Methods of Runge-Kutta type for differential-algebraic equations of index \(3\) with applications to Hamiltonian systems. (Méthodes du type Runge-Kutta pour des équations différentielles algébriques d’index \(3\) avec des applications aux systèmes Hamiltoniens.) (French. English summary) Zbl 0869.65046
Genève: Univ. de Genève, Fac. des Sciences, 168 p. (1994).
The main scope of this thesis is to study the application of partitioned Runge-Kutta methods to semi-explicit index 3 differential-algebraic equations (DAE’s) in Hessenberg form. The emphasis is on stiffly accurate methods and we restrict ourselves to initial value problems. The organization of this thesis is as follows:
In Chapter I we review some fundamental notions and results related to DAE’s and to their numerical treatment. After having described some common basic structures of DAE’s we then discuss the important concepts of solvability and index. Next we give some current examples of higher index DAE’s. Further we present some available techniques to reduce the index of a problem. We then review some common numerical methods used for the solution of DAE’s. Finally a brief overview of the scope of this thesis and the main convergence results is given.
In Chapter II we give theoretical results related to semi-explicit index 3 DAE’s in Hessenberg form. After characterizing the set of consistent values and the index of the problem, we then derive the Taylor expansion of the exact solution by means of a “rooted-tree” type notation. In this setting the theory of \(B\)-series due to Hairer and Wanner is extended to such problems giving birth to the new denominated DA3-series theory.
Chapter III deals with the application of (projected) partitioned Runge-Kutta methods to semi-explicit index 3 DAE’s in Hessenberg form. Results about the existence and uniqueness, the influence of perturbations, the local error, and the global error of the numerical solution are given. A short discussion on the application of simplified Newton iterations to the arising nonlinear system ends this chapter.
The next two chapters are similar to Chapter III with the addition of some numerical experiments. In Chapter IV we restrict ourselves to the direct application of pure Runge-Kutta methods to semi-explicit index 3 DAE’s in Hessenberg form. A proof of a conjecture related to the superconvergence of stiffly accurate Runge-Kutta methods is given, together with an application of this result to the convergence analysis of these methods for stiff mechanical systems. In Chapter V we mainly deal with the application of a special class of partitioned Runge-Kutta methods to Hamiltonian systems with holonomic constraints. These methods are superconvergent and preserve the symplectic structure of the flow and all underlying constraints as well.
In Chapter I we review some fundamental notions and results related to DAE’s and to their numerical treatment. After having described some common basic structures of DAE’s we then discuss the important concepts of solvability and index. Next we give some current examples of higher index DAE’s. Further we present some available techniques to reduce the index of a problem. We then review some common numerical methods used for the solution of DAE’s. Finally a brief overview of the scope of this thesis and the main convergence results is given.
In Chapter II we give theoretical results related to semi-explicit index 3 DAE’s in Hessenberg form. After characterizing the set of consistent values and the index of the problem, we then derive the Taylor expansion of the exact solution by means of a “rooted-tree” type notation. In this setting the theory of \(B\)-series due to Hairer and Wanner is extended to such problems giving birth to the new denominated DA3-series theory.
Chapter III deals with the application of (projected) partitioned Runge-Kutta methods to semi-explicit index 3 DAE’s in Hessenberg form. Results about the existence and uniqueness, the influence of perturbations, the local error, and the global error of the numerical solution are given. A short discussion on the application of simplified Newton iterations to the arising nonlinear system ends this chapter.
The next two chapters are similar to Chapter III with the addition of some numerical experiments. In Chapter IV we restrict ourselves to the direct application of pure Runge-Kutta methods to semi-explicit index 3 DAE’s in Hessenberg form. A proof of a conjecture related to the superconvergence of stiffly accurate Runge-Kutta methods is given, together with an application of this result to the convergence analysis of these methods for stiff mechanical systems. In Chapter V we mainly deal with the application of a special class of partitioned Runge-Kutta methods to Hamiltonian systems with holonomic constraints. These methods are superconvergent and preserve the symplectic structure of the flow and all underlying constraints as well.
MSC:
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |
| 34E13 | Multiple scale methods for ordinary differential equations |
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 34A34 | Nonlinear ordinary differential equations and systems |
| 34A09 | Implicit ordinary differential equations, differential-algebraic equations |
