Nonlinear Hamiltonian Partial Differential Equations (PDE) describing nonlinear wave propagation phenomena (Klein-Gordon, Schroödinger equation in quantum physics, ocean wave propagation, Euler, etc...), and stochastic ordinary and partial differential equations. In such situations the exact solutions of the equations endow with many physical properties that are consequences of the geometric structure: Preservation of the total energy, momentum conservation or existence of ergodic invariant measures. However the preservation of such qualitative properties by numerical methods at a reasonable cost is not guaranteed at all, even for very precise (high order) methods.
1. The invention of new efficient numerical schemes for geometric PDEs cannot be dissociated from the most recent progresses in analysis (stability phenomena, energy transfers, multiscale problems, etc...).
2. The reverse is also true.
The principal aim of the geometric numerical integration theory is the understanding and analysis of such problems: How (and to which extend) reproduce qualitative behavior of differential equations over long time? This theory - whose principal tool is to consider a numerical scheme as a mathematical object in its own right - is now well established and has known many successes in the derivation and analysis of the modern numerical methods used in molecular dynamics and celestial mechanics simulations for example.
The extension of this theory to geometric PDEs is a fundamental ongoing challenge, which requires the invention of a new mathematical framework bridging the most recent techniques used in the theory of nonlinear PDEs and stochastic differential equations with the practical constraints of numerical simulation. Due to the formidable variety of possible behaviors of in the infinite dimensional world of PDEs, the analysis of numerical schemes from the qualitative point of view can be considered as being one of the next major challenges in scientific computing for the next 20 years.