Mid-Term Report Summary - GEOPARDI (Numerical integration of Geometric Partial Differential Equations)
The goal of this project is to develop and analyze numerical schemes for the approximation of evolution equations possessing strong geometric properties. Typical examples are given by Hamiltonian Partial Differential Equations (PDE) such as wave equations in nonlinear propagations problem or Schroedinger equation in quantum physics, for which a physical energy remains constant along the exact solution. In such situations, the preservation of the geometric properties of the original system by the numerical method is not automatically guaranteed, though absolutely crucial to reproduce physical phenomena (conservation of an energy, adiabatic invariants, etc...) over long time, and at a reasonable numerical cost. The goal is hence to understand the global behaviour of numerical methods applied to Geometric PDEs, viewed as (infinite dimensional) discrete flows.
In the finite dimensional case – the original equation being hence an ordinary differential equations – the situation is now well known. Such systems arise for example in Molecular dynamics or celestial mechanics. In these cases, the use of symplectic integrators ensures the existence of a modified energy that remains constant along the numerical trajectories. This is the result of the “backward error analysis” developed in the nineties. In the infinite dimensional case, such an analysis fails and resonances phenomena are observed, due to the presence of high oscillations in the system.
The extension of this theory to an infinite dimensional context is a formidable ongoing challenge and requires the invention of new mathematical tools. The understanding of these phenomena is fundamental to develop new efficient schemes in many area of physics: from the simulation of Bose-Einstein condensats to ocean waves propagations or Maxwell equations in electromagnetism.
During the first period of this project, the result obtained concerned the analysis of the long time behavior of different kind of equations: the Langevin equations in Molecular dynamics, the Hamilton-Jacobi equation by using technics from discrete weak-KAM theory, and the stability analysis of solitons for the Schroedinger equation in quantum dynamics. The originality of the project is to mix extensive numerical simulation to rigorous analysis. This allowed to obtain new rigorous results in various fields such as the existence of quasi-periodic solutions for the Euler equation in Fluid mechanics, some mathematical step towards a complete justification of weak-turbulence theory in physics, as well as the analysis of Landau damping effects for Plasma physics.