The mixed composability is studied and a method of synthesis IDA-PBC was developed. All the obtained results are numerically illustrated in the manuscript.Ces travaux de thèse traitent de l'approximation en dimension finie de système de dimension infinie. La classe considérée est celle des systèmes hamiltoniens à ports. Nous étudions dans un premier temps les systèmes d'équations différentielles ordinaires. Sur la base d'un intégrateur énergétique, nous définissons une classe de dynamiques passives discrètes qui est invariante par interconnexion. Nous obtenons alors des conditions de stabilité (LMI) pour des dynamiques en réseau en présence de retards et d'incertitudes, et proposons une méthode de synthèse énergétique stabilisante. Ces développements ont été validés expérimentalement par la mise en oeuvre d'une commande énergétique sur un convertisseur de puissance (Buck). Nous étudions ensuite le formalisme hamiltonien en dimension infinie. Nous proposons une approximation qui combine une semi-discrétisation et un intégrateur énergétique. La composabilité mixte est étudiée et une méthode de synthèse IDA-PBC a été développée. L'ensemble des résultats obtenus sont illustrés numériquement dans le manuscrit

Proceedings of the ECCOMAS Thematic Conference on Multibody Dynamics 2015

This volume contains the full papers accepted for presentation at the ECCOMAS Thematic Conference on Multibody Dynamics 2015 held in the Barcelona School of Industrial Engineering, Universitat Politècnica de Catalunya, on June 29 - July 2, 2015. The ECCOMAS Thematic Conference on Multibody Dynamics is an international meeting held once every two years in a European country. Continuing the very successful series of past conferences that have been organized in Lisbon (2003), Madrid (2005), Milan (2007), Warsaw (2009), Brussels (2011) and Zagreb (2013); this edition will once again serve as a meeting point for the international researchers, scientists and experts from academia, research laboratories and industry working in the area of multibody dynamics. Applications are related to many fields of contemporary engineering, such as vehicle and railway systems, aeronautical and space vehicles, robotic manipulators, mechatronic and autonomous systems, smart structures, biomechanical systems and nanotechnologies. The topics of the conference include, but are not restricted to: ● Formulations and Numerical Methods ● Efficient Methods and Real-Time Applications ● Flexible Multibody Dynamics ● Contact Dynamics and Constraints ● Multiphysics and Coupled Problems ● Control and Optimization ● Software Development and Computer Technology ● Aerospace and Maritime Applications ● Biomechanics ● Railroad Vehicle Dynamics ● Road Vehicle Dynamics ● Robotics ● Benchmark ProblemsPostprint (published version