Abstract : The aim of the work described in this thesis is the construction and the study of structure-preserving numerical integrators for differential equations, which share some geometric properties of the exact flow, for instance symmetry, symplecticity of Hamiltonian systems, preservation of first integrals, Poisson structure, etc.
In the first part, we introduce a new approach to high-order structure-preserving numerical integrators, inspired by the theory of modified equations (backward error analysis). We focus on the class of B-series methods for which a new composition law called substitution law is introduced. This approach is illustrated with the derivation of the Preprocessed Discrete Moser-Veselov algorithm, an efficient and high-order geometric integrator for the motion of a rigid body. We also obtain an accurate integrator for the computation of conjugate points in rigid body geodesics.
In the second part, we study to which extent the excellent performance of symplectic integrators for long-time integrations in astronomy and molecular dynamics carries over to problems in optimal control. We also discuss whether the theory of backward error analysis can be extended to symplectic integrators for optimal control.
The third part is devoted to splitting methods. In the spirit of modified equations, we consider splitting methods for perturbed Hamiltonian systems that involve modified potentials. Finally, we construct splitting methods involving complex coefficients for parabolic partial differential equations with special attention to reaction-diffusion problems in chemistry.