Laurent O. Jay

Numerical solution of highly oscillatory ordinary differential equations

One of the most difficult problems in the numerical solution of ordinary differential equations (ODEs) and in differential-algebraic equations (DAEs) is the development of methods for dealing with highly oscillatory systems. These types of systems arise, for example, in vehicle simulation when modelling the suspension system or tyres, in models for contact and impact, in flexible body simulation from vibrations in the structural model, in molecular dynamics, in orbital mechanics, and in circuit simulation. Standard numerical methods can require a huge number of time-steps to track the oscillations, and even with small stepsizes they can alter the dynamics, unless the method is chosen very carefully.

Symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems

This article deals with the numerical treatment of Hamiltonian systems with holonomic constraints. A class of partitioned Runge–Kutta methods, consisting of the couples of s-stage Lobatto IIIA and Lobatto IIIB methods, has been discovered to solve these problems efficiently. These methods are symplectic, preserve all un-derlying constraints, and are superconvergent with order

A Note on Q-order of Convergence

2s - 2

An SQP method for the optimal control of large-scale dynamical systems

. For separable Hamiltonians of the form

Convergence of a class of runge-kutta methods for differential-algebraic systems of index 2

H(q,p) = \frac{1}{2}p^T M^{ - 1} p + U(q)

Structure Preservation for Constrained Dynamics with Super Partitioned Additive Runge--Kutta Methods

the Rattle algorithm based on the Verlet method was up to now the only known symplectic method preserving the constraints. In fact this method turns out to be equivalent to the 2-stage Lobatto IIIA–IIIB method of order 2. Numerical examples have been performed which illustrate the theoretical results.

Improved Quasi-Steady-State-Approximation Methods for Atmospheric Chemistry Integration

To complement the property of Q-order of convergence we introduce the notions of Q-superorder and Q-suborder of convergence. A new definition of exact Q-order of convergence given in this note generalizes one given by Potra. The definitions of exact Q-superorder and exact Q-suborder of convergence are also introduced. These concepts allow the characterization of any sequence converging with Q-order (at least) 1 by showing the existence of a unique real number q ∈ [1,+∞] such that either exact Q-order, exact Q-superorder, or exact Q-suborder q of convergence holds.

Numerical Optimal Control of Parabolic PDES Using DASOPT

We propose a sequential quadratic programming (SQP) method for the optimal control of large-scale dynamical systems. The method uses modified multiple shooting to discretize the dynamical constraints. When these systems have relatively few parameters, the computational complexity of the modified method is much less than that of standard multiple shooting. Moreover, the proposed method is demonstrably more robust than single shooting. In the context of the SQP method, the use of modified multiple shooting involves a transformation of the constraint Jacobian. The affected rows are those associated with the continuity constraints and any path constraints applied within the shooting intervals. Path constraints enforced at the shooting points (and other constraints involving only discretized states) are not transformed. The transformation is cast almost entirely at the user level and requires minimal changes to the optimization software. We show that the modified quadratic subproblem yields a descent direction for the l1 penalty function. Numerical experiments verify the efficiency of the modified method.

A Discussion of Low-Order Numerical Integration Formulas for Rigid and Flexible Multibody Dynamics

This paper deals with convergence results for a special class of Runge-Kutta (RK) methods as applied to differential-algebraic equations (DAEs) of index 2 in Hessenberg form. The considered methods are stiffly accurate, with a singular RK matrix whose first row vanishes, but which possesses a nonsingular submatrix. Under certain hypotheses, global superconvergence for the differential components is shown, so that a conjecture related to the Lobatto IIIA schemes is proved. Extensions of the presented results to projected RK methods are discussed. Some numerical examples in line with the theoretical results are included.

Inexact Simplified Newton Iterations for Implicit Runge-Kutta Methods

A broad class of partitioned differential equations with possible algebraic constraints is considered, including Hamiltonian and mechanical systems with holonomic constraints. For mechanical systems a formulation eliminating the Coriolis forces and closely related to the Euler--Lagrange equations is presented. A new class of integrators is defined: the super partitioned additive Runge--Kutta (SPARK) methods. This class is based on the partitioning of the system into different variables and on the splitting of the differential equations into different terms. A linear stability and convergence analysis of these methods is given. SPARK methods allowing the direct preservation of certain properties are characterized. Different structures and invariants are considered: the manifold of constraints, symplecticness, reversibility, contractivity, dilatation, energy, momentum, and quadratic invariants. With respect to linear stability and structure-preservation, the class of s-stage Lobatto IIIA-B-C-C* SPARK methods is of special interest. Controllable numerical damping can be introduced by the use of additional parameters. Some issues related to the implementation of a reversible variable stepsize strategy are discussed.